11. The Project of Formalizing Logic
12. Boole’s Philosophical and Mathematical Aims
13. Boolean Semiotics
13.1 Appellative and Descriptive Terms
13.2 Part-to-Whole Operations
13.3 Relations and Propositions
14. Fundamental Laws of Thought
15. Primary and Secondary Propositions
16. Boolean Rules of Inference
16.1 Development of Functions
16.2 Interpreting Logical Equations
16.3 Elimination of Variables
16.4 Reduction of Systems of Propositions
16.5 Interpretation and Reduction of Secondary Propositions
16.6 Examination of Aristotelian Logic
17. Probability Theory
Scholastic logic dominated European thought all the way through the eighteenth century. Even the arch-critic of Scholastic metaphysics, Immanuel Kant, acknowledged that logic was a complete science, with nothing new to be added.
In the nineteenth century, some logicians expanded the traditional set of categorical propositions A, E, I, O with a complementary set where the subject is negative. For example, the counterpart of All As are Bs
is All not-As are Bs.
Here the negative is attached to the subject term, so it does not signify the negative mood, but only the complement or privation of A. Then All not-As are Bs
is fully equivalent to All Cs are Bs
where C is defined as an entity which is not A. These complementary propositions therefore added nothing to the structure of logic. The nineteenth-century emphasis on negative subjects was grounded in the assumption that the negation in an ordinary simple proposition pertained to the predicate rather than the copula. Thus A is not B
would be parsed as A is ¬B
rather than ¬(A is B), using modern notation. This convention effectively reified negatives, so that they could be considered as objects. Such a conception would make possible what we might call a set theoretic model of propositional logic.
In retrospect, it is perhaps unsurprising that mathematicians should attempt to express logic as a calculus. After all, the four categorical propositions were distinguished mainly by their quantification, and Scholastic logicians had already developed the habit of referring to individuals as members of some set that represented the universal. It was a small leap, then, to take this conceptual shorthand literally, and objectify all terms, positive or negative, substantive or accidental, universal or particular. By treating all terms as representing objects, indistinguishable in category, one could derive those rules of logic that were independent of the ontological content of the terms, and thus have an ontologically neutral logic.
This could only be achieved by hypothesizing a purely object-oriented ontology, so such logic was not in fact ontologically neutral. Such a formalism also ran the risk of losing certain subtle distinctions preserved in classical logic. This logic
would have a purely extensional notion of reference, similar to the classical appellatio, and a purely existential notion of being. For a system that pretended to express the very laws of thought, it restricted man to a highly materialist conception of reality, leaving no place for intensional meaning. Unsurprisingly, this extensional logic
eventually became the dominant paradigm among English philsophers, while a richer intensional logic prevailed on the Continent. Russell would later complain that intensional logical calculus was impracticable, as if this were an indictment of the notion of intensional meaning, rather than a consequence of the intrinsic limitations of logical calculus. It was inconceivable to Russell that there should be any reality beyond that which could be expressed by a calculus.
George Boole (1815-1864), a self-educated mathematician of lower class background, developed the first calculus of propositional logic, an ingenious system that replicated syllogistic inferences and could analyze multiple propositions and variables. Boole was cautious in his proposal to supplant classical logic with a calculus, paying due deference to the former, but finding that his allegiance to the truth compelled him to accept the latter as expressing more fundamental laws of thought. He did not believe it to be a priori necessary for the laws of thought to be mathematical, but he held that this happened to be the case. He first presented his theory in Mathematical Analysis of Logic (1847), followed by Calculus of Logic (1848). Finally, in An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probability (1854), he revised his earlier calculus and extended his argument to show how a theory of probability could be developed.
Boole’s ultimate aim was mathematical, namely to give a quantitative theory of probability, building on the work of Pascal, Laplace and others. This goal practically required him to turn to questions of conditional logic (then called hypothetical,
now called modal
), which in turn required the more fundamental categorical logic. If Boole was to provide a calculus for probability, it seemed he would also need to give a calculus for categorical and hypothetical propositions.
Probability in mathematics deals with objects and events, so Boole sought to reduce all ontological terms to objects, and all categorical propositions to expressions of facts or events. Yet Boole did not think he was providing a logical calculus only for mathematics, but sincerely believed that his calculus expressed the fundamental laws of thought, which were of universal applicability. Thus all of reason would be made in the image of mathematics. Boole did not consider this to be an artifice of his, but a discovery of objective reality, i.e., that the laws of thought really do correspond to a particular kind of algebra. We will explore his claims, considering both the calculus itself and its supposed basis in fundamental laws of thought.
In An Investigation of the Laws of Thought (1854), Boole gives a formal exposition of his mature theory from first principles. It begins as an apparently philosophical endeavor, then turns to an exposition of his logical calculus, showing its ability to replicate the results of classical logic, and even to make seemingly non-syllogistic inferences. Following the empiricist zeitgeist, Boole presumed to make logic an empirical
science based on observation. The only difference between logic and other empirical sciences, he argued, was that in logic the things observed are thought processes, and a single observation suffices to establish a general truth. This sounds a lot like an a priori science, and only cultural preferences can account for why Boole felt the need to fit logic under the rubric of empirical science. Here was an early expression of scientism,
that bane of the nineteenth century where all of reality is forced onto the Procrustean beds of empirical science. The standard of whether a field of study was respectable
or not was measured by whether it earned the moniker of ‘science,’ a term that had only recently gained currency in its present usage. Respectability, in turn, meant matching the epistemic and metaphysical prejudices of the Enlightenment and its heir, liberalism.
Although men can directly know or intuit many general truths of logic (e.g., the principle of non-contradiction, or the validity of a certain syllogism), it is not so easy to determine which principles are more fundamental. Boole believed that if logic is to become a science, it must be able to determine which general rules are more fundamental. ‘Fundamental’ rules were defined as those principles from which others may be deduced. An apparent weakness of Boole’s theory, we shall see, is that the order of deduction is not always unique. Sometimes we may deduce A from B and B from A, so how can we know which is more fundamental? In classical logic, we might appeal to ontological considerations, but Boole wishes to minimize the role of ontology in his logic.
According to Boole, logic deals with relations among things and relations among facts or events, with facts or events being represented by propositions. Boole abstracts from any assumptions about the nature of causality, expressing a relation between events as Event A is an invariable consequent of B.
This minimalist account of causality mirrors that of Hume. The fact of event A being an invariable consequent of event B corresponds to the truth of the proposition affirming A
being an invariable consequent from the truth of the proposition affirming occurrence of B.
This is similar to what Russell would later call material implication.
While Boole is perfectly free to define this sort of relation between events, his logic cannot be considered a complete account of the laws of the thought unless at some point it is capable of expressing more developed notions of causality. Even empirical science has shown (in medicine, for example) that correlation is not always the same as causality, and we consider understanding of fundamental principles to follow along lines of causality, not correlation. A further problem is how we know when something is invariably consequent. If, as Boole says, general truths can be apprehended in logic by a single observation, it would seem that we can simply declare something to be invariably consequent. Yet if Boole wants to ground logic in empiricism, so that it can only make statements about empirical facts (or about relations between such statements), we arrive at the problem of induction, for no finite quantity of observations will suffice to establish a general truth, as long as there remains the possibility of new observations of similar phenomena. His material implication is grounded in an inferential impossibility, hardly an auspicious start for his logic, but such are the problems that arise when trying to reduce logic to an empirical science with an impoverished ontology.
Boole proceeds to describe inference in the proper sense, which is the logical derivation of a full proposition from another full proposition. In classical logic, inference is represented syllogistically, with a conclusion being inferred from two premises. Boole calls the premises of an inference the given
relations among elements (terms), while the conclusion is the implied
relation. That is, when we explicitly declare the relations given by the premises, we are implicitly declaring the relation expressed by the conclusion. This is true whether we are conscious of the conclusion or not, so there is no need to invoke psychology here.
In classical logic, inference is achieved by a combination of two processes, syllogism and conversion. Conversion is a process whereby a logically equivalent form of a proposition is given in order to reverse the order of the terms. Once propositions are in the proper form, they can be subjected to one of the appropriate syllogisms, where a middle term shared by the two premises may be eliminated. In Boole’s calculus, this process of elimination is simulated, but in a way that allows multiple eliminations across many propositions, without breaking it down into syllogistic steps. This is an important practical advantage over syllogisms, yet Boole claims something more. He believes that the mathematical laws governing his calculus are the true fundamental laws of logic. He does not pretend to know why this is so, nor does he claim it is a priori necessary for the laws of thought to be mathematical, but only that this happens to be the case. From our perspective, it is no surprise that the laws are mathematical, for he is giving the laws of his calculus, not the laws of logic in general. As we shall see, Boole’s calculus cannot handle everything that classical logic does, though it does accomplish a great deal. Boole’s logic is concerned only with the extensive aspect of terms, which is why it can be modeled as a mathematical calculus. It does not deal with the intensive aspects of terms; indeed no calculus can do so, unless we postulate additional qualitatively-based rules of inference, but such a logic is arguably no longer a calculus.
Boole’s ultimate goal is to give a theory of probability, which is mathematical only if probability is quantifiable. The data, or the given, are the probabilities of each event, which are computed by dividing the number of cases favorable to the event’s occurrence by the total number of cases, favorable or unfavorable to the event’s occurrence. All cases are assumed to be equally likely. The quaestium in probabilistic logic is the probability of some other event, which is inferred from the data. According to Boole, the data can be given either causally (e.g., understanding how a rolled die works) or from repeated observation of trials. In the first case, Boole seems to restore causality to his logic, notwithstanding his earlier disdain, though here it is just one possible means of establishing a material implication. The other method, repeated observation, runs into the problem of induction, but Boole appeals to a limit theorem, whereby truly stochastic events will tend toward some fraction reflecting the event’s probability, after many trials. This is a mathematical and physical thesis, which appears to be built into his logic at a basic level. It is difficult to avoid these kinds of assumptions when making a quantitative probability theory, as opposed to the more generic classical modal logic of necessity and contingency.
Boole’s approach to semantics is minimalist, since he wishes to take little heed of the semantic content of signs. In his words, he wishes to consider language specifically as the instrument of reason,
not in its general function as the expression of thought. In this way he distinguishes logic from psychology. Logic is concerned only with those signs that represent concepts and rational operations, and Boole claims that by studying the laws of these signs we effectively study the laws of reasoning. This thesis implies a faith in the ability of signs to convey concepts and rational operations without distortion that would affect the logical structure of an argument. Boole will develop his own system of signs that purportedly has these qualities.
A written sign is defined by Boole as follows: A sign is an arbitrary mark, having a fixed interpretation, and susceptible of combination with other signs in subjection to fixed laws dependent upon their mutual interpretation.
Boole held that signs represent conceptions of the mind, which in turn represent things. This is nothing other than the tripartite theory of semantics posited by St. Augustine and followed by medieval Scholastics. The laws of signs are assumed to reflect the laws of the mind, which in turn reflect the connections and relations of things. Syntax corresponds to semantics, which corresponds to ontology. The respective sciences are grammar, logic and metaphysics. Boole, however, wishes to exclude semantics from his logic, resorting to psychology only in the beginning to discern his laws of thought,
after which he will derive everything else by the power of his calculus. Indeed, he uses the classical conception of semantics as a justification for dispensing with any discussion of psychology or ontology, and discussing symbols only.
Boole discerns three kinds of elements of rational language. The first is literal symbols
(e.g., x, y, z) that represent things as subjects of our conceptions.
[1] The second is signs of operations
(e.g., ‘+’, ‘-’) for operations by which conceptions are combined. Third is the sign of identity, ‘=’, which Boole uses in lieu of the copula. We see right away that this is a highly restrictive logic, as identity is only one of many different possible functions of the copula. Boole will attempt to justify this by abolishing ontological category distinctions among terms. Absolutely everything—universal or particular, substance or accident—will be treated as a subject (or, equivalently, an object, since the subject-predicate distinction is abolished). Boole’s calculus cannot be considered analogous to modern predicate calculus, as he denies any distinction betwen subject and predicate.
Boole’s schema for making every term (x, y, z) a subject involves some grammatical play that is ontologically dubious. Adjectives or accidents are reduced to substantives by using the appended expressions being
or thing.
For example, instead of saying, Water is fluid,
we are to say, Water is a fluid thing.
Instead of Leaves are green,
we could say, Leaves are green things.
Fluid things
and green things
are not to be considered logical compounds, but primitive substantives. I have noted in my Introduction to Ontological Categories (Part II) that adding being
to a term does not change its ontological category, for ontological categories are predications of being. Being
is not some generic thing of which the various substances and accidents in the world are particular instances. To say, A leaf is a green being
does not change green
into a substance, since being
has no category. How about a green thing,
as Boole suggests (in his example of white things
and sheep
)? By thing,
Boole effectively refers to an indeterminate, generic substance, like prima materia. So green thing
means prime matter that is green. All well and good, but this does not abolish the fact that green is a quality. The being of green thing
is determinate only in its qualitative aspect, not its substantive aspect. Therefore, declaring an object to be a white thing
determines it qualitatively, but not substantively. If Boole’s decree of equivalence is valid, then there is no logical distinction between qualitative and substantive determination. This, of course, is not so, for we can conceive of these determinations as distinct in kind, which suffices to establish an a priori distinction.
For example, consider categorical syllogisms of the form:
M is P.
S is M.
S is P.
If S is an object (substance) and M is a property (accident), then P must also be a property. This is because a substantive is not predicable of any property. A leaf is green,
is not the same as Green is a leaf,
nor does it help if we say Green things are leaves,
as that would not be generally true. Boole is well within his right not to deal with predication, but then he cannot claim to have a comprehensive logic, as it does not encompass all rational thought. Modern logicians have recognized this deficiency, and therefore supplement Boole’s logic with a predicate calculus, to be discussed at length later. We should note that, among the nine classical categories of accident, at most five are logically distinct forms of predication: quality, quantity, space, time, relation. The other four are types of relations.
Boole also collapsed the distinction between universal and particular, a simplification that has persisted in modern mathematical logic, as it effectively reduces universals to set theory, a discipline that would be formally defined by Cantor in 1874. In Boole’s formulation, we can define the object x = all men,
where all men
is the class man.
A class is just a collection of individuals to which a name or description is applied. This is a useful construct in mathematics and computer science, but it is not a universal. The power of universal concepts comes in the fact that I can apply them to particulars I have not yet encountered or even imagined. I do not need to have contemplated every possible triangle in order to understand the universal triangle.
Rather, once I grasp the universal, I can confidently examine any new object and determine whether it is a triangle. The universal triangle
is not the collection of all objects that are triangles. The proof of this is that I can understand triangle
without knowing all conceivable triangles, or all actual triangles. We inhabit such a small corner of the universe that we could not credibly claim to understand any universal concept if it really were the synthesis of all individuations. Further, it is possible to know every actual instance of an object without exhaustively specifying the universal, since it is always conceivable to make more of the same kind.
Boole overextends this set theoretic concept of class, making it cover universals (the range of all
is problematic), as well as a single individual; the empty set is nothing,
and the universe is all beings.
This goes too far even for set theory (as later defined by Cantor and his successors), as it collapses the distinction between element and set. Such oversimplification does, however, have the advantage of making a very compact calculus. It is no wonder, then, that modern Boolean logic
(a simplified system of binary truth values and connectives, far removed from Boole’s calculus) has dominated computer science, as it makes use of a minimal number of operators and types of operands. What is good for programming efficiency, however, is not necessarily good for a philosophically comprehensive logic, one that deals with the objects of the intellect, namely concepts. Computer logic is best suited for manipulating quantifiable attributes of concretized objects, such as material things represented by symbols. It is useful in mathematical and physical problems dealing with the concrete or material, or their mathematical images.
In Boole’s simplified ontology, which has no predication and no categorical distinctions, the order of terms in a sentence is irrelevant. In other words, for any objects x and y, xy = yx. For example, sheep that are white
= white things that are sheep,
and navigable rivers that are estuaries
= estuaries that are navigable rivers.
Note that each of these expressions is not a statement, but a set theoretic description of a class of objects. Boole’s identity
means only that the two classes or sets have the same members, not that they are conceptually identical. Boole thus tacitly makes discrete objects, not concepts, the subject of his logic. As a result of his logic’s lack of ontological distinctions, he may use variables x, y, z, etc. to replace substantive, adjective and descriptive phrases.
Note that Boole’s collapsing of ontological distinctions does not make his logic ontologically neutral, but rather assumes an object-only ontology. Such an ontology is suitable in set theory and programming, but is hardly suitable for establishing comprehensive rules of rational thought. His claim that terms can be placed in any order (not talking about syntax here, but the semantic behind a given syntax) fails to capture an important distinction. While it makes no difference whether we say plump white sheep
or white plump sheep,
it does make a difference whether we say white sheep
or sheepy whites,
the latter being ontologically incoherent. Boole would make the latter expression white things that are sheep,
and perhaps write off the whole substance-accident distinction as an idiosyncrasy of our grammar. Yet the expression white things
preserves the substance-accident distinction, folding both categories into a single, more specific substantive term. Yet the substantive thing
is substantial only in grammatical form. It is not a defined class of substance, but at best inconceivable prime matter. This does not mean thing
is unintelligible; we understand it as a placeholder for some substance to be determined. When I say ‘white things,’ I intend white sheep or white clouds or white pillows or anything whatsoever that is white, whether I have explicitly thought of it or not.
This is conceptually distinct from sheepy whites,
where I am erroneously considering white
as a substantive and trying to make sheep
into an accident by a superficial grammatical change. This is not tenable, for reasons outlined in Introduction to Ontological Categories (Parts I and II). Boole’s logic is not wrong for ignoring this distinction, but it is simply less comprehensive. He is within his rights not to deal with problems of predication and categorical distinction, but this does not abolish the rational validity of such aspects of reality. We must regard his calculus as dealing with a particular problem set: that of classes or sets of objects. This is still a very broad scope, so it is no wonder that Boole’s logic has such wide application.
Another way of putting it is that Boole considers predicates not as accidents inhering in a substantive (or related to several substantives), but in their special function of defining a genus or species of substantives. For him, black cats
and sleeping cats
merely define subsets of cats,
which is only part of what these expressions signify to us in ordinary language. It is generally valid, in fact usually necessary, to use accidents to define a species or genus of substantives, at least implicitly. This is not the sole significance of accidents or predicates, however, nor even their primary significance. Thus Boole’s logic is really a logic of classes, and not a substitute for predicate logic.
From the considerations discussed, Boole concludes that the following law of signs is permissible:
We are permitted, therefore, to employ the symbols x, y, z, &c., in the place of the substantives, adjectives and descriptive phrases, subject to the rule of interpretation, that any expression in which these symbols are written together shall represent all the objects or individuals to which their several meanings are together applicable, and to the the law that the order in which the symbols succeed each other is indifferent.
The application of logical symbols to adjectives and descriptions will often lead Boole to speak of a variable x as signifying the presence or possession of some property x. Strictly speaking, however, the variable x refers to the group of objects possessing the property in question, not the property itself considered as a universal accident.
Combining terms (e.g., xy) is effectively set theoretic intersection, so the order of terms is indifferent. Boole says this law of signs is actually a law of thought, though he will only expound his fundamental laws of thought later. Indeed, he is not always clear on the distinction between laws of signs and laws of thought.
From the above rule of interpretation, it follows that if two symbols in a term signify the same thing, then the term is identical to that which has one of the symbols. In other words, white white things
means the same as white things.
Boole is able to express this intuitive principle algebraically. Let white things
= x and white
= y. Then xy = x. In Boole’s syntax, we put the qualifier (x) first, but it does not matter, since he treats everything as an object. Thus xy = yx for all x, y. Really, then, both x and y mean white things
in his syntax. xy means white things that are white things.
Since x and y are the same thing, xy = x implies that xx = x (substituting x for y). Boole gives this a more succinct algebraic notation x2 = x.[2]
In Boole’s object-oriented logic, terms can be coupled by logical operators, which are syntactically equivalent to arithmetic operators in his pseudo-binary formalism. He symbolizes and
and or
as +, and except
as −.
Students of modern set theory may be surprised that Boole seems to recognize no fundamental distinction between and
and or.
After all, the set of all things that are white or green
is not the same as the set of all things that are white and green.
We would represent the first as the union of two sets (that of white things and that of green things), while the latter would be the intersection of the same two sets. Boole considered both and
and or
to signify an aggregate conception. For example, if I say dogs and cats,
I am juxtaposing the objects dogs
and cats.
When I say dogs or cats,
I am doing the same thing. So, Dogs and cats are animals,
signifies the same thing as, Dogs or cats are animals,
and Boole uses the same symbol + to signify both and
and or.
He thus treats ‘or’ only in its conjunctive aspect, not in its disjunctive aspect. This is because + couples objects, not predications.
Boole explicitly supposes that conjunction, in strictness, implies that the combined classes are quite distinct, so that no member of one is found in another.
Thus his + symbolizes an exclusive union, joining terms whose intersection is null. Jevons and Frege sharply criticized this privilege of exclusive over inclusive unions, but we shall see that it is a valid choice by Boole, in line with his aims for his formalism.
Boole uses the minus symbol to signify except.
So animals − cats
would mean all animals except those that are cats.
As Corsi and Gherardi (2017) note, this is an inclusive subtraction,
for the validity of the expression A − B presupposes that B is a subset of A.[3]
A single relational symbol =, signifying is
or are
, is used to construct propositions, which have the form of algebraic equations. Boole reduces predicates to substantives by rephrasing Caesar conquered the Gauls
as Caesar is he who conquered the Gauls,
making the copula (is
) replace the active verb. Thus all sentences can be formulated in terms of Boole’s object-only syntax. Two axioms apply to the manipulation of propositions.
1st. If equal things are added to equal things, the wholes are equal.
2nd. If equal things are taken from equal things, the remainders are equal.
These principles are familiar from elementary algebra, as a means of manipulating systems of equations. We may write these algebraic rules as follows:
x = y ⇒ x + a = y + a
x = y ⇒ x − b = y − b
Of course, we cannot use algebra on real numbers to prove things about Boole’s more limited calculus. Recall that equal
in Boole’s calculus means identity (according to his notion of identity). So x = y means x and y are different names for what is existentially the same thing. (Boole only considers identity in the order of concrete existence, not conceptual identity.) It is only under this definition of identity that we are to understand that x + a = y + a (where + means and
or or
), and that x − b = y − b (where ‘−’ means except
).
A third rule is that if two classes x and y are identical, the members of each class possessing a common property z will likewise be identical. That is:
x = y ⇒ zx = zy
However, the analogy with algebra does not hold for the reverse implication.
zx = zy ⇒ x = y
This is a false implication, as we can show clearly from numerical set theory. Let x = {2, 4, 6}, y = {3, 6}, and z = {5, 6}. Then zx = zy = {6}, yet x and y are not identical.
Earlier, Boole had derived from his rule of interpreting signs (which he considered a law of thought) that x2 = x holds for any x whatsoever. Only two real numbers, 1 and 0, obey this relation considered algebraically. Thus Boole’s logic may be treated as an algebra (on a subset of reals) only on the assumption that 1 and 0 are the only values admitted by the symbols x, y, z, etc. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them.
Boole will compose propositions from the three classes of symbols discussed and nothing else: substantives (x, y, z), two conjunctions ( disjunctive union [+] and exception [−]), and identity (=). The juxtaposition of variables zx might be considered a third conjunctive operation, namely intersection. Boole acknowledges that the conjunctions if, either, or may be used to express relations among propositions, but he will attempt to show that these relations may be expressed equivalently by symbols analogous to those constituting propositions.
Having already reduced verbs and adjectives to substantives, Boole remarks that pronouns are just a form of substantive, adverbs modify verbs, and prepositions, expressing circumstance or relation, give precision and detail to the meaning of the literal symbols.
This gives short shrift to the role of pronouns as substituents, generating the richness and paradox that would challenge Russell. Prepositions, likewise, can be rich in logical import, especially as modifiers of time (before, after
).
Boole did not claim to discover a merely useful calculus, but pretended to have uncovered the fundamental laws of thought and mathematicized them. If this claim is to be given credence, he must account for all valid logical thought, not just a subset of it. Boole has the unfortunate habit of designating as meaningless or unscientific anything that his logic does not grasp. Thus he effectively reduces the statement, Sheep are white
to, Sheep are a subset of white things,
even if the latter is not precisely what I intend by the first statement. His laws of thought
would forbid many ordinary thoughts as illogical, without due cause.
Confident, nevertheless, that he had discovered the philosopher’s stone, Boole purported to derive his algebraic rules of logic from fundamental laws of thought. In doing so, he effectively advances a theory of linguistics or psychology.
The office of any name or descriptive term employed under the limitations supposed is not to raise in the mind the conception of all the beings or objects to which that name or description is applicable, but only of those which exist within the supposed universe of discourse. If that universe of discourse is the actual universe of things, which it always is when our words are taken in their real and literal sense, then by men we mean all men that exist; but if the universe of discourse is limited by any antecedent implied understanding, then it is of men under the limitation thus introduced that we speak.
Boole’s widest possible universe of discourse
is much too narrow to comprehend all of human thought, limited as it is only to things that actually exist. This is an early precursor of claims by later logicians that terms corresponding to no actual existent have no referent. Yet it is manifest that we may speak intelligibly of things that do not exist actually, but only potentially or conceviably, even of things that are demonstrably fictitious. To deny this is so would be to make an unjustified confinement of the function of thought to representations of physical reality. The term man,
insofar as it refers to the concept rational animal,
may embrace creatures on worlds we have not yet conceived, and innumerable potential men, many of whom may never become real. If we were to allow such breadth of meaning to our concepts, however, any attempt to reduce their analysis to set theory might fail, as the quantifiers may become ill-defined, for the universe of discourse is inconceivably infinite. Boole’s logic is really a quantificational logic, so we must define some scope to all in order for it to work. This scope need not be limited to actual existents.
Boole considers the function of adjectives to be not merely attribution, but selection according to a prescribed principle or idea.
By saying white men,
we direct our attention to a select group of men. If we reversed the order of terms, the result would be the same, considering all white things, then focusing on those white things which are men. Thus the order of conception is a matter of indifference, and the commutative law xy = yx reflects a law of thought, without regard for whether x or y are grammatical substantives or adjectives. Moreover, repeated application of the same selective act makes no difference to the conception. If we considered first white things, then those white things that are men, and finally those white things that are men
that are white, this last step makes no change to the conception. This law of thought is expressed as the second law of symbols: x2 = x. Boole will often call this the law of duality.
Likewise, it matters not in which order we combine in union two distinct sets of objects, as expressed in the third law of symbols: x + y = y + x. We should note, however, that Boole’s pretension that there is no logical difference between subjects and attributes depends on + signifying a disjunctive union. We can never use + to combine an attribute (white
) with a subject (men
) of which it is predicable. The law xy = yx applies indifferently to subjects and attributes only because it is a quantificational law, and does not encompass all aspects of thought, which include relations of existential dependence (i.e., white must inhere in a subject, but subjects may exist without whiteness or color at all). Nonetheless, we may follow Boole within the confines he sets, for a logic that deals with the actual world and the quantification of its objects is already a quite powerful logic, if not comprehensive.
Boole notes that, in algebra, the only two numbers that satisfy x2 = x are 1 and 0, which also have their own peculiar laws and 1y = y and 0y = 0 for all values of y. If the symbols 1 and 0 are used in Boole’s logic to obey the logical laws represented by these same algebraic expressions, then 0 must mean Nothing and 1 must mean Everything, i.e., the universe of discourse.
Since juxtaposition of symbols signifies intersection, 0y = 0 for all y if 0 is a class with no members, for then there can be no individuals common to 0 and y, no matter what y is (even if y is 0!). The equation ay = a would not hold for all y if a has at least one member, for then there could exist some y that did not contain at least one member of a, in which case ay would not equal a. Boole is effectively introducing the existence of the empty set as an axiom. As he defines it, 0 the empty set is the intersection between itself and all sets, so it is a subset of all sets, just as in modern set theory.
Note, however, that 0 in Boole’s calculus is not the additive identity. The expression x + 0 is not definable, because + signifies disjunctive union. If 0 is a subset of every set, it cannot be in disjunctive union with anything.
The identity 1y = y holds for all values of y if 1 contains all members of all sets y. 1 is therefore the universe of discourse, i.e., the set of all individual objects that can be analyzed in Boole’s logic. This Everything, for him, is the set of all actual existents. We could make our Everything even broader if we expanded the scope of what was permissible within all
y, i.e., if we allowed potential, imaginary, or fictitious entities.
Recall that the introduction of the classes 1 and 0 was motivated by the principle x2 = x and its analogy with algebra. In Boole’s ontology, 1 or Everything
or the Universe
means all objects that exist.
0 or Nothing
means a class that does not contain any objects that exist. If we apply the law x2 = x to 1 and 0, so 12 = 1 and 02 = 0, this means everything that is everything,
i.e., all objects that exist that are among all objects that exist, is existentially identical with everything,
and nothing that is nothing
is identical with nothing.
This is far from obvious in the second case, but Boole will not allow an exception in his algebra. The awkwardness of treating the empty set equivalently with other sets is not peculiar to Boole, but persists in modern set theory.
If 1 represents the universe of objects, then 1 − x is the complement of x, that is, the set of all objects that are not of class x. If x is men,
then 1 − x is the class of not-men.
Note that the empty set 0 is a subset of both 1 − x and x. This does not contradict the principle of non-contradiction, but is another way of affirming it, since 0 has no members and is not itself an object in the universe 1.
In a radical maneuver, Boole argues that the law of contradiction, long thought to be a primitive axiom, is nothing more than a consequence of the more fundamental law of thought x2 = x (law of duality). The algebraic proof:
x2 = x
x − x2 = 0
x(1 − x) = 0
The last statement, which asserts that nothing can be both x
and not x,
is nothing other than our familiar law of contradiction. Boole is claiming that this law is not fundamental, but is derivative of the law of duality
by which white things that are white
is the same as white things.
The logic underlying the first step of the algebraic proof is sound. If x things that are x things
is the same as x things,
then the set of all x things
excluding x things that are x things
would be a nullity. That is, x − x = 0, so x − x2 = 0.
The second step of the proof seems a bit dubious at first. Why are we justified in dividing out
the x, going from x − x2 to x(1 − x)? Just because this is valid in ordinary algebra does not mean it is valid in this particular calculus of the laws of thought. Consider the general case a(1 − c). This is the set of all a’s that are non-c’s.
Is this the same as a − ac? This last expression would mean all a’s except those a’s that are c’s. This is clearly identical with the previous set, so the identity holds and the proof is valid.
Could we have made the derivation in the reverse direction? Starting with the principle of contradiction, we have:
x(1 − x) = 0
x − x2 = 0
x2 = x
The first step is valid, since a − ac and a(1 − c) are existentially identical. How about the last step? We start with all x things except those x things that are x things are nothing.
Does it necessarily follow that x things that are x things
is identical with x things?
If we tried the algebraic approach of adding x2 to both sides, we would have x − x2 + x2 = 0 + x2, but the right side is an invalid expression, as 0 cannot be in disjunctive union with anything. Since the minus sign signifies exclusion, x − x2 = 0 means that if we exclude the members of x2 from x we are left with nothing. This means x2 must be identical with x or a superset of x. The latter possibility is precluded by Boole’s notion of exclusion, which is separating a part from a whole, undoing a disjunctive union. Thus the expression a − b, to be valid, presupposes that b is a part or subset of a. Thus we have x2 = x. Since the deduction of x2 = x from the law of non-contradiction is just as valid as the reverse, Boole has not proven that one law is more fundamental than the other; rather, they are logically equivalent.
Would we get this logical equivalence if we were not confined to Boole’s object-only ontology? Let us use predicate logic for a moment, to express that a white white fence
is the same as a white fence. As a first attempt, let us say P(x) means A fence (x) is white (P).
(1) Let P(x) = P(y) where y = P(x)
(2) Then P(x) = P(P(x))
Already in (1) we have a problem, because we are treating a proposition P(x) (x is P
) as if it were a term y. Pressing on regardless, if we accept (1) as a premise, it follows that x = y, so x = P(x), which is absurd. The problem is that we were trying to express something absurd, namely ‘A fence is white,’ is white.
Rather, we wish to say, A fence that is white, is white.
We may express this by introducing a noun phrase operator [ ], so that [f(x)] is defined to mean an x that is f.
Then we can formulate an analogue to the law of duality.
f[f(x)] = f(x)
‘f’ may now represent a predicate such as white.
The left side of this equation may be read as An x that is f, is f.
The distinctness of the [ ] operator prohibits us from inferring that f(x) = x, which would be unintelligible. In imperfect analogy with Boole’s x2 = x, we may express the left side’s two operations as f2(x) and infer:
| f2(x) = f(x) | x2 = x |
| f(x) − f2(x) = 0 | x − x2 = 0 |
| f(x)[1 − f(x)] = 0 | x(1 − x) = 0 |
This last step is invalid, however, for we cannot factor out the f(x), because f2(x) = f(x) ≠ f(x) ⋅ f(x). Indeed f[f(x)] is not simply predication applied twice, for we need the [ ] operator to convert the atomic formula f(x) into a subject.
So, in predicate logic, the principle of contradiction is not formally derivable from the non-iterative nature of a single predicate. That result was an artifact of Boole using predicates as objects. Boole’s proof is valid only in an object-only ontology, which refuses to distinguish between subject and predicate. I will later argue that modern predicate logic has its own limitations, on account of its failure to distinguish adequately between universals and particulars, regarding the former as mere collections of particulars.
The propositions we have discussed so far are what Boole calls primary,
as they relate things to things, i.e., the objects represented as variables x, y, z. There are also secondary propositions, which express the logical relationships among the truth and falsity of primary propositions. If A, then B,
means that in all cases where primary proposition A is true, proposition B will also be true. No implication regarding causality or temporality is thereby intended. A or B
means A is true or B is true or both A and B are true. Either A or B
means A is true or B is true but not both.
Primary propositions are represented as algebraic equations, where the logical variables x, y, z, etc. denote terms (subjects or predicates). In secondary propositions, each letter X, Y, Z represents a primary proposition, i.e., a logical equation in terms of x, y, z. The = sign, which is the existential copula in primary propositions, signifies logical equivalence in secondary propositions. Following a device used by Richard Whately in his Elements of Logic (1826), Boole notes that for every primary proposition X we may define an object x signifying all cases in which X is true. In this way, we can define a primary proposition corresponding to every secondary proposition. Thus x = 1 is equivalent to stating X is always true, while x = 0 is equivalent to stating that X is always false. X and Y
corresponds to xy = 1. X or Y (exclusive)
corresponds to x(1 − y) + y(1 − x) = 1. Since multiplication is set intersection, the expression x(1 − y) means x’s that are not y’s.
Inclusive union may be represented fully as: xy + x(1 − y) + y(1 −x). Boole notes, however, that this is equivalent to the asymmetric expression x + y(1 − x). We can confirm this ourselves:
xy + x(1 − y) + y(1 −x)
= xy + x − xy + y − xy = x + y − xy = x + y(1 − x)
Thus X or Y (inclusive)
indicates either that X is true or that Y is true and X is false. Alternatively, we could have chosen y + x − yx = y + x(1 − y). Then X or Y (inclusive)
also indicates that Y is true or that X is true and Y is false.
For if-then statements, Boole uses the indefinite variable v. y = vx can represent All men (y) are mortal beings (x),
for vx represents some mortal beings,
as v is indefinite. If Y then X,
where X means Q is mortal
and Y means Q is a man,
is true if and only if y = vx for some unspecified v. Here the if-then implication is understood to indicate that the set of cases where Y is true is a subset of the set of cases where X is true. This set theoretic notion makes sense insofar as X and Y may be characterized by sets of possible cases in which each are true. There is no notion of priority in causality or temporality. We need not suppose that the truth of X causes Y to be true, or that X must be true before Y comes to be true, only that in all cases where Y is true, X is also true.
Boole’s scheme for handling universal or quantified propositions needs closer scrutiny. All men are mortal,
in Boole’s object-only ontology, can only mean, All men are mortal beings.
He introduces the symbol v to signify a class indefinite in every respect but this, viz. that some of its members are mortal beings.
It is far from obvious that this is a valid definition for a literal sign, since v is at once definite and indefinite, yet it plays a critical role in Boole’s logic. The expression vx, where x stands for mortal beings, means some mortal beings,
according to Boole. v acts as an indefinite quantifier, yet Boole regards it as a class variable, no different from x, y, etc.
y = vx
where y = men
and x = mortal beings,
means that the set of all men is the intersection of some indefinite set of objects v with the set of mortal beings x. Boole had to specify that v contains mortal beings in order for the vx not to equal zero.
Even if we were to admit that Boole’s definition of v is valid, it does not follow that the meaning of All men are mortal
is truly contained in the expression y = vx, though the two expressions might be logically equivalent. Setting aside the fact that classical universals are not exhaustively specified by their individuals and thus cannot be represented as determinate sets or classes, there remains the problem that there could be many different sets v that give the same result. y = v1x, y = v2x, y = v3x, etc. would all be equally equivalent to All men are mortal,
which has no such multiplicity, so it is not saying the same thing. Further, it is not enough for v to contain some mortal beings, if these mortal beings are not the same as those who are men. To make the latter specification would be question begging and Boole’s y = vx would have no more logical content than y = yx (Men are men who are mortal.
) Yet to neglect to make this specification, we have, Men are things-some-of-which-are-mortal that are also mortal beings.
Allowing the dubious definition of the class v as things-some-of-which-are-mortal that are also mortal beings.
Allowing the dubious definition of the class v as things-some-of-which-are-mortal,
which has a quantifier built into it, though it supposedly substitutes for a quantifier, need not suffice to be a superset or genus of men, since the class things-some-of-which-are-mortal
need not include all mortal beings. It is entirely possible, then, for an indeterminate set that contains some mortal beings not to have any men, so vx ≠ y in that case. We must assume that v contains all mortal beings, but if that is the case, we hardly need v, but only x with the quantifier some.
Boole’s formalism for quantification, then, is a highly contorted method of dubious validity, which succeeds only when it tacitly includes concepts from classical quantificational logic.
Boole’s treatment of negative universal propositions involves moving the negative to the predicate. That is, No x’s are y’s
should be rendered as All x’s are not y’s
In Boolean notation, we would have: y = v(1 − x). This expression is not subject to any more criticism than the affirmative form. It may seem strange for Boole to insist that the negative must go with the predicate, since hitherto subject and predicate have had similar logical functions for him. Acccording to Boole, No men are perfect beings,
does not declare a class called no men
and assert that they are perfect beings. Rather, we are saying of all men
that they are not perfect beings.
This is fair enough, as Boole here prioritizes the intended thought over literal rendering of customary expressions.
When developing his rules of inference, Boole will operate with primary propositions only, for he considers that all secondary propositions correspond to primary propositions as we have described, so they may be treated with the same algebraic rules as primary propositions. One of the striking features of Boole’s calculus is that it may have uninterpretable expressions as intermediate steps in inferences. Anticipating objections to such dependence on uninterpretable expressions in reasoning, Boole holds that only the following is necessary for valid reasoning:
1st, That a fixed interpretation be assigned to the symbols employed in the expression of the data; and that the laws of the combination of those symbols be correctly determined from that interpretation.
2nd, That the formal processes of solution or demonstration be conducted throughout in obedience to all the laws determined as above, without regard to the question of the interpretability of the particular results obtained.
3rd, That the final result be interpretable in form, and that it be actually interpreted in accordance with that system of interpretation which has been employed in the expression of the data.
The second element is crucial, as it allows intermediate results that are uninterpretable, but nonetheless valid for getting to an interpretable conclusion, as long as we apply the laws of the symbols correctly. Once we have established the laws that the symbols x, y, z may be treated as quantities admitting only values of 0 and 1, we may proceed to treat them as such quantitative symbols and perform whatever operations on them are permissible in the calculus. This is valid only because we derived these algebraic laws from the logical interpretation of symbols. How, then, can it yield uninterpretable intermediaries?
Boole admits these uninterpretable expressions would be useless if they could not be made intelligible, so he proposes a method by which the final result may be given an interpretable form. This method of reducing uninterpretable formulas by reordering and canceling terms is what Boole calls development.
Any function f(x), in which x is a logical symbol, or a symbol of quantity susceptible only of the values 0 and 1, is said to be developed, when it is reduced to the form ax + b(1 − x), a and b being so determined as to make the result equivalent to the function from which it was derived.
The developed
version of the expression uses a and b to subdivide the class f(x) into groups that have or lack the property x. To prove that there always exist such an a and b for any f(x), Boole exploits the fact that literal symbols such as x can only take the values 1 or 0. Assume f(x) = ax + b(1 − x). When x = 1, f(x) = f(1) = a. When x = 0, f(0) = b. Thus we have:
f(x) = f(1)x + f(0)(1 − x)
Regardless of whether x is 1 or 0, the right-hand formula will give the same value as f(x), i.e. f(1) or f(0). This suffices to show that the expression will equal f(x) no matter what form f(x) takes. Thus any f(x) can be developed.
Developing functions of more than one symbol will result in 2n terms, where n is the number of logical variables. To develop f(x, y), we first treat y as a constant, and develop in terms of x:
f(x, y) = f(1, y)x + f(0, y)(1 − x)
Then we take f(1, y) and regard it as a function of y:
f(1, y) = f(1,1)y + f(1, 0)(1 − y)
and similarly:
f(0, y) = f(0,1)y + f(0, 0)(1 − y)
Substitute these last two formulas into the first equation:
f(x,y)=[f(1,1)y + f(1,0)(1 − y)]x + [f(0,1)y + f(0,0)(1 − y)](1 − x)
= f(1,1)xy + f(1,0)x(1 − y) + f(0,1)(1 − x)y + f(0,0)(1 − x)(1 − y)
From this form, we can easily see what the development of functions with more variables would look like. In each term, we take a possible permutation of products of the n variables or their complements, and the coefficient of each term is the function evaluated at 1 for each positive variable in the permutation or 0 if that variable’s complement is taken.
Development can be applied even to formulas with no obvious interpretations, such as
f(x, y) = (1 − x)/(1 − y)
f(1,1) = 0/0, f(1,0) = 0/1, f(0,1) = 1/0, f(0,0) = 1
Here the coefficients include values 0/0 and 1/0, the interpretation of which is not obvious. Boole will later show them to be logically significant, as he must, if he considers developed equations to be interpretable.
It is possible to develop functions in terms of variables they do not contain. For example, 1 − x is developed in terms of x and y:
f(x, y) = 1 − x
f(1,1) = 0; f(1,0) = 0; f(0,1) = 1; f(0,0) = 1
1 − x = f(x,y) = 0xy + 0x(1 − y) + 1(1 − x)y + 1(1 − x)(1 − y)
= (1 − x)y + (1 − x)(1 − y)
As an even simpler example, the symbol 1 can be developed with respect to x:
1 = x + (1 − x)
Boole says a developed function is not necessarily interpretable, though equations of developed functions are always reducible to interpretable equations. For example, x − y is developed as:
x(1 − y) − y(1 − x)
Boole says this is not interpretable, because the second term is not contained in the former. Recall, however, that x − y, to be interpretable, requires supposing that y is a subclass of x. Thus the second term of the developed form vanishes (there are no y’s that are not x’s), and the developed form is in fact interpretable, though not immediately.
In any case, it is clear that equating a developed formula to one of the permitted values 1 or 0 will always result in interpretable equations. Using the example above and equating it to 0 yields:
x(1 − y) = 0 and
y(1 − x) = 0
In developed functions, the functions evaluated at 0 or 1, e.g., f(1,1), f(1,0), are called coefficients, while the rest of each term, consisting of various permutations of variables and their complements, e.g. xy, x(1 − y), are called the constitutents. Any single constituent satisfies the law of duality t2 = t, equivalent to t(1 −t) = 0. The same law holds for any sum of a series of constitutents. The product of any two distinct constituents is 0, and the sum of all constituents equals 1. This reflects their mutual exclusivity and their completeness in spanning all possible objects in the universe. If the variables represented possible events rather than objects, we might consider these quantitative relations to express joint probability.
Development
expresses formulas in terms of constituents
xy, x(1−y), etc., which subdivide the universe of discourse into all classes defined by the variables under consideration, for there is no object which may not be described the presence or absence of a proposed quality.
This presumes a law of the excluded middle for existential statements. These constituents are all interpretable, since they are the intersections (products) of interpretable logical symbols x, y, z, etc. and their negatives or complements. Developed functions, however, have coefficients such as f(1,1,1), f(1,1,0), which may make the function impossible to interpret immediately. Any function, however, when set equal to 1, 0 or some logical symbol w, constitutes an equation that can be made interpretable.
Since a conclusion must be interpretable, it must be expressible in one of the three admissible forms for an interpretable equation: f(x,y,z,…) = 0, f(x,y,z,…) = 1, f(x,y,z,…) = w.
Equations of the first type are rendered interpretable by developing the formula, eliminating terms whose coefficients vanish, and setting each term with a non-vanishing coefficient equal to 0. An equation reducible to axy + bx(1 − y) = 0 is interpreted as two equations: axy = 0 and bx(1 − y) = 0, or more succinctly xy = 0 and x(1 −y) = 0. These last two are the conclusions. Boole calls this the conjoint denial
of the existence of certain classes, or more exactly, of these classes having any members.
To interpret equations of the second form, f(x,y,z,…) = 1, we consider only the non-vanishing components, for these alone contribute to the sum. The sum of these terms equals 1. This conclusion is a disjunctive affirmation,
declaring that all existing things belong to exactly one of the classes defined by the non-vanishing terms.
The third form, f(x,y,z,…) = w, is the most general, because it requires us to interpret a variable w in terms of other variables. A formula developed with respect to w will have the form:
Ew + E'(1 − w) = 0
where E and E' are functions of x,y,z,, etc. Solving algebraically for w yields
w = E'/(E' − E)
In general, the coefficients of the terms in such complex expressions can be 1, 0, 0/0, or else something that does not satisfy the law of duality (e.g. 1/0). Boole defines the following rules of interpretation depending on the coefficient of each term:
1st, if the coefficient is 1, then the term is interpreted as the entire class defined by the constituent.
2nd, if the coefficient is 0, then no part of the class defined by the consitutent is taken.
3rd, if the coefficient is of the form 0/0, that coefficient should be interpreted as taking an indefinite class of the constituent class, i.e., taking some
members of that class.
4th, if the coefficient is definitely not 1 and not 0, so that it cannot satisfy the law of duality, then its corresponding constituent must be equated to 0.
To justify the third rule, Boole uses the example, Men (y) who are not mortals (x) do not exist,
which is the equation:
y(1 − x) = 0
y - yx = 0
yx = y
Since division is not defined for letter symbols, we cannot say y/y = 1 or any other value, but it is an undefined symbol that is equal to x. Thus if we develop x in terms of y, we might render y/y interpretable.
x = f(y) = f(1)y + f(0)(1 − y) = y + (0/0)(1 − y)
Mortals are all men plus some indefinite number (possibly zero) of non-men. Whether that number is none, some, or all non-men, Boole says, that is compatible with the premise Men who are not mortals do not exist.
Yet Boole does not prove this compatibility with his calculus. Rather, he invokes our common sense reasoning to convince us of this compatibility, and then uses the result of that reasoning to prove his calculus. The calculus as such does nothing to elucidate logic, but is dependent on our prior grasp of logic.
Nonetheless, once these rules of interpretation are accepted, it is possible to take any logical equation and convert into an interpretable form with respect to any other variable it contains. This often entails subdividing it into multiple equations, each of which is a distinct conclusion
following from the initial equation. For example, to interpret x = yz + yw in terms of y, we start with the uninterpretable y = x(z + w) and develop it as a function of x, z, w, getting the coefficients by evaluating f(x, z, w) at values of 1 for each positive variable and 0 for its complement. Ignoring terms with coefficients of 0 yields:
y = (1/2)xzw + xz(1 − w) + x(1 − z)w + (1/0)x(1 − z)(1 − w) + (0/0)(1 − x)(1 − z)(1 − w)
The terms with uninterpretable coefficients 1/2 and 1/0 are to have their constituents equated to 0, per the fourth rule. This would be justifiable only if the corresponding constituents really make no contribution to y. We are inferring that: xzw = 0 and x(1 − z)(1 − w) = 0 simply and absolutely, regardless of any connection with y. If these constituents are truly 0, then we are justified in excluding the terms from the equation.
y = xz(1 − w) + x(1 − z)w + v(1 − x)(1 − z)(1 − w)
0/0 is replaced with the indefinite class variable v, representing a non-empty set that is intersected with the constituent. This equation is the direct conclusion,
while the other equations xzw = 0 and x(1 − z)(1 − w) = 0 are independent relations.
If Boole’s method of interpretation never fails, as he claims, we should account for why the uninterpretable coefficients always correspond to constituents that equal zero. We recall that a premise of his calculus is that the only admissible values are 1 and 0. Yet his admission of 0/0 already seems to compromise that premise, as the indefinite class v is not a mere ambiguity between all
or nothing,
but takes an indefinite positive number of objects from the constituents, more than none, yet not necessarily all. On the other hand, it cannot be pure chance that Boole’s fourth rule always works.
Boole offers an algebraic proof, exploiting the fact that any coefficient that is not 1 or 0 fails to satisify a(1 − a) = 0. Suppose we express w, some class of objects, in terms of coefficients ai and constituents ti
w = a1t1 + a2t2 + a3t3 + …
Further suppose that a1 and a2 do not satisfy the law of duality, but all other coefficients a3, etc. do satisfy it. If we multiply both sides of the equation by itself, we get:
Left-hand side: w2 = w
w satisfies the law of duality, since it is a class of objects.
Right-hand side: a12t1 + a22t2 + a3t3 + …
This follows from all the constituents t satisfying duality, since they are definite classes, and from the cross terms titj all vanishing, since the constituents are mutually exclusive classes.
w = a12t1 + a22t2 + a3t3 + …
Subtracting this equation from the first yields:
(a1 − a12)t1 + (a2 − a22)t2 = 0
The other terms have vanished because the law of duality holds for a3 etc. We can multiply the equation by t1 or t2 (recalling that t1t2 = 0) to get:
a1(1 − a1)t1 = 0
a2(1 − a2)t2 = 0
Since a1 and a2 are neither 1 nor 0, it follows that t1 and t2 must equal 0 for the equations to vanish. This completes the proof, which is clearly generalizable to larger numbers of uninterpretable coefficients, due to the mutual exclusivity of constitutents. The proof relies on the law of duality and the mutual exclusivity of the constitutents, so the law of non-contradiction and the disjoint union operator are essential to the interpretability of the calculus.
How is it that 0/0 but not 1/0 satisfies the law of duality? Start with y = vx, where v,x,y satisfy duality. Then consider the cases:
0 = v0
1 = v0
The first equation is valid for any v, since the intersection of the empty set with anything is the empty set. By the same token, it tells us nothing about v with respect to x and y, so it is an indefinite class. Note that this equation would hold even if v were empty, contrary to the usage of v in universal propositions, which precludes emptiness. All men are mortal
can be represented as y = vx only if v is non-empty.
The second equation can never be valid, since the intersection of 0 with anything can only be 0, not 1. Thus v = 1/0 does not correspond to any set of real objects (nor the empty set).
Since the logical symbols v,x,y, can be anything satisfying the law of duality, it follows that v = 0/0, being a valid set, satisfies the law of duality, while v = 1/0 does not. If we were to naively square either of these coefficients, we would think that both satisfy the law of duality, but this only shows the perils of applying rules of general algebra for operations not defined in Boole’s calculus (i.e., ratios).
An important characteristic of Boole’s calculus is its ability to eliminate any single variable from a logical equation. Boole considers this operation to be analogous to eliminating the middle term in a syllogism, except in his method we only need to have a single premise, rather than two premises with a common term. Thus he purports to have found a mode of inference that is not reducible to a syllogism.
To eliminate a variable x from some equation of the form f(x,y,z,….) = 0, we have only to evaluate it setting x = 1 and x = 0, so that
f(1,y,z,…)f(0,y,z,…) = 0
To prove this equation is always true for any f(x,y,z,….) = 0, we have only to develop f with respect to x:
f(1,y,z)x + f(0,y,z)(1 − x) = 0
or in shortened notation: f(1)x + f(0)(1 − x) = 0
We solve for x and (1 − x) in terms of f(1) and f(0):
x = f(0)/(f(0) − f(1))
1 − x = -f(1)/(f(0) − f(1))
We substitute these into x(1 − x) = 0 and get:
-f(0)f(1)/|f(0) − f(1)|2 = 0
f(1)f(0) = 0
f(1,y,z)f(0,y,z) = 0
Thus the proof follows from the law of noncontradiction (or, equivalently, the law of duality). Note that the resultant equation is of the form f'(y,z,…) = 0, so that we could successively eliminate variables until we have a product of all coefficients f(1,1,1)f(1,1,0)f(1,0,1)… f(0,0,0) equaling zero.
On its face, by elimination we are making inferences from a single premise, f(x,y,z,…) = 0, though we are also tacitly invoking the logical axiom x(1 − x) = 0.
The universality of the proof of elimination suggests to Boole that we may even eliminate the indefinite class variable v.
All men (y) are mortals (x).
y = vx
y − vx = 0
f(v,x,y) = 0
f(1,x,y)f(0,x,y) = 0
(y − x)y = 0
y − yx = 0 (by law of duality)
y(1 − x) = 0
Men who are not mortal do not exist.
This is not syllogistic inference, but mere conversion into a negative, as Boole himself acknowledges. So there is nothing remarkable about such inferences
from a single premise. We may likewise infer They who are not mortal are not men.
1 − x = (0/0)(1 − y)
Similarly, from y = v(1 − x), colloquially No men are perfect,
but more exactly, All men are not perfect.
, we can derive yx = 0, Perfect men do not exist.
We can develop a contrapositive as before, in this case 1 − x = y + (0/0)(1 − y). This means Imperfect beings are all men with an indefinite remainder of beings, which are not men.
The only amplification over classical logic is the set theoretic interpretation.
These inferences
effectively restate the original premise without asserting a new fact. Whether you say No men are mortal
or Mortal men do not exist
amounts to the same thing, except phrased negatively. It is a true inference only insofar as the principle of non-contradiction is introduced as a second premise. In syllogistic reasoning, by contrast, the conclusion is a new fact:
All men are mortal.
Socrates is a man.
∴ Socrates is mortal.
The conclusion is a distinct fact from those declared in either premise. Granted, one may say the conclusion is implicit in the premises, but that is true with any logical deduction.
Boole’s abandonment of a distinction between subject and predicate appears as a strength when his calculus enables him to make deductions from multivariate equations. This is due to his ability to reverse the order of terms at will.
Wealth consists of things transferable, limited in supply, and either productive of pleasure or preventive of pain.
Taking the either… or
as intending an inclusive OR relation, Boole parses this as:
w = st{pr + p(1 − r) + r(1 − p)}
where w = wealth, s = things limited in supply, t = things transferable, p = things productive of pleasure, and r = things preventive of pain. If we were to take the either… or
expression as exclusive, we would delete the pr term.
By eliminating r and developing the result, we get
w = stp + (0/0)st(1 − p)
Interpreted: Wealth consists of all things limited in supply, transferable and productive of pleasure, and an indefinite remainder of things limited in supply, transferable, and not productive of pleasure.
Does this conclusion reflect a novel power of Boole’s calculus, or could we infer this using classical term logic?
The composite premise may be expressed as:
All W is S.
All W is T.
Some W is P.
Some W is R.
No W is neither P nor R.
The fifth sentence may be expressed as:
If W is not P, then W is R.
If W is not R, then W is P.
We may also infer from conversion of the third and fourth sentences:
Some S is W.
Some T is W.
Boole’s conclusion contains the following information:
All W is S.
All W is T.
Some W is P.
Not all W is not P.
In the last sentence, we take the indefinite remainder
as encompassing some or none. We should express this less ambiguously as It is not the case that all W is (not P).
The first three sentences are also in the initial premises, so we have only to consider whether the last is a novel inference.
Some W is P
implies that Some W is not (not P).
This implies that it is not the case that All W is not (not P).
Granting that this inference is not syllogistic, but mere conversion, this only shows that we have not arrived at any novel inferential powers. In fact Boole’s conclusion is somewhat weaker than what was known prior to elimination. His initial formula tells us precisely which non-productive of pleasure things are included, namely, those that are preventive of pain. Nonetheless, we can see the convenience of dealing with multiple terms by connectives rather than having many two-term propositions.
Elimination alone cannot get us to profitable inferences akin to syllogisms, much less surpass syllogistic logic. We must be able to take systems of equations and infer new equations using the new calculus, taking advantage of shared variables in the premises.
Boole’s calculus, we have noted, is really limited to dealing with objects, and so far his examples have had the logical variables represent classes or kinds of objects. Thus we should expect its inferential power to be limited to that which we have called Syllogism #10, where all terms represent kinds. Even here, however, we must caution that Boole’s classes are sets of existent objects, which is more limited than the classical notion of a universal substance. Aristotle and the Scholastics recognized that the validity of a syllogism depended on the type of predicability that existed between terms, and developed some ad hoc rules for distinguishing between genuine and fallacious syllogisms, the latter having the form of a true syllogism, yet invalid due to different senses of the copula in the various propositions. Boole solved this problem by avoiding it, and declaring that all predicates were expressible as groups of objects, and the copula always represents identity. Thus he made his system set theoretic before set theory was even developed.
In this limited system, it is possible for Boole to assume the validity of his inferences as long as the manipulations of equations follow the rules of his algebra, and the result is developed into an interpretable form. In fact, he discovers two methods of reducing systems of equations: multiplying by an arbitrary constant, and adding the equations. Both methods are followed by elimination and development. As is typical, he confines discussion to primary propositions.
Both methods of reduction work only on logical equations of the form V = 0, where V is some function of logical variables f(x,y,z…). There is no loss of generality, since any logical equation may be made into this form by transposition of terms.
Take two equations of developed
functions: V1 = 0 and V2 = 0. In the first method, we define an arbitrary constant c, so V1 + cV2 = 0. Since both functions are developed, they are expressed as a sum of terms for all possible constituents, i.e., all possible products of x, y, z…, 1 − x, 1 − y, etc. For a particular constituent t, let A be its coefficent in V1 and B its coefficient in V2. Then the term for that constituent in V1 + cV2 = 0 will be (A + cB)t.
The coefficient A + cB vanishes if and only if A and B are both 0. Boole offers a proof by contradiction. Suppose A + cB = 0. We then take the three cases. First, if A = 0 but B is not zero, then cB = 0 and c = 0. But this contradicts the supposition that c is an arbitrary constant, which can take any value. If B = 0 and A is not 0, the coefficient reduces to A = 0, which is a contradiction. Lastly, if neither A nor B vanish, the equation reduces to c = -A/B, again a definite value, contradicting the supposition that c is arbitrary. This supposition is essential if we regard c as defining a class of equations for all possible values of c. Then A + cB =0 is true for that class of equations V1 + cV2 only if it holds for all values of c.
Arbitrary constants are not logical symbols, so they do not satisfy c(1 − c) = 0. Boole contends that they may be used because equations involving the symbols of logic may be treated in all respects as if those symbols were symbols of quantity, subject to the special law x(1 − x) = 0, until in the final stage of solution they assume a form interpretable in that system of thought with which Logic is conversant.
Boole is not saying that the constants may be treated as logical symbols. Rather, it is that the logical symbols may be treated as though they were quantities, subject to laws of addition and multiplication, so you may do anything to them you would do to a quantity, including multiply by an arbitrary constant. It does not matter that this is uninterpretable, as long as the result is interpretable.
As an example of this method, we are given the following premises.
1st. That wherever the properties A and B are combined, either the property C, or the property D, is present also; but they are not jointly present.
2nd. That wherever the properties B and C are combined, the properties A and D are either both present with them, or both absent.
3rd. That wherever the properties A and B are both absent, the properties C and D are both absent also; and vice versa, where the properties C and D are both absent, A and B are both absent also.
Representing A, B, C, D as the variables x, y, z, w, we have:
xy = v(w(1 − z) + z(1 −w))
yz = v(xw + (1 − x)(1 − w))
(1 − x)(1 − y) = (1 − z)(1 − w)
In each of the first two equations, Boole eliminates v and uses development with respect to two variables so our three equations are now:
xy(wz + (1 − w)(1 − z)) = 0
yz(x(1 − w) + w(1 −x)) = 0
(1 − x)(1 − y) = (1 − z)(1 − w)
He multiplies the last two equations by arbitrary constants c and c', in order to eliminate w. Reduction is really an elimination technique. Solving for x gives:
x = (1 − y)z + (0/0)(1 − z)
Interpreted: Wherever the property A is present, there either property C is present and B is absent, or C is absent.
Boole often refers to his classes in terms of the presence or absence of properties, though even here the implication is reference to sets of objects, for wherever
means whichever objects possess or do not possess a given property defining a class. Note that the last term has an indefinite remainder 0/0 of objects where C is absent.
This inference from the initial three premises can be replicated using classical logic.
(1) Suppose A.
(2) If B is present, then C or D is present, but not both. (First premise; 1)
(3) Suppose B and C are present.
(4) Then D is not present. (2, 3)
(5) Then A is not present. (Second premise, 3+4)
(6) (5) contradicts (1), so (3) is false if (1) is true; i.e., B and C are not both present.
(7) Suppose B is present and C is not present.
(8) Then D is present (2, 7)
(9) Suppose C is present and B is not present. No inferences can be made.
(10) Suppose neither B nor C are present.
(11) If A is present, then D is also present. (Third Premise, 10)
(12) D is present. (1, 11)
Statements (7)-(10) and (12) define all the possibilities for when A is present. B and D are present and C is not present,
(7-8) OR C is present and B is not present
(9) OR D is present and B and C are not present.
(10, 12) Boole’s conclusion gives us only the second of these three parts plus an indefinite remainder of things that are not C. Classical logic gives us more specific information about this indefinite remainder: B and D are present, and C is not present,
OR D is present, and B and C are not present.
Granted, this information could be easily inferred from Boole’s conclusion, but again we have no evidence of his logic’s inferential superiority.
The second method of reduction also begins with equations V1 = 0, V2 = 0. Both methods are generalizable to arbitrarily many equations. In the case of the second method, less tedious than the first, this ability to deal with multiple equations argues for the computational superiority of Boole’s formalism. These equations can be added to form a single equation:
V1 + V2 + … = 0
For each constituent t, its coefficient in each Vi can be added: (A + B + …)t. If the coefficients A, B, etc. are assumed to be all non-negative, then the combined coefficient vanishes only if all the individual coefficients vanish. If we allow negative coefficients, we can render them positive by taking the squares of the logical equations and summing them, exploiting the law of duality:
V12 + V22 + … = 0
Developing this function, then using the law of duality and the mutual exclusivity of constituents, i.e. titj = 0 for i ≠ j, it can be shown that this equation is equivalent to V1 = 0, V2 = 0, etc.
The method of squaring is sufficiently robust that it may be applied even to equations that do not obey the law of duality, i.e., equations involving the indefinite class variable v, interpretable as some objects
(inclusive of all
, but not none
). Recalling that All X are Y
is expressed by Boole by making X the intersection of Y and some indefinite class, it follows that X = Y is equivalent to the two equations X = vY and Y = vX. Eliminating v gives:
X(1 − Y) = 0
Y(1 − X) = 0
Likewise, vX = vY is equivalent to;
X(1 − Y) = 0
Y(1 − X) = 0
Thus all valid equations can be converted into equations having the form V = 0 and obeying the law of duality.
X = vY ⟷ X(1 − Y)= 0
X = Y ⟷ X(1 − Y) + Y(1 − X) = 0
vX = vY ⟷ vX(1 − Y) + vY(1 − X) = 0
X =1 ⟷ 1 − X = 0
Secondary propositions make judgments about the truth or falsity of primary propositions X, Y, etc. As mentioned earlier, secondary propositions use the same calculus as primary propositions. They are analogous insofar as we may define x as a set of all cases where some proposition is true.
Interestingly, Boole interprets such classes of cases as taking pieces of time whenever
a certain proposition is true or false. Thus x + y means the time when X is true aggregated with the time Y is true. Again + is interpreted to mean a disjoint union or exclusive OR.
x − y means the time X is true, excluding the portion of time where Y is true, i.e., X is true and Y is not true. x = y means the time X is true is identical with the time Y is true, so X and Y are logically equivalent. Note that the identity in time does not imply a strict identity between X and Y, only mutual material implication. It seems unnecessary for Boole to insist on time as the thing
that is subdivided into classes x, y, etc. He might have been just as well served by a conceptual space or range of possibilities or scenarios, except we must recall he has limited logic to dealing with actual existents.
In his original Mathematical Analysis of Logic, Boole did use cases
or conjunctures of circumstances
instead of time when interpreting secondary propositions, but found that he could not define what is meant by a case.
He now declares that whatever is involved in the term [
case
or conjuncture of circumstances
] beyond the notion of time is alien to the objects, and restrictive of the processes, of formal Logic.
Recall that Boole limits the objects of primary propositions to existent objects. He did not, however, model primary propositions on divisions in space (though we may visualize it that way using Venn diagrams), since there is nothing prohibiting two objects from occupying the same space. It is strange, then, that he should model secondary propositions, whose objects are judgments, on divisions of time. This is workable only on the assumption that judgments are successively true or false at different points in time, and in no other respect can be distinguished. This requires him to posit an ontological category, namely time, prior to his logic, though he was reluctant to do so in the case of space. Alternatively, x, y, z may be taken as denoting events that make the propositions X, Y, Z true. This may be a preferable interpretation, in light of the relativity of simultaneity.
It is easy to see that the laws of Boole’s algebra can be applied to secondary propositions. xy = yx holds.
For whether we select mentally, first that portion of time for which the proposition Y is true, then out of the result that contained portion for which X is true; or first, that portion of time for which the proposition X is true, then out of the result that contained portion of it for which the proposition Y is true; we shall arrive at the same final result, viz., that portion of time for which the propositions X and Y are both true.
Likewise, one can see that x(y + z) = xy + xz holds. X is true and (either Y is true or Z is true)
is equivalent to Either (X is true and Y is true) or (X is true and Z is true).
More importantly, we have the law of duality x2 = x, i.e., X is true and X is true
whenever X is true, and the law of non-contradiction x(1 − x) = 0, i.e., there is no time that X is true and X is not true. In this interpretation 0 represents never,
i.e., no time, and 1 represents always,
i.e., all time. Always
is defined with respect to the universe of discourse. By default, it is eternity, though this would be further limited if it were known that the actual universe is limited in time. He offers the defense that, without any change in our faculty of reasoning, space might have been other what we perceive it to be, but this is not so for time.
Boole does not consider it demonstrable that space and time are forms of the human understanding
(alluding to Kant). He does believe to have proven that the formal processes of reason do not require objects to be manifested in space; they would remain appplicable, with equal strictness of demonstration, to forms of existence, if such there be, which lie beyond the realm of sensible extension.
Thus space is not essential to the analysis of primary propositions, but no similar proof can be made regarding time. Indeed, he is correct to suggest that logic at least in some respects supposes time, for the law of contradiction is not violated if a proposition is true and false at different times. Yet propositions may also be considered atemporally, as with abstract statements: The sum of the angles of a triangle equals two right angles.
This statement is true when the triangle is in a plane, but not when it is in a curved manifold. The truth and falsity of this statement need not be at different times, for an example of each kind of triangle might exist at the same time, or we may consider triangles abstracted from all time, since no ideal triangles, with perfectly straight sides, etc., exist in concrete reality.
By modeling secondary propositions with divisions of time, Boole forces us to think in terms of extension. This is appropriate for quantificational propositions, though we have noted it is perilous to confine all
to concrete existents. This is especially dubious when dealing with something as abstract as the truth or falsity of a judgment. Boole requires us to think of something as actually existent when we posit it as true, and to always consider the copula as specifying time. The = sign in equations of secondary propositions, oddly enough, does not mean that the truth-making events on both sides are identical, but only that their time of occurrence is the same. Thus = represents a mutual material implication. This is contrary to the = sign in primary propositions, which really does mean the things on both sides are identical, not merely occupying the same space.
To say that X is categorically, or always, true, one need only specify that x = 1. To say X is always false, we specify that x = 0, or 1 − x = 1, i.e., the negation of X is true. This use of 1 and 0 to mean true and false is what most people today recognize as Boolean logic, and the following expressions will be familiar if translated into modern operators such as AND, OR, and XOR.
x(1 − y) + y(1 − x) = 1 [Either X is true or Y is true; i.e. X XOR Y]
xy + x(1 − y) + y(1 − x) = 1 [X is true or Y is true or both are truee; i.e. X OR Y]
x + (1 − x)y = 1 [X is true, or X is not true and Y is true; equivalent to X OR Y, above.]
For if-then statements, we use: y = vx. If Y is true, then X is true, but the converse need not be true. The case-based or set theoretic interpretation of the intersection v and x makes this similar to material implication. We are only saying that whenever Y is true, X is also true, without saying anything about formal or efficient causality. The indefinite class v here signifies some indefinite set of times, so vx means an indefinite portion (some or all) of the times that x is true.
If an equation has some terms with v and other terms without v, the direction of implication flows only from terms with coefficient 1, so no implication flows from the v terms (with indefinite values 0/0). For example,
y = xz + vx(1 − z)
If Y is true, then either X and Z are both true
or X is true and Z is false.
For the reverse implication, however, we ignore the v term, which implies nothing, and have only: If X and Z are both true, then Y is true. Only the terms with coefficient 1 on one side of the equation may be taken as the antecedent, while all terms on the other side of the equation are the consequent. This ability to interpret equations as if-then statements lets us drop any explicit reference to time. Nonetheless, the materialist interpretation underlies the interpretation of what if-then means, which is material implication, somewhat more limited than the classical significance of antecedent
and consequent.
We need not hold that the fact underlying the truth of X and Z is a consequence of the fact underlying the truth of Y. We might hold that our knowledge of the truth of X and Z is a consequence of knowing the truth of Y, but even this is derivative of the core interpretive tenet that X and Z is true whenever Y is true. We make no judgment as to why this is so; it might be happenstance, or there might be some intrinsic logical necessity (e.g., due to subordination of X and Z to Y). Since Boole’s logic is restricted to what actually occurs, happenstance implications cannot be ruled out, especially when dealing with sets of only one or a few objects.
To solve systems of secondary propositions, Boole derives the following rule using his calculus. First, eliminate v from each equation containing it. Then:
Eliminate the remaining symbols which it is desired to banish from the final solution: always before elimination reducing to a single equation those equations in which the symbol or symbols to be eliminated are found. Collect the resulting equations into a single equation V = 0.
Then proceed according to the particular form in which it is desired to express the final relation, as–
1st. If in the form of a denial, or system of denials, develop the function V, and equate to 0 all those constituents whose coefficients do not vanish.
2ndly. If in the form of a disjunctive proposition, equate to 1 the sum of those constituents whose coefficients vanish.
3rdly. If in the form of a conditional proposition having a simple element, as x or 1 − x, for its antecedent, determine the algebraic expression of that element, and develop that expression.
4thly. If in the form of a conditional proposition having a compound expression, as xy, xy + (1 − x)(1 − y), &c., for its antecedent, equate that expression to a new symbol t, and determine t as a developed function of the symbols which are to appear in the consequent…
5thly. Interpret the results by [rule discussed above for terms with coefficients of 1 and other terms].
If it only be desired to ascertain whether a particular elementary proposition x is true or false, we must eliminate all the symbols but x; then the equation x = 1 will indicate that the proposition is true, x = 0 that it is false, 0 = 0 that the premises are insufficient to determine whether it is true or false.
Boole applies his method to some multi-premise arguments, yielding the correct conclusions. He admits that as an exhibition of the power of the method, the above examples are of slight value,
but the same principles would apply to more complicated arguments. Yet even for a six-premise argument, he omits the preliminary steps of eliminating v and developing the result, for developing even a three-variable formula is tedious, requiring the evaluation of eight terms, though most of these vanish. Thus his calculus involves many more steps than what we could deduce for ourselves quie easily without it. While it is true that, in principle, his calculus should be able to make deductions from any number of premises with any number of variables, the number of steps increases with 2n, where n the number of variables in a given premise. On the other hand, it is only to be expected that a method for exhaustively determining deductions from a set of premises must go through all logical possibilities.
Boole proceeds to illustrate his methods by treating some proofs about the existence and attributes of God by Clarke and Spinoza. Here he is concerned only with articulating their premises formally and examining whether their conclusions can be properly deduced from those premises. We should note that his calculus works even when he opines that at least one of the premises is doubtful, false, or hypothetical. This shows that his confinement of logic to existent objects is by no means necessary, and his logic of secondary propositions may be applied even to abstract subjects. In the course of developing and reducing Clarke’s theses, he deduces two conclusions that are at variance with each other when interpreted in ordinary language, though not in the sense of formal logical contradiction. This is remedied by asserting an implicit premise, and it is a credit to the clarity of Boole’s system that such unstated premises can be identified.
Boole is less impressed with Spinoza’s arguments, finding that there are many implied equivalences of terms, though it is not clear that he has always parsed Spinoza correctly. We should note that Boole’s method is still sound in principle, even for analyzing abstruse metaphysical arguments, insofar as these arguments rely on deductions from premises that relate concepts by the operations of conjunction (AND and OR), negation, and material implication. It is not necessarily equipped for analyzing the premises themselves, for this depends on a comprehension of the concepts they contain. Boole escapes the need to know the category of each term, insofar as Spinoza’s argument depends on the classification of things or substances that are in se, etc. Yet his identification of implied premises or inconsistent propositions depends on whether he has interpreted Spinoza’s terms correctly. This is no discredit to Boole’s system, since any calculus of logic, if it is to represent any intelligible argument, must begin with an understanding of the terms, so we know which classes should be represented as mutually exclusive, etc. Strikingly, much of Boole’s critique of Spinoza makes little or no use of his calculus.[4]
Boole regards the Aristotelian and Scholastic systems of logic as attempts to classify all allowable forms of inference and refer them to some axiomatic principle. They deal with two kinds of propositions: the categorical, and the hypothetical or conditional. Oddly, Boole finds this distinction to be nearly identical with that of primary and secondary propositions in the present work.
Most logical treatises devoted far more attention to categorical propositions, which can be analyzed by conversion
(reversing order of terms, while maintaining logical equivalence) and syllogism. Boole will show where conversion and syllogism fall in his own system, and thus elucidate the true significance of these processes.
The categorical propositions are distinguished by the universal (all
) or particular (some
) quantifiers, and the application of the affirmative or negative regarding the presence of each term. It would be more proper to speak of positive
and privative
rather than affirmative
and negative,
Boole admits, since we apply these modifiers to the terms, not the copula, but he has already used positive
and privative
elsewhere in a different sense. No Y's are X's,
really means All Y's are not-X's,
so No Y
does not constitute a third kind of quantified term. Boole’s schema requires that we be allowed to treat not-X and not-Y as terms, rather than negate the proposition as a whole, so he necessarily invokes De Morgan’s amplification of categorical propositions to eight kinds.
y = vx
y = v(1 − x)
vy = vx
vy = v(1 − x)
1 − y = vx
1 − y = v(1 − x)
v(1 − y) = vx
v(1 − y) = v(1 − x)
For particular propositions, i.e., those using the quantifier some,
conversion is a matter of simple transposition. vy = vx becomes vx = vy, which is to say that Some Y's are X's
is equivalent to Some X's are Y's.
This assumes that X and Y are of the same ontological category. Other conversions are handled by eliminating v and developing the formula in terms of the new subject variable. Thus y = v(1 − x) becomes x = (0/0)(1 − y, which is to say, All Y's are not-X's
is equivalent to All X's are not-Y's.
Both statements might be expressed more succinctly as xy = 0.
Conversion, then, is but a particular application of a more general process that can be applied to propositions of innumerable terms, namely to eliminate variables and develop with respect to whichever variable we wish to make the subject. Even with two-term propositions, the calculus can yield deductions other than what are conventionally recognized as conversions. While these additional conclusions are merely equivalent forms of the same proposition, we can hardly expect otherwise when deducing from a single premise.
Boole regards the processes of elimination and development to be more fundamental than classical conversion. To declare simply that All Y's are X's
is equivalent to All not-X's are not-Y's
(or No not-X's are Y's
) conceals the more elementary manipulations of logical symbols that yield this result. Boole does not consider his calculus a mere useful computational device. Grounded in the laws of thought, his processes of elimination and development, with all their oddity and intermediate uninterpretability, are the real basis of the equivalence of converted propositions.
Syllogism, similarly, is a particular kind of reduction of systems of propositions, dealing with a set of two two-term propositions with a common middle
term. Boole’s method of reduction can handle arbitrarily many propositions with arbitrarily many terms. As we have noted, however, multi-term propositions can be expressed as a set of two-term propositions, so, in principle, all the deductions available to Boole’s calculus should be attainable syllogistically. Nonetheless, the method of reduction has the distinct advantage of being a uniform process that can be applied to any system of propositions, without having to memorize (or rediscover) the various rules of syllogisms. This process, to be sure, can be quite tedious, but its uniformity lends credence to Boole’s claim that he has expressed something fundamental about inference in general.
All syllogisms in general may be expressed in one of two forms. First, suppose the middle term is affirmed in both premises or denied in both premises, so that it can be represented as y in both.
vx = v'y
wz = w'y
The quantifier variables v, v', w, w' may take the values of 1 or an indefinite class, to represent the various universal or particular categorical propositions. The general solution of syllogisms of this form requires that we eliminate y and develop the resulting formula with respect to x, 1 − x and vx. In all three cases, we get a long formula consisting of a z term and a 1 − z term, with coefficients expressed as functions of the the quantifier variables and their negatives. All these coefficients have at least one indefinite remainder
0/0 term. To give a conclusion only in terms of z or 1 − z rather than both, we must impose conditions that make one of the coefficients vanish. This leads to the following rules of inference.
If at least one of the extremes (x or z) has a universal quantifier (v = 1 or w = 1), then change the quantity and quality of that extreme, and equate the result to the other extreme.
x = v'y
wz = w'y
Conclusion: v(1 − x) = wz
vx = v'y
z = w'y
Conclusion: w(1 −z) = vx
If both middle terms have universal quantifiers (v', w' = 1), then change the quantity and quality of either extreme, and equate the result to the other extreme unchanged.
vx = y
wz = y
Conclusion 1: 1 − x = wz
Conclusion 2: 1 − z = vx
In the second form of syllogism, the middle terms are unlike in quality, i.e. y in one premise and 1 − y in the other. The same rules as the first form apply to the second form.
To show that his method is more general than that of syllogism, Boole remarks that the variables v, v', w, w' could represent any definite class property whatsoever, rather than indefinite classes or 1 (quantifiers), and the inferences would still hold. Indeed, when all variables are considered as representing definite classes, the calculus is no more complex than necessary.
Syllogisms only solve systems of equations of a particular form, those with a common middle term. Boole notes that not all sets of logical equations need have this form. Yet, as we have noted, in order to make truly constructive inferences, it is necessary to have at least two premises. For these to be combined, they must have at least one variable in common. We cannot make any inference from the premises x = y, z = w, or more complicated expressions of dissimilar variables. Boole correctly notes that syllogism is a species of elimination. He then considers the questions (1) whether all elimination is reducible to syllogism, and (2) whether deductive reasoning consists only of elimination.
I believe, upon careful examination, the true answer to the former question to be, that it is always theoretically possible so to resolve and combine propositions that elimination may subsequently be effected by the syllogistic canons, but that the process of reduction would in many instances be constrained and unnatural, and would involve operations which are not syllogistic.
So there are no inferences that Boole’s logic can make that syllogistic logic cannot, but Boole’s logic can work with propositions expressed in a broader variety of forms.
As to second question, he says it is arbitrary to confine reasoning to elimination; it encompasses all methods founded on laws of thought. Thus he would include development of single equations as true deductive reasonings, regardless of their triviality.
Boole recognizes that the older logic had simpler principles, but a truth is not more fundamental on account of its simpliicity of expression, but by the nature and extent of the structure which they are capable of supporting.
Again, he appeals to the more general applicability of his method as evidence that it is more truly grounded in the fundamental laws of thought.
Hypothetical syllogisms, involving if-then propositions, are treated more irregularly in classical logic. Boole notes that at least one form of hypothetical syllogism, effectively a modus ponens, is not truly syllogistic.
If A is B, C is D.
But A is B,
Therefore C is D.
If we express this in terms of secondary propositions, we can see that there are only two terms rather than the three required for syllogism.
X is true, Y is true,
But X is true,
Therefore Y is true.
A true hypothetical syllogism should have a form such as:
If X is true, Y is true;
If Y is true, Z is true;
If X is true, Z is true.
Given that secondary propositions obey the same calculus as primary propositions, we may add quantifiers sometimes
or always
to the conditions, and come up with similar rules of inference as for categorical syllogisms.
A primary objective of Boole’s work is to develop a logic of probabilities. As we have mentioned, the calculus lends itself readily to a probabilistic interpretation, as the sum of all disjoint sets, i.e., the classes represented by logical variables, is defined to be 1. Thus we may interpret each set as a class of possible events, and following Laplace, the probability would be the size of that set as a fraction of the universe of possible events (1).
Probability statements are expressed as propositions about the occurrence or non-occurrence of events. We might consider this a primary proposition insofar as we are speaking of the presence or absence of an event (considering event as a thing), or as a secondary proposition insofar as we consider it to be whether it is true or false that the event occurs (considering event as occurrence or action, expressed by the copula). The syntactic equivalence of primary and secondary propositions holds for modal or probabilistic statements just as it did for categorical statements.
x = 1 means the event x occurs. x = 0 or 1 − x = 1 means x does not occur, so 1 − x can be interpreted as the negative or non-occurrence of x. We can join events with the standard connectives.
x(1 − y) + y(1 − x) = 1
This means, Either x occurs or y occurs (but not both).
Multiplication signifies a conjunction of events, but even negatives or non-occurrences can be subjects of such conjunction, as in the example above, which identifies the conjoint occurrence of x and non-occurence of y as the first possibility or term. If x and y could both occur together, we would add xy as a third term to the equation. Addition remains purely disjunctive, so the terms must be mutually exclusive possibilities, expressible as unique combinations of occurrence or non-occurrence of the events x, y, z, identical in form to the constituents
of logical equations. Since each term represents a non-overlapping possibility, the sum of the probabilities for all terms equals 1. In the logical equation above, 1
signifiies the universe of possible events, so its probability is also 1, as required since both sides of the equation represent the same class of events.
To represent the numerical probabilities of events, we need a parallel set of numerical equations with variables for these probabilities. If the probability of x occurring is p, then the probability of its non-occurrence is 1 − p. If the probability of y is q, and x and y are independent, then the probability of their conjoint occurrence is pq. If the logical equation above is assumed, so that either x or y must occur, but both cannot occur, we have the numerical equations:
p(1 − q) + q(1 − p) = 1
p − pq + q − qp = 1
p + q = 1
The probabilities p and q add to 1, exactly as we should expect if either x or y must occur. If we allowed both to occur, we would have p + q + pq = 1
Our ability to express formulas in terms of logical variables (x, y, z) or numerical probabilities (p, q, r) runs parallel to the logical and numerical interpretations of probability theory. The logical relations denote the various mutually exclusive combinations of events, while the numerical relations add or multiply probabilities for disjoint or conjoint possible events. Note that Boole’s choice of disjoint union for addition is especially apt when applied to probability theory. It is only because logical terms joined by + represent mutually exclusive possibilities that we are justified in having this same + represent addition when applied to the corresponding numerical probabilities.
Conditional probability, i.e., the probability of Y given X, is presented in the usual Laplacian way: P(XY)/P(X), i.e., the probability of XY divided by the probability of X. This equals P(Y) only if X and Y are independent. X and Y may each represent complex formulas for compound events, and the probability relation will still hold. The requirement of independence of the variables x, y is a necessary assumption in probability theory for relations such as P(xy) = P(x)P(y) to hold. If the occurrence of x or y affects the probability of the other, then we can only say that P(xy) = P(xP(y|x), the multiplicand being the probability of y given x. This is just a manipulation of the formula for conditional probability.
The calculus of logic can deal with dependent events no less than independent events. Perhaps we know P(x) = p and P(xy) = q, but do not know the dependence relation of x and y. We cannot simply assume that P(y) = q/p. Instead we treat xy as a compound event u, and develop the logical equation xy = u with respect to y. We can do this, in fact, with any logical equation, no matter how complex. In general, we would get an equation of the form:
w = A + 0B + (0/0)C + (1/0)D
The terms are grouped by coefficients 0, 1, 0/0, 1/0. A, B, C and D each consist of the sums of various constituents of the logical variables s, t, …, each representing some compound event. The same constituent does not appear twice in this equation. As always, we can discard the uninterpretable 1/0 terms, so the universe of possible events is spanned by A, B, C. Let A + B + C = V; let Vs represent the sum of constituents in V that contain s (rather than 1 − s), and so on for other logical variables. Thus:
Vs/P(s) = Vt/P(t) = V
Since the B terms are of probability 0, we only need to consider the A and C terms as contributing to the numerical probability, but the indefinite 0/0 is here taken as an unknown constant c.
w = (A + cC)/V
V = (A + cC)/P(w)
This last equation may be set equal to the above equations for Vs, etc., forming a system of equations to solve. Where c appears, it is interpreted as the probability of w given C: P(Cw)/P(C).
Returning to the simple equation xy = u, where x and y are dependent events, this method yields:
P(y) = q + c(1 − p)
where p = P(x) and q = P(xy). c is interpreted as P((1 − xy)(1 − x)y)/P((1 − xy)(1 − x)). Multiplying the binomials and eliminating terms with the law of duality, this simplifies to:
c = P((1 − x)y)/P(1 − x)
Thus c is the (unknown) probability that y occurs, given that x does not occur. Substituting this expression for c into the original equation yields:
P(y) = P(xy + P((1 − x)y)
There is nothing profound about this form of the equation, as we have simply subdivided the probability of y into its joint concurrence with x and not-x. Boole’s method is more useful when the equation is expressed in terms of c, which is treated as an unknown parameter. This led Boole to believe that all probabilities must be expressed in terms of conditional probabilities, a thesis we have supported elsewhere on philosophical grounds.
The same method may be applied to hypothetical syllogisms, so we determine the probability of the conclusion from the probabilities of the premises being true. A premise of a hypothetical syllogism is of the form: If Y is true, then X is true.
By assigning a fractional probability p to the truth of that statement, Boole means that there is a probability 1 − p that X is false when Y is true. Again, general solutions are expressed in terms of an arbitrary constant representing an unknown conditional probability.
Probabilities of truth may be applied even to premises of categorical syllogisms, no less than hypothetical syllogisms, but we must note that certain propositions cease to be equivalent when they are regarded as probable rather than certain. To illustrate, Boole considers these secondary propositions as premises:
(1) Either Y is true, or both X and Y are false.
(2) If X is true, then Y is true.
Ordinarily, these statements would be considered logically equivalent. (1) is true if (2) is true and vice versa, and the same for their falsity. However, when the truth of (1) is allowed to have a fractional probability p, this logical equivalence breaks down, as we shall see. This indicates that probabilistic treatment of propositions, where we subdivide the universe into events where one or another proposition is true or false, is not generally the same as when we analyze binary cases of truth and falsity non-probabilistically. That is to say, probability space is not the same as that which is spanned by the whenever
considered above.
Let x and y be the events that render X and Y true. Then the primary propositions corresponding to the secondary propositions above are:
(1) y + (1 − x)(1 − y) = 1
(2) y = vx
Suppose the probability of the truth of (1) is p.
P[y + (1 − x))(1 − y)] = p
Then we wish to find the probability of the truth of (2). Since (2) is a conditional statement, what we want is the probability of y occurring given that x occurs. This is just the conditional probability:
P(xy)/P(x)
Using the techniques of elimination and reduction of probabilistic equations, Boole derives P(xy) = 1 - p + cp and P (xy) = cp, where c is P(x|(y + (1 − x)(1 − y)). Thus the probability of (2) is:
cp/(1 − p + cp)
The conditional probability c, which reduces to P(xy)/P(y + (1 − x)(1 − y)), is unspecified or indefinite, so the probability of (2) is indefinite, and not generally equal to p. How is it that two logically equivalent statements can have different probabilities of truth?
This paradox arises because one of the propositions, namely (2), is a conditional statement, and thus modally different from the other when we are considering statements probabilistically. The logical equivalence of (1) and (2) exists only when we suppose one of these statements to be true or false, i.e., that P(1) = p = 1 or 0. If we do that, then P(2) will have the same value. In the present problem, however, we are making no supposition as to the value of p, and we recall that all computed probability values are conditioned by our state of knowledge. Knowing the probability of the disjunctive statement (1) being true tells us nothing about the relative probabilities of the constitutents that can make it true, and the absence of this knowledge introduces an uncertainty in the probability of (2) that is distinct from that of (1). The difference in internal structure between (1) and (2) now becomes relevant, as we are exploring the degree of correlation between parts of each secondary proposition.
If we were to solve the reverse problem, where the probability of (2) is some known quantity q, P(1) would be 1 − c' + c'q, where c' is the unspecified P(x). This result is somewhat simpler in form, due to the simpler form of (2). Again, if we set P(2) = q = 1 or 0, then P(1) will have the same value.
These unequal and asymmetric results should caution us that logical equivalence of secondary propositions does not entail identity. Even the equivalence can break down when the statements are considered probabilistically. Although one might intuit that the probabilities of logically equivalent statements must be equal, since one can not be true without the other being true, we must recall that all probability is conditioned by knowledge, so there is no such thing as P(1) or P(2) simply and absolutely. When we try to compute the probability of one proposition in terms of the probability of another, we are taking the latter proposition’s probability as a given or datum, while the other is a conditioned quaesitum. Thus we should not in general expect any symmetry between probabilities.
The probabilistic calculus may be applied to complex problems in causality. Unlike Laplace, who required causes to be mutually exclusive, Boole allows for different causes to be compatible contributing factors. He also moves beyond the determinist paradigm, where a cause invariably produces its effect, and allows that a cause may produce its effect only some of the time. Moreover, he abstracts from the question of whether an effect is truly consequent to its cause or if it merely coincides with it. Thus Boole’s causality in the most generic sense is if-then material implication. He formulates the question of two compatible imperfect causes in what has come to be known as his Challenge Problem:
The probabilities of two causes A1 and A2 are c1 and c2 respectively. The probability that if the cause A1 presents itself, an event E will accompany it (whether as a consequence of the cause A1 or not) is p1, and the probability of that if the cause A2 presents itself, that the event E will accompany it, whether as a consequence of it or not, is p2. Moreover, the event E cannot appear in the absence of both the causes A1 and A2.
If the two causes are mutually exclusive, the probability of E is simply c1p1 + c2p2. If they are not necessarily exclusive, then the calculus can be employed to derive a more general solution in terms of these quantities. We know that the probability of E must in general be greater than its concurrence with either cause, i.e., greater than c1p1 and c2p2, since in general we allow that neither cause alone is necessary for E to occur. It also must be less than or equal to the sum of its conjoint occurrence with either cause, i.e. c1p1 + c2p2, if by assumption the event cannot occur without either of the two causes. The general solution of P(E) involves a cubic equation, but only one of the three roots falls within these limits. In the simple case where p1 = p2 = 1, the solution for P(E) reduces to: 1 − (1 − c1)(1 − c2). This is the case where a cause invariably produces its effect, i.e., determinism.
Boole’s probabilistic calculus is highly fruitful, and is the one aspect of his method that yields results not attainable by standard approaches. It allows for a broadened notion of causality, where a cause may sometimes fail to produce its effect, and can deal with problems where there is no numerical solution except in terms of intervals, as in more complicated variants of the Challenge Problem. Keynes later analyzed and implemented Boole’s approach in order to show that probabilities in general do not admit of Archimedean ordering, because of unknowns. This would render impossible so-called rational choice
theory insofar as it supposes that a probabilistically optimal choice is univocally identifiable in a given knowledge state.
A fuller treatment of Boole’s calculus was given by T. Hailperin (1986), who showed the correctness of its results, especially in probability theory.[5] Boole did not always rigorously demonstrate that his use of algebra was valid when applied to logical equations, much less that this was derivable from his fundamental laws of thought. The fact that his calculus works, replicating the results of Aristotelian syllogisms and Laplacian probability theory, indicates that it does indeed reflect some genuine truths of quantificational logic. Although set theory had not yet been formalized, Boole’s object-only ontology lends itself readily to set theoretic interpretation. His method of developing functions, where each constituent represents a possible logical combination of truth or falsity for each variable, is a natural choice for subdividing the universe of discourse. The broad applicability of this calculus to metaphysical and probabilistic arguments, abstracting from the truth of propositions, indicates that this universe can be much larger than that of actually existent objects.
[1] In Boole’s original 1847 work, Boole used selection
operators to represent the mental act of selecting a class of objects on the basis of some shared property. His 1854 calculus dispensed with selection operators and simply dealt with classes directly.
[2] This idempotent law or law of duality,
which Boole found to be of profound importance, had been discovered independently by Leibniz. The early discoveries by Leibniz and Jacob Bernoulli (1685) of parallels between algebra and logic had no direct influence, however, on Boole or other logicians of his time.
[3] Giovanna Corsi, Guido Gherardi. The basis of Boole’s logical theory.
arXiv:1710.01542, 4 Oct 2017.
[4] Boole generally finds that Spinoza equivocates or is unclear in his definitions and axioms. He has much higher esteem for the logical clarity of Clarke, but finds some of his premises to be unjustified assertions. In particular, he finds fault with the impossibility of infinite succession,
as applied to causation, demonstration, and subordination. Apparently the impossibility of our forming a definite and complete conception of an infinite series, i.e., of comprehending it as a whole, has been confounded with a logical inconsistency, or contradiction in the idea itself.
This is an unfair accusation, as should be clear from the fact that metaphysicians from Aristotle onward have admitted and treated infinite series in other contexts, but find infinite regression to be impossible when applied to contexts where things posterior in sequence are absolutely dependent on or derived from the existence of the prior. Boole considers all a priori proofs of God to be futile, though he seems to confound purely ontological arguments with a posteriori metaphysical proofs (relying on insights into existence, causality, etc.). He sees more promise in the ancient approach of probable induction from evidences of design in the world.
[5] Theodore Hailperin. Boole’s Logic and Probability, 2nd ed., Amsterdam, North-Holland, 1986.
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