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The Incompleteness of Formal Logic

A Philosophical Criticism of Symbolic Calculus

Daniel J. Castellano (2009-12)

II. The Semantics of Logic

Signs
Signification and Reference
Formalization
Logic, Logistic and Calculus

Much like ordinary language, formal logic deals in symbols. Many of these symbols may seem foreign to the layman, either on their own or in combination with other signs. This lack of familiarity can obscure the reality that formal logic is just another language, though it pretends to be otherwise. An important part of our endeavors will be to strip away the intimidating notation of formal logic, and arrive at the meaning behind the symbols, which we can determine by examining how symbolic logic is actually used. We will find that insofar as logic is used to do real proofs, it will not suffice to ignore the meaning of its symbols, as though reasoning were a purely mechanical process. If we are to reason truly, and not just in name, we must have understanding, and this involves the apprehension of meaning behind a symbol or sign.

We have already discussed the relationship between language and logic, or syntax and meaning, at length in a previous work. In the present discussion, which is directed toward modern logic, we will make use of modern linguistic concepts to give our examination more precision, and to be consistent with the thought expressed by some modern logicians. These concepts are: syntax, semantics, pragmatics, reference, and meaning, which we shall define presently.

Syntax is the relationship among symbols. When dealing with written language, this means the sequence in which symbols are written, generally from left to right, as well as other positioning, such as superscripts, subscripts, etc. It may also refer to more complex relationships, such as nesting within parentheses, numerators and denominators, and other delimiting punctuation, but these are often reducible to a simple horizontal sequence of symbols, as is the case with Polish notation. We are concerned with the positioning or sequence of symbols only insofar as they involve a change in how we interpret symbols, alone or in combination. In the most general sense, however, syntax involves even those sequential relationships among symbols that do not involve a change in meaning.

Semantics refers to the relationship of language to things other than symbols. This is meaning (semantikos) in the broadest sense. If symbols signify something, semantics relates the symbols to the thing signified. The thing signified may be some ontological entity, or at least the conceptualization of that entity. Certain types of linguistic objects, such as sentences, do not merely represent ontological entities, but express judgments about them, as we discussed at length in a previous work. If we are to speak of the semantics of a language as a whole, we must give an account of judgments, which have an intellectual dimension. Semantics takes us beyond a purely mechanical parsing of symbols, and requires us to apprehend real ideas and judgments behind these symbols. Semantics, contrary to its name, does not directly refer to the meanings or ideas themselves, but rather the relation of each meaning to a symbol. When we “argue over semantics,” we often merely dispute whether a word ought to be assigned one meaning or another. To the extent that our choice of linguistic symbol is arbitrary, such disputes are idle quibbles. What is not a quibble, however, is the notion that it makes a big difference what we intend by a given word. A word is not a word at all if it does not signify anything, so semantics is linked by necessity to symbology and syntax.

Different arrangements of symbols may be understood to give the same symbols a different collective meaning. The rules of syntax in a given language often correspond to rules of semantics, but this is not always the case. Some languages, for example, permit adjectives to come before or after the modified noun, without any change in meaning. Thus some changes in syntax do not cause changes in meaning. We are limited seemingly only by our imagination as to how many different syntaxes we can construct, but the relationships among meanings are limited by logical coherence. If we are to construct a logical syntax, it ideally ought to be able to express any logically coherent thought, and at the very least, it must not permit incoherent expressions. The construction of a language without incoherence is a formidable challenge, if it is even possible; in practice we content ourselves with languages that make incoherent expressions readily identifiable. “Green dreams sleep loudly,” is syntactically correct, though logically incoherent, but our understanding of each term and the significance of their syntactic relationships enables us to identify the expression as incoherent. Expressions that do not even have the syntactic form of a judgment, such as “yellow Socrates sweetly” are even more evidently incoherent, since there is not even a pseudo-judgment to scrutinize.

Peirce, Dewey and Heidegger also identified what they called pragmatics, which means the relations of symbols to speakers, listeners, and social contexts. In our discussion of non-indicative moods, we found special emphasis on the relation to the will of the listener and the speaker. A question is inconceivable unless we allow the possibility of conversation. The interactivity that is implicit in language seems to bring human psychology into the mix, but we need only assume that there are beings that understand, and in some cases, will. We do not need to assume anything more about the psychology of the individual. It is possible, of course, that the limitations of our psychology blind us to the possibility of other non-indicative moods that might exist, but we can hardly deny those modes of judgment we do express. Against a fact there is no argument.

Some thinkers distinguish “reference” from “meaning.” Reference is the denotation or extension corresponding to some linguistic object, while meaning is the connotation or intension. This distinction attempts to separate logic from psychology, i.e., the “objective” reference on the one hand, with the subjective “meaning” on the other. Such a strong division is impossible, since logic is about the structure of reality as it can be thought, and is an intrinsically intellectual activity. Some important thinkers, nonetheless, have insisted on this distinction, drawing from it some unfortunate consequences. Wittgenstein acknowledged a distinction between syntax and semantics, but did not allow that we could talk about semantics, the relationship between a symbol and its object. In his view, there could only be statements of fact and tautological statements about syntax. Since statements about semantics fit neither category, these are unacceptable. As Nicholas Capaldi points out (The Enlightenment Project in the Analytic Conversation, p. 195), this elimination of semantic statements allowed naturalist philosophers to ignore metaphysical or epistemological questions that would undermine their naive attempt to reduce everything to natural science (as if this contained no metaphysical or epistemic assumptions). Wittgenstein here, it seems, was susceptible to that modern tendency to declare intelligible statements meaningless because he did not wish to deal with them. However persuasive may seem some abstract argument that statements about semantic relationships are impossible, we have an invincible response: it is possible, for we have done it. How’s that for empiricism? To his everlasting credit, Wittgenstein eventually rejected scientism and naturalism.

Quine rejected Wittgenstein’s notion of tautological statements as distinct from statements of fact, as this presupposed a distinction between analytic (tautologically true) statements and synthetic (derived from multiple theses). Here Quine was a worse offender than Wittgenstein, when it comes to declaring the intelligible meaningless, for the distinction between analytic and synthetic is perfectly understandable (see Kant). It is as if Quine expected some statements to have objective existence and others did not. Indeed, Quine rejected meaning (intension) and upheld reference (extension) as though the two were separable. The same man who rejected intelligible semantic statements nonetheless clung to this incoherent notion of reference without meaning. Can we really split this atom? We will ignore for the moment the evident truth that Quine’s critique itself necessarily involves statements about semantics.

Suppose we defined the reference of the term ‘Socrates’ to be the relation between the word ‘Socrates’ and the actually existing man Socrates. It would be difficult to describe such a relationship beyond the formal definition we have just given, but then many relations have no intelligible aspects other than their two subjects. In what way are the subjects related? They can only have a relationship if someone understands the word ‘Socrates’ to refer to the man Socrates. If no one understands ‘Socrates’ to refer to the man Socrates, then there would be no basis for any reference relationship, since linguistic symbols are arbitrary. Without intension, there is no reference.

While this argument shows that reference depends on meaning, the two may still have some conceptual distinction. Reference is the relation that holds between a word and its object when it is understood in a certain way. What does it mean to understand a symbol in a certain way? It means that when we see or hear the symbol, we conceive of a certain concept that we assume (by convention) the symbol is intended to convey. We only interpret such symbols as we believe were conveyed with intent. If we believed some ten leaves arranged themselves by blind chance, we would not think there was any meaning unless we supposed some providence (i.e., intent) in the occurrence. We interpret symbols that we consider motivated by the intent of the speaker, writer or software programmer. We assume that the source of the symbol follows our convention, and use this convention as a tool to help us divine what the communicator intended. Meaning is always conceptual and intentional, and when we ascribe a conceptual meaning to a word, we establish a relationship between the word and its object. How? The concept itself points to the object. In other words, the word is linked to an object via the conceptual meaning associated with the word.

The primacy of meaning is more clearly seen in the case of universal terms, such as ‘cat.’ The term ‘cat’ refers to no particular real object; indeed, it would be intelligible even if cats were extinct or mythical. Yet we can still understand the concept of “cat,” so the term has meaning. Meaning does not seem to depend on reference to an actually existing object. However, those who wish to insist that there is only objective reference, not meaning, must give an unconventional account of universal terms. They must say that ‘cat’ refers to the collection of all objects that have such-and-such feline properties. They must discard universals since they do not fit into their notion that there is only meaning and not reference. Of course, universals are perfectly intelligible, and they are the conceptual means by which we ordinarily interpret terms such as ‘cat.’ By excluding intelligible concepts, analytic philosophers are not practicing logic, but metaphysics, and building this impoverished property-object ontology into their system. As we have shown, ontology is indeed necessary to make logic intelligible, but we cannot eliminate any ontological concepts that do not involve a contradiction. When we perform such an elimination, we have crossed the line from logic to metaphysics, for we are considering how the actual world must be, rather than considering only what is a priori conceivable. The whole point of the analytic enterprise was to narrow the scope of intellectual endeavor to the natural sciences (and mathematics, the black sheep of empiricism), so metaphysical questions could be declared meaningless, or at least safely ignored. This was a wise approach, considering the naivety of the metaphysics employed in the natural sciences.

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Signs

A sign, or semeion in Greek, is that which represents something other than itself. A sign can be visible, audible, or gestural; what is essential is that it represents something other than the sensory phenomena that constitute it. Some signs are natural: for example, when we see smoke, that is a sign of fire, since by nature smoke is only present when there is fire, or at least the correlation is frequent enough that we associate smoke with fire. In the case of natural signs, one natural phenomenon is associated with another because of a natural correlation in their occurrence. No convention or definition needs to be established; the object represented by the sign is evident from a knowledge of nature.

Other signs are not natural: we may use the color red to signify a fire hazard. This association does not follow from a natural relationship, nor is it exclusive of other significations we could have chosen to give the color red. The object represented by the sign cannot be inferred from the nature of the sign alone. There is an element of arbitrary choice or convention in determining the object represented by the sign. In order for the sign to effectively represent its object, both the person employing the sign and the person receiving the sign must agree on what the sign is conventionally supposed to represent. The sign serves as a surrogate for the object, only when the users of the sign understand it to represent that object. Intellectual activity is essential to non-natural symbols, which is why they are employed only by intelligent creatures, and only humans show facility in arbitrarily defining new symbols, while other animals adopt conventions by instinct or imitation. Non-natural signs are called symbols. Symbols may be processed or manipulated by unintelligent machines such as computers, but they do not act as symbols unless an intelligent being interprets them. The computer does what it does with symbols out of physical necessity, but it is up to us to decide that the output represents an arithmetic calculation, or whatever problem we use the computer to solve.

Semiotics is the science of signs, natural and unnatural, while linguistics is that subset of semiotics that deals with language. Properly speaking, linguistics (from Latin lingua, “tongue”) ought to refer to spoken language only, but we generally apply it to written and gestural languages as well, though these are derivative of spoken language. We use it primarily to describe non-natural signs, though we might call natural signs a sort of “natural language.” For the purposes of logic, any system of non-natural signs might be considered a language. Indeed, we may prefer an unfamiliar written symbology to the ordinary spoken word, in order to impose greater rigor or precision on what is signified. However, the symbols of formal logic are still non-natural signs, and it is up to intelligent beings to assign them their objects.

In classical logic, a sign was understood primarily as a spoken word with conventional meaning, hence the medieval definition vox articulata ad significandum instituta. This kind of sign would be what we might call a linguistic symbol, but the medieval sign must have a meaning. As we discussed in Logic and Language, individual letters and syllables do not have any meaning, except to the extent there might be some onomatopoeia. Meaning can be found only in words, or implicitly in parts of words (such as roots and affixes) that can signify words, and be used to create a composite meaning in a word. These parts are called morphemes. Medieval logic did not grant morphemes the status of signs, since their significance was subsumed in that of the word as a whole. The medieval sign, then, seems to be nothing other than a word, or more precisely a term,[1] which may consist of more than one word (as in ‘great white shark’). In Scholasticism, there was no theory of reference, but signs or terms were assumed to stand as surrogates for the things they signify, without really explaining how they did that. The Scholastics did understand, however, that the sign was distinct from the thing signified, and that it represented its object only by convention.

Peirce was the first of the modern philosophers to perform an extensive study of signs, and his work was essential to the development of a symbolic logic. Peirce catalogued over 59,000 types of signs, and considered them from three perspectives. Signs could be considered of themselves, in relation to their object, and in relation to their interpreter. These three properties correspond to syntax, semantics, and properties, the three branches of semiotics mentioned previously. Indeed Peirce’s work was instrumental in the development of modern semiotics, encompassing all signs, not just linguistic symbols. This tripartite approach, applied to words, was not unknown to classical thinkers. In fact, they correspond closely to the Latin trivium of grammar, logic, and rhetoric. Grammar deals with language syntactically, while logic is concerned with argument, so the meaning of terms becomes important. Rhetoric is clearly concerned with the effect words will have on the hearer, including all the psychological dimensions of communicating. These correspondences to modern semiotics are imperfect, however, especially in the case of logic, which is concerned more with the structure of arguments than with the meaning of individual terms. Classical logic did include a concern for the meaning of terms, but modern logic has become strictly formal, treating terms as dumb placeholders or variables. There is still a place for semantics in modern logic, of course, since it uses symbols with designated meanings, and Russell would even attempt a theory of reference. The need for semiotics and pragmatics was understood by Peirce himself.

Peirce distinguished three major classifications of symbols: first, icons, which had the character of the thing they signify. This would include pictures, sculptures, and onomatopoeic utterances. These are basically natural signs, since they do not depend on any convention, but their reference is contained in their physical properties. A second class, indices, are indicative of some object, but do not depend on the presence of any interpreter. This would be the smoke that indicates fire, and other indicative natural signs. Lastly are symbols, which cease to be signs in the absence of an interpreter, since their reference is defined by some conventional interpretation. These are non-natural signs, and include all the signs of language and formal logic.

Husserl denied that purely indicative signs could express anything. Both indicative and significative functions are necessary for a sign to express something. He further observed that even those words that do have a significative function have a broader extension than their role of signifying or expressing. Nonetheless, the thing signified, or meaning, is not to be considered a species of sign or part of a sign. The thing signified is generically distinct from the sign. The signification or meaning is an element of an intentional act. It is the conceptual term toward which expressions (signs that signify) point. For Husserl, all signification was purely intentional. We may say that Husserl’s signification is what we would call meaning, while reference without meaning would be what he called the indicative function.

We see some correspondences among these models that lead us to believe they were asserting the same truths in different terms. In classical logic, material signs were distinguished from conventional signs, and only these latter were considered true signs. Peirce divided natural signs into icons and indices, while the signs of medieval logic were now called symbols. He recognized that symbols only acted as signs by virtue of interpretation, so they were necessarily conventional. Husserl observed that purely indicative signs, which Peirce called indices, did not signify or express anything. Only signs with a significative function could do this, and the thing signified was purely intentional. This explicitly added a psychological dimension to Peirce’s symbols, which was already present, as they depended on interpretation, a psychological act. We see, then, that any symbol of language or logic necessarily has an intensional meaning conventionally defined.

A strange notion has arisen in modern symbolic logic, whereby some signs are considered “primitive” while others are defined. As the discussion above should make clear, even the most primitive symbols can act as symbols only by virtue of interpretation. There is no material basis for preferring the symbol ∧ to mean “and” and the symbol ∨ to mean “or”; these are conventions. Naturally, it makes a big difference to our logic which symbol means which. Declaring such signs to be “primitive” in the sense of lacking any meaning would be fallacious; we could hardly interpret sentences using these symbols if we intended no meaning by this. Attempts to reduce these symbols to pure syntax would strip them of any truly logical function. These signs may be considered “primitive” in another sense, namely that we do not formally define them in terms of other symbols, in order to avoid an infinite regress or circular definition. This problem is not unique to symbolic logic, but is common to all languages. Not every word is understood in terms of being defined by other words. The dictionary definitions of ‘and’ and ‘or’ are not particularly helpful, as they contain far more complex terms, while even a child can grasp what is meant by ‘and’ and ‘or’. Primitive symbols lack definition not in the sense of having no meaning, but in that they are not defined using other symbols in the language.

In symbolic logic, determining which signs are primitive and which are defined is a matter of how you choose to set up your calculus. No sign is intrinsically primitive, but we may find some signs to have more intuitive, uncontroversial meanings, so that there is no need to define these formally. If we used ordinary language terms such as ‘and’ and ‘or,’ there would be no need for definition, though we may clarify whether ‘or’ is intended in an inclusive or exclusive sense. However, when we introduce foreign symbols, such as ∧ or ∨ no intuition will help us here (though they are intended to visually represent conjunction and disjunction, if we know to read from the bottom up), so we need to define them in terms of intuitive concepts of ordinary language. If we give a purely syntactic definition, then sentences using these symbols cannot express anything. If our logic is to help us understand reality, we better have intelligible concepts behind even primitive symbols.

The primitive symbols of sentential logic may include some or all of the following: the connectives ¬ (‘not’), ∨(‘or’, inclusive), ∧ (‘and’), → (material implication), and parentheses ‘(’ and ‘)’. Some of these signs can be syntactically defined in terms of the others using truth tables. These “definitions” are really just demonstrations of logical equivalence. None of these symbols are semantically empty, however. The practice of using a particle ¬ (‘not’) to express negation can lead to a misleading understanding of the relationship of negation to being, as we discussed in Logic and Language regarding the negative mood. Material implication is hardly intuitive at all, as it actually resembles conditional statements more than inference. We will scrutinize all the logical connectives in due time, to ascertain the semantic underpinnings and ontological assumptions of modern logic.

One possible system of symbolic logic would be to take the signs ¬, ∨, ‘(‘, and ‘)’ as primitive, and then to syntactically define in terms of these: ∧, →, ↔ (logical equivalence), ↓ (exclusive ‘or’), | (such that). Such syntactic definitions are misleading, as they suggest that these concepts are derivative of those signified by the primitive signs, when they are merely logically equivalent to some permutation of those signs. It is viciously circular to use such an approach to “define” logical equivalence. We will examine each syntactic definition with care, and examine if there is a basis for preferring one or another concept as semantically more primitive.

We will look with special care at the quantifiers ∀ and ∃, which hide a lot of ontological assumptions behind them. Quantificational calculus uses sentential signs (¬, ∨), predicate symbols (P, Q, R, S...) and arguments or variables (w, x, y, z...) to which the quantifiers ∀ and ∃ may be applied. The universal quantifier ∀ may be defined in terms of sentential signs; i.e., ∀x(Fx) = ¬ ∧x(¬Fx). If this is a definition (‘=’), then ∀ means nothing more than the combination of sentential symbols shown. Of course, in actual practice they do intend something more than this, when they use the symbol ∀ as a universal quantifier. We need to specify the range of ∀ (every), and this is not done by sentential logic. We cannot reduce quantificational logic to sentential logic by syntactic definitions.

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Signification and Reference

Syntactic definitions simply define symbols in terms of other symbols, so that they serve as a shorthand for some longer string of symbols. Yet no symbol can act as a symbol unless it refers to something other than itself. Eventually the symbols must tell us something about an object outside of our symbolic formalism if the formalism is to signify anything at all. It is all well and good to say that P ↔ Q refers to (P → Q) ∧ (Q → P), but if we have no idea what the second sentence signifies, neither can we understand the former. A logic that teaches us about anything other than symbols must have signification.

In order to discuss this subject clearly, we must distinguish between the reference and designation of a sign. Reference is the relation between a sign and object or fact. Sometimes the object or fact may itself be called the reference. Even natural signs (icons and indices) may have references. Symbols may also have references, yet because they are defined by an intellectual act, they also have a conceptual dimension. The word ‘cat’ is intended to direct my attention to the concept of a cat, and this concept is the designation of the symbol. The concept itself may indicate some fact or object, and the symbol refers to this object via the concept. Thus, with symbols, as J. Ferrater Mora observes, reference and designation are not independent. The designation is what we would commonly call the meaning of a word.

We often use the word ‘signification’ as a synonym for ‘meaning,’ as we have done thus far in this book. However, the term ‘signification’ has been used by philosophers in several other senses as well. The signification of some symbol X could be (per Mora): (a) the object denoted by X, (b) a psychic process in a subject (the act of thinking “X”), (c) a distinct entity altogether, (d) reducible to use of the term ‘X’. Sense (a) is simply the reference of X, (b) is what we call the intellectual act that generates the concept corresponding to ‘X’. Only (c) really accounts for meaning, since different people may think of the same concept, so the concept is not bound to any particular psychic process. Sense (d) refers to Quine’s attempt to dispense with meaning, leaving only the reference and the symbol.

Frege understood the need to distinguish between reference and meaning, yet he characterized this as a distinction between the objective and the subjective. The reference is the actual object denoted by a symbol, while the meaning is our subjective image of that object. By reducing meaning to psychological phantasms, Frege put all of the objective aspects of signification into the reference. This separation is problematic, for we have seen that reference is dependent on meaning, and further, there are many words whose meanings do not correspond to determinate objects. The insistence that a reference must always be a determinate object or set of objects prejudges the metaphysical question of universals, and ignores how we actually use words. When I say ‘cat,’ I do not intend “the set of all objects having such-and-such feline qualities.” Rather, I have an abstract model or concept against which particular objects are tested. The universal is conceptual, but not on that account subjective. Frege, and later Russell, would build their metaphysical preconceptions about the reality of universals and particulars, and build that into their logic. A true a priori logic, by contrast, should be able to handle any conceivable metaphysical system that does not involve a contradiction.

Russell followed Frege in the attempt to distinguish reference from meaning as though the first were objective and the latter were subjective. Russell took this reasoning to greater extremes, contending that expressions that had no referent were meaningless. He defended this wildly counterintuitive position by characterizing any signification that did not indicate an existent object as purely “subjective.” By requiring all expressions to have an existing referent, Russell brought metaphysics into his logic even more forcefully than the psychologism he criticized. If we were to take Russell seriously, we could never know if a statement is logically sound unless we check that its referent actually exists. This confuses a priori and a posteriori investigations, and robs logic of much of its power, making it largely superfluous to empiricism. This, of course, is what Russell and other naturalists wanted, since they only needed logic to rationalize science, the only domain with objective knowledge, in their view. Their logic and Russell’s theory of reference only formalizes their metaphysical prejudices, and declares all other systems to be meaningless. We will not repay the analytic philosophers with their counterfeit coin, but we will articulate a true a priori logic that does not declare any intelligible metaphysical system to be meaningless, not even naturalism.

First, we offer a brief critique of Russell’s theory of description. Russell observed that we may know a thing either directly by acquaintance (as in the Spanish conocer) or indirectly through “denotive phrases.” In order for a symbol to be a name, it must have a real referent. Thus when we declare ‘the king of France,’ we implicitly declare that there is at least one real object that is the king of France. If there is no such object, the name has no referent and thus is not the name or symbol at all. Strawson noted that there was nothing intrinsically meaningless about the expression, since it could be true depending on when it is spoken. Russell countered that Strawson was bringing “egocentric” subjectivity into the analysis of expressions. I will remove that criticism by considering the expression ‘the king of France in 2008’, which clearly cannot have any real referent, regardless of who speaks it or when. Pace Russell, this expression is perfectly intelligible, conveying a meaning that is independent of individual psychological idiosyncracies. Anyone can conceive of France having a king in the year 2008, though that did not actually happen. (Those who are ignorant of French affairs might even regard it as possible, but this would bring in subjectivity.) It will not do to complain that conceiving of something requires an intellect, and therefore subjectivity, for the same is true of any symbol defined by convention. Besides, only an indeterminate intellect is required, so we do not depend on any specific psychology. The referent (object of reference) is the concept, which is objective as it does not depend on individual psychology.[2]

Russell refused to allow concepts as referents, though we have seen that they always mediate reference in non-natural symbols, even when the referent is a concrete object. When there is a concrete referent, there is no need to regard the concept as a referent, since we have a palpable object to act as the terminus of the symbol’s signification. However, even in these cases, the concept is truly a referent, since it is something other than the symbol. We may not always conceive of the concepts corresponding to symbols in exactly the same way, so there is always some subjectivity in meaning, as language imperfectly conveys our intuition, but we are able to convey a great deal of objective conceptual content to one another, as is evident from our ability to communicate effectively to one another, even in abstract terms.

The enterprise of reducing logic to the logic of science founders on the fact that there is no naturalistic explanation of the abstraction process, as Nicholas Capaldi observes. In order to present knowledge as a naturalistic process, and avoid appeal to an intellectual subject, philosophers of the early twentieth century shifted attention from interpretive epistemology to the epistemology of language, as if language were some objective fact that could be detached from the interpretation of an intellectual subject. We see this agenda in Russell’s theory of reference, which would reduce linguistic expressions to mere labels for physical objects, since he refuses to admit the existence of anything else. His theory fails because it ignores the reality that artificial symbols must have intellectual meaning in order for them to perform the function of reference. In fact, it is concepts, rather than physical objects, that are indispensable referents for a symbol to be a symbol. Many analytic philosophers have realized this weakness in their theory, and have sought to reduce all psychological acts to neuroscience, an endeavor still in its infancy. Some thinkers, most notably in computer science, still persist in speaking about language, communication, and information as if symbols could have any signification without someone to interpret them.

While medieval philosophy did not have a theory of reference as such, it did have the concept of supposition (suppositio), which in several of its varied senses, overlaps with the notions of reference and signification. The suppositio is that which is “supposed” to exist (but might not actually) by virtue of declaring a term. The relation of supposition is between the signification of a term and the entity that term designates (referent). Mora discusses several of these senses (defined by Peter of Spain). The suppositio discreta is a referent that is an individual or discrete entity. The suppositio communis is the referent of a universal term. (We must recall that medieval philosophers attributed varying degrees of reality to universals.) These were subdivided into suppositio naturalis and suppositio accidentalis. A natural supposition is when the term itself has a capacity for supposition, when an accidental supposition is when the supposition of a term is determined by something added to the term. Accidental suppositions can be simple (simplex) or personal (personalis). The simple supposition is when the term designates a universal entity. The personal supposition is a common supposition, where the term designates something less than the universal. These can be the suppositio determinata, a determinate supposition, or indeterminate (suppositio confusa), where the term is designated by the sign of universality.

William of Ockham allowed that terms might be used improperly, so he distinguished the suppositio impropria from the suppositio propria. Proper usage can have personal, simple or material suppositions. These are three ways of understanding a term, rather than three different classes of referents. A term can be considered a neutral entity (personal), spoken (simple) or written (material). The personal supposition is then divided into discrete and common.

Walter Burleigh divides suppositions into proper and improper, then proper suppositions are material or formal. The formal supposition is the entity or referent, while the material is the word as a word (i.e., having syllables, etc.). This is a precursor to the use versus mention distinction of modern linguistics.

With such a broad menu of options, as summarized by Mora, we see no reason to confine ourselves to the impoverished notion of signification espoused by Russell and others who would grant reality only to existent references. The analytic philosophers resort to the old sophist argument (described by Plato in the Cratylus) that a false name signifies nothing at all, but is like a clanging pot, meaningless. Plato’s Socrates sagely responds: if that were so, it would be impossible for people to speak falsely, or at any rate, their falsehoods would not be understood. Plato understood that the meanings of words were held in common by the users of a language, and that interpretation, what we would call pragmatics, was essential to establishing any meaning for conventional symbols.

We have seen that various Scholastics used the term suppositio in widely divergent senses, so we would do well to standardize the meaning, while retaining the content of their insights. The basic concept of a suppositio, in most cases, is that which is supposed to correspond to the name. Supponere in Latin means to substitute, so suppositio is the thing substituted by the name. When dealing with non-natural signs, we have only convention to guide us to the supposition. Those who are ignorant or disdainful of such convention may apply an improper supposition to a symbol, that is, one contrary to convention. If we permitted everyone to use symbols however they please in any instance, communication would be impossible, and we would all be effectively talking to ourselves, as if language did not exist. For reasons of pragmatics, therefore, we insist on a conventional supposition, or if we use an unconventional supposition, that we define this in conventional terms. This practice of confining ourselves to proper suppositions introduces an element of objectivity into language, since the user of a symbol is bound to respect a norm outside his own preference or psychological state.

Focusing on proper suppositions, we divide these into the material and the formal. The material supposition is the physical conditions that materially constitute the declaration of a term. We may declare a term in three ways: mentally (thinking the term in our head), verbally (spoken aloud), or in writing. There are other methods as well, such as gestural language, electronic signals, etc., but the same general principles apply. In the case of verbal and written declarations, it is easy to see how the qualities of the utterance and the shapes of the characters form the material basis of the declaration. With thought, however, the concept is more muddled, since in one and the same mental act we seem to declare the linguistic symbols and think of its referent. It would seem that the distinction between material and formal supposition is only virtual, not actual. At any rate, when we think a term in our head, we mentally reproduce the material qualities: i.e., the sound of the syllables, or the shape of the letters. This sensory phantasm is not itself the concept, as is evident from the fact that we can hear and mentally repeat foreign terms whose significance we do not know. When we have facility and familiarity with a term, however, declaring a term and thinking of its concept appear to be one and the same act, allowing us to rapidly think in language, all the while understanding its significance.

The formal supposition is the signification of a term in the most generic sense. No clear distinction is made between meaning and reference, though the language of the Scholastics suggests that the formal suppositions are entities. However, they are supposed entities, not necessarily actually existing entities. That is, they are entities as conceived, not as actualized in physical space. This is exactly what we should expect in logic, which gives the laws of thought. The referent of a symbol is always conceptual, and only possibly actual. The question of actuality, like that of the relation between the conceptual and the actual, is a question of metaphysics. We do not need to know about actual reality in order to determine if a statement is logically coherent, or even to build abstract arguments that are valid. In fact, some statements such as counterfactual conditionals explicitly suppose the unreality of our supposition. To use modern terminology, the suppositio is the referent as it is conceived. There is no neat distinction between referent and meaning, since the declaration of a term cannot guarantee the existence of the thing signified, but only supposes it formally. If the thing signified actually does exist, then that thing may be called the referent in the modern sense. Otherwise, the formal supposition corresponds solely to what we would call meaning, though we have seen that this meaning is not purely subjective (as in improper suppositions), nor is it merely the sensory phantasms of the word in our mind (material supposition).

Formal suppositions apply only to terms, that is, words that correspond to conceivable ontological entities. We could not speak of the formal supposition of prepositions or conjunctions, except when we consider these as part of some term. We will need other tools to analyze the semantics of such words. For now, we will focus on the classes of formal suppositions, which correspond to the ways in which a word may be applied to a referent. We have some discretion in the order in which we choose to subdivide formal suppositions, but we will follow the basic scheme of Peter of Spain, which is comprehensive as it spans all conceivable ontological classes.

A term may just be a label for a discrete object, or it may be held in common by several referents. In the case of a discrete supposition, the term acts as a mere token for some object; it is much like an icon, save that its designation is assigned by arbitrary convention. Words with discrete suppositions do not really allow us to analyze anything, since they apply no concept to the referent — the concept behind the word is only the image of the referent. If all suppositions were discrete, Russell might be justified in regarding conceptual meaning as nothing more than our subjective perception of a referent. Yet a language consisting solely of such terms would be analytically barren, as it would just replace a collection of objects with a collection of names, and we would be no better off at understanding anything than if we had no language at all.

It is from terms with a suppositio communis that logic derives its analytical power. These terms always invoke universals. Even modern philosophers who are disdainful of universal substances must make use of universal accidents or properties; otherwise we could never say that disparate objects have a common property, an observation which is fundamental to scientific theorizing. However, they have inconsistently tried to characterize universals as though they were discrete, as we shall examine when we study predicates, classes and the like. Terms with a common supposition do not merely affix labels to objects, but transcend objects, pointing to concepts that could apply to innumerable objects, or none at all, without adding to or diminishing their meaning, coherence, and intelligibility. Once we admit common suppositions, referent objects in physical space will not suffice, for reference is always mediated by some universal concept.

We pass over the distinction between natural and accidental suppositions, since we are dealing only with formal suppositions, so there are no ‘natural’ suppositions here. We may subdivide common suppositions into ‘simple’ and ‘non-simple’ (avoiding the awkward Scholastic term ‘personal’). Simple suppositions refer to the universal simply as universal, a pure concept detached from any group of conceivable individuations. Non-simple supposition considers the universal as it may exist in a determinate (‘these dogs’) or indeterminate (‘some dogs’) group of objects. A determinate supposition (‘these dogs’) may have the same referent as a discrete supposition, yet the reference is mediated by a universal applied to various objects, so it is truly a common supposition. It makes a real semantic difference whether I call a group of dogs ‘these dogs’ or simply apply a label ‘Zygax’. In the first case, I am applying a universal concept to various objects, thereby declaring some commonality among them. By applying a label to them, I make no such declaration of commonality, and could have just as easily applied it to wildly disparate objects. Further, I am labeling the dogs as though they collectively formed a single entity. It is a mistake, then, to regard terms with determinate suppositions such as ‘these dogs’ as though they only had discrete objects as referents, with no conceptual referent. We will examine modern promotion of such an error in set theory.

Indeterminate suppositions (‘some dogs’) are even more manifestly mediated by universals, so that even symbolic logic acknowledges that ‘there exists some’ is expressible in terms of the universal quantifier (∀, ’for every’). In Logic and Language, we discussed how symbolic quantifiers failed to capture universals conceptually and logically. In the present work, we will examine the use of such quantifiers in more detail. (Part IV) The referent of ‘some dogs’ is not any determinate object or group of objects, and there is no way to conceive of this referent without recourse to the universal concept ‘dog’; otherwise it would be rendered determinate.

We see from this overview of the problem of reference that reference and meaning cannot be neatly distinguished when dealing with non-natural symbols. In the case of terms with common suppositions, concepts are absolutely indispensable, since they mediate reference. This is why the Scholastics seldom bothered to distinguish between the meaning and the referent of a term, and it is also why modern symbolic logic, using Russell’s theory of reference, is at its weakest when dealing with universals. We will explore these shortcomings in painstaking detail when we examine modern predicate calculus and set theory.

Several types of reference can be analyzed using what we have said about suppositions. Coreference occurs when two or more terms have the same referent. This is trivially the case when the terms are synonyms: since they have the same formal supposition, they will apply to the same range of possible referents. However, we may also have coreference in grammatical contexts where one term is a substituend for an antecedent. For example, in the sentence, ‘I opened the door, and then closed it,’ the terms ‘the door’ and ‘it’ have the same referent, not because they have the same formal meaning, but because ‘it’ is a coreferential term whose logical function is to refer to the same referent as its antecedent. The use of such coreference, we shall see much later (Part VII), will be instrumental in resolving paradoxes of autoreference.

A term may be called autological or autoreferent when the term may serve as its own referent. This is often phrased as a term that ‘refers to itself.’ Failure to adequately examine the use of coreference (‘itself’) leads to paradox. We may further note that, speaking more precisely, autological means the formal supposition can apply to the material supposition, or the material supposition exemplifies (instantiates) the formal supposition. Complementing the notion of autological is heterological or heteroreferent, which means that the referent is always something other than the symbolic term itsel. This means that the formal supposition does not apply to the natural supposition, or that the material supposition neither exemplifies nor instantiates the formal supposition. If we do not explore the coreference that is latent in the concepts of autoreference and heteroreference, we will find paradoxes in questions like, “Is ‘heterological’ heterological?” This class of paradox will be discussed in Part VII.

In order to apply some notion of reference to basic ontological terms (e.g., ‘individual’, ‘property’, ‘that which is’) that do not refer to any entity, but are purely categorical, José Ferreter Mora introduced the concept of transreference (transreferencia), discussed in El ser y sentido and Fundamentos de Filosofia. To establish this concept, he introduced his own peculiar notions of coreference and heteroreference distinct from what we have defined. They are nonetheless useful in expanding the notion of reference from that of the analytic school.

According to Mora, a name is referential in the full sense when that which it names exists; i.e., there is an actually existing referent. We have already criticized this notion of reference as confusing physical and mathematical questions with a priori logic. However, we may proceed with this analysis of reference as long as we do not consider that reference contains the fullness of a term’s signification, especially when the term has a common supposition. Mora examines the case of the name of a fictional character, ‘Aquiles,’ whose referent does not exist. However, there are actual men who are capable of being named Aquiles, and, within the context of the fiction in which he appears, Aquiles may be considered a man of flesh and bone. For either reason, ‘Aquiles’ may be said to “co-refer” (in Mora’s special sense) to a man of flesh and blood. The insistence on this peculiar sort of reference is part of Mora’s attempt to make the concept of reference broad enough to handle all terms. Rather than simply admit that a concept can be a referent, or that the conceptual is more essential to signification, we are forced to adopt this equivocal notion of reference, where something is considered a (co-)referent simply because it might exist in some context, even though we know in actuality it does not. There is no small element of truth in this analysis, for to conceive of a thing is to conceive of it as real in some context, so we may call this conceptual object a referent. Yet I would submit that the semantic relationship between a symbol and its conceptual signification is not altered by the actual presence or absence of its physical referent. I may speak of something as if it were physically real, not knowing this was actually not the case, but my meaning would not thereby be changed. If this were so, no one could understand any assertion unless he already knew if it were true, which would defeat much of the purpose of communication. Nonetheless, there is a place for Mora’s co-reference in a priori logic, since a person may deliberately intend a term as co-referent, not considering the referent to actually exist, as in fiction. A clearer way to express this, however, is to say that the formal supposition is not an existent, but only hypothetical. The very notion of supposition already captures this since it is something that is supposed to exist, yet we do not necessarily know whether it actually does exist. If we expand the notion of reference to include things that do not exist, we might as well frankly admit that we are restoring the conceptual as necessary to signification.

Mora applies his notion of co-reference to descriptive phrases as well. ‘The Emperor of La Mancha’ does not refer to any emperor that is or has ever been. However, ‘La Mancha’ truly does refer to La Mancha, and ‘The Emperor of…’ is an incomplete expression that would refer to someone if he existed. Thus, ‘the Emperor of La Mancha’ co-refers to an emperor, according to Mora, but not that of La Mancha. If the (co-) referent of ‘the Emperor of La Mancha’ is not the emperor of La Mancha, it would seem that Mora’s co-reference is reference in name only. It would be one thing to say ‘the Emperor of…’ could refer to some indeterminate emperor, but in what sense does the whole description refer or co-refer to any emperor? The awkward notion of co-reference, which seems to simultaneously affirm and deny the presence of a referent, would be better replaced by a candid admission that the “co-referent” of ‘the Emperor of La Mancha’ is a conceptual supposition, not an existent, or “co-existent” entity. To say that ‘the Emperor of La Mancha’ co-refers to other emperors by virtue of ‘the Emperor of…’ would be to consider ‘the Emperor of…’ as abstracted from the rest of the description, and then to say it could refer to something if it were not for ‘La Mancha’. This is simply to say that part of the expression has a referent, yet the expression as a whole does not. Mora’s system retains the weakness that we cannot know if something is a referent or co-referent without knowing whether it actually exists. No such difficulty exists if we consider a conceptual supposition.

Mora also introduces heteroreference with his own peculiar meaning. A common name that can be true or false of ones, some or all entities is heteroreferent. As long as there is at least one existing thing that could be a referent, the expression is heteroreferent. Heteroreference differs from direct reference only in that the referent is not explicitly specified. We would say the supposition is not discrete, but common. We noted that common suppositions are not handled well by a logic that only recognizes reference and not meaning as objective signification. Mora tries to remedy this deficiency by augmenting Russell’s theory of reference with ‘heteroreference.’ We do not consider ‘being a French general’ as a concept that could conceivably apply to any man, but only as an expression that might correspond to some existent that is its referent. Trying to squeeze out the mediation of concepts between the expression and any potential referent is disingenuous, of course. In actuality, we decide whether or not an object is a referent of a common term by comparing that object to the conceptual essence represented by the term. We have an idea of what it is to be a French general, and apply it to various objects. Without a conceptual meaning to the term, it is difficult to see what basis there is for applying it to some object and not others. If we say they have similar attributes, we are acknowledging universal accidents. Once we acknowledge that there is an a priori basis for considering whether a term applies to an object, we have acknowledged conceptual meaning, which can be applied even to things we are not sure exist. If we try to avoid this conclusion by saying, for example, that ‘French general’ is nothing more than an arbitrary label for all the people we have chosen to call French generals, then there would be no objective way for me to determine, fifty years from now, who is now a French general. We are able to make this determination in advance, even regarding people about whose existence we are uncertain, because we can apply an a priori concept to objects we encounter physically or mentally.

Mora defines the aforementioned terms to contrast with ‘trans-reference’ which is the reference of basic ontological terms such as ‘property’, ‘individual’ and so on. There is no entity that is purely and simply an ‘individual’ or ‘property’, so there is no direct reference in these terms. We can agree with this analysis, since even a priori such terms do not refer to any conceivable entity, though they might be considered as common terms, at least equivocally. I say ‘equivocally,’ allowing that the most fundamental ontological terms are not generic universals, though they are used in a semantically similar way (i.e., with common suppositions). The fundamental ontological terms, according to Mora, do not refer, co-refer, or hetero-refer to anything that they name or describe. However, they do categorize objects of (co- and hetero-) reference. Just as coreference and heteroreference are dependent on the existence of some direct reference, so is transreference dependent on the other three forms of reference. Mora observes, paradoxically, that transreference is the least fundamental mode of reference, since it cannot exist if the others do not exist, yet conceptually, it is the most fundamental since all the other referents fall under some ontological class. This paradox results from the mingling of a priori logic and a posteriori metaphysics. Considered a posteriori, we may require (depending on our theory of metaphysics) an individual, or discrete supposition, to be actual in order for a universal, and ultimately some ontological class, to be actual. However, considered a priori, we may consider ‘properties’ without defining any particular property, and consider universals without asserting the existence of a particular individual, as we have in fact done in the Introduction to Ontological Categories, following countless other philosophical writers. I have no idea what a ‘trans-referent’ would be; it seems that the notion of transreference, like that of coreference and heteroreference, is designed to compensate for the fact that direct reference cannot account for much of reality, all the while retaining the metaphysical assumption that direct referents are fundamental, and the semantically bizarre assumption that the signification of abstract terms depends on the existence of direct referents. He accomplishes this by examining other kinds of signification, and calling them ‘references’, justifying this by showing their relationship to direct reference. However, there is no necessity of direct reference in these other forms of signification, and we could do just as well without it. Otherwise, we have the same problem as before: supposedly I cannot know what a term signifies unless some instance of its referent exists.

A theory of signification based solely on reference would cheat us of most of our intellectual life. Mora sought to rehabilitate Russell’s theory by broadening the notion of reference, yet we find that this is achieved much more coherently by simply acknowledging intensional meaning as an essential aspect of the signification of a symbol. Further, we shall see that a logic whose semantics is solely referential and descriptive fails even within its own domain. We cannot fully account even for terms with discrete referents without appealing to a notion of meaning. The theory of reference was intended to strip from logic the possibility of any metaphysics other than a naturalism where there are only existent objects, to be analyzed in set theoretic fashion. This dishonest gambit would enthrone naturalism without any possible means of arguing its merits, as all other metaphysical theories have been written out of logic. This theory-laden formal logic fails even within its own domain, showing that the sciences are not intellectually autonomous. Before we embark on an exhaustive discussion of the failures of such logicism, we shall examine some of the reasons that have led philosophers to formalize logic in the first place.

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Formalization

Since the time of Plato and Aristotle, if not earlier, philosophers have realized that the structure of ordinary languages does not always coincide with logical structure. Before we can construct a truly logical language, it would perhaps be useful to construct a meta-language that allows us to specify and compare the structures of various languages. We can thereby analyze the form of the expressions of a language, perhaps without regard for the semantic content (meaning and referent) of each term. The formal structure of a language is syntactic; it is of logical interest only insofar as it is intended to relate logical relationships among concepts or ontological relationships among referents. Moreover, the rules of syntax may depend on the meaning of a term, or the ontological category of its referent. Attempts to separate syntax from meaning are purely artificial; it is only when we fully grasp the meaning of terms, at least with regard to their ontological category, that we can determine logically acceptable rules of syntax. (I discussed in Introduction to Ontological Categories why a priori ontology, as opposed to a posteriori metaphysics, is essential to logic.) Once we have established such rules, then we can blindly apply syntactic analysis to parse sentences, or even program a computer to do so, but at the end of such analysis, the sentence can only become a statement if the interpreter recognizes its meaning.

Formalization is a useful tool that helps us express logical statements with a clear, concise and unambiguous syntax, so we can analyze statements and arguments simply by examining the syntax. However, syntax is only a representation of logical structure; it does not add any semantic signification in virtue of itself. Syntax is subservient to logic to the extent it is logical, and we are able to determine the validity of an argument from its syntax only because we have first agreed on a signification in accordance with logical principles. The usefulness of knowing a priori any formal syntax is determined by its grounding in sound logical principles. Only then can we trust that syntactic validity corresponds to logical validity. However, even when a system accurately represents some aspects of logic, it may omit others, so that we cannot take the unrepresentability of a statement in that formalism as evidence of incoherence or meaninglessness. Such is in fact the case with most modern formalizations of logic, which dismiss universals and a priori metaphysical concepts out of hand, and then presume to show these concepts are useless or meaningless because they have no place in this syntax. We will cast aside such blinkered thinking, and not allow the limitations of a particular formalization to blind us to valid inferences we may understand in ordinary language. Formal logic, if it is to be intelligible as logic rather than syntax, must be expressible in human language corresponding to actual thoughts, or else it cannot be a law of thought. We can use formalization to organize our thinking more rigorously, with less equivocal terms, making it easier to see if a deduction is valid, but strictly speaking, a formalization should not make possible any inference that would not have been possible without it. If it is limited (as are perhaps all known ordinary languages and formalizations), there will be inferences it cannot represent, but such are not thereby considered invalid.

Classical logic used ordinary language, not as it was given, but in a highly structured formal way. Greek terms acquired hitherto unknown technical meanings, defined with a precision unintended in lay speech. In the West, Latin terms were invented to correspond to Greek, or to introduce new concepts unknown in classical Latin. The Latin of medieval Scholastics, often scorned for its barbarous style, departed from classical Latin in terms of usage and syntax in order to achieve a technical meaning. Perhaps the “worst offender” in this regard was John Duns Scotus, whose writing is often unintelligible even to those familiar with Scholastic Latin, because of his hyper-precise idiosyncratic use of terms, earning him the title of Doctor Subtilis. These developments made medieval Latin a living language, and an early effort at formalization.

Modern formalization of logic uses its own special set of characters, rather than borrow from an existing ordinary language, for connections showing the logical relationships among terms. Single-letter characters are used to represent fixed or variable terms. Symbolic logic is generally unconcerned with the semantic content of terms, but treats them as though they were discrete objects, in analogy with elements of a set. This treatment is valid when dealing with terms that truly do represent discrete objects, but has less happy results in other contexts. Having a property-object ontology, it does not handle universals well, and paradoxes such as Russell’s antinomy derive in part from its failure to deal with substance adequately.

Since modern logic deals only with the form of statements, as if the formalization were logic itself, it abstracts from the conceptual content of terms. In this sense, it is more generic than classical logic, which contains definite ontological theses (though modern logic is far from ontologically neutral, and fails even metaphysical neutrality). Considered purely as a formalization, symbolic logic does not need even axioms such as non-contradiction and the excluded middle. From this perspective, classical logic is but one of many possible logics. The belief that there will therefore be logic contrary to classical logic is absurd, since the meta-logic used to interpret different symbolic “logics” itself depends on classical Aristotelian logic. This error results from a failure to distinguish between syntax and logic. Other “logics” are syntactically possible in modern formalization, but they are not truly logical. At best, they make sense as a sort of calculus, whose fundamental or meta-logical structure of inference is grounded in classical, or true, logic. We will explore the distinction between a logic and a calculus in more detail shortly.

While classical logic is a subset of modern logic in terms of what is syntactically possible, conversely modern logic is a subset of classical logic in terms of what it can prove. Since modern logic regards classical laws of thought as mere arbitrary axioms, it cannot prove anything from these a priori. In fact, there is no true a priori reasoning in symbolic logic without axioms. Further, its limited ontology leaves it incapable of proving anything about anything beyond discrete objects. Refusing to explicitly admit sound ontology, it is often unable to recognize ontologically incoherent terms as such.

Those modern philosophers who have relied on formalization as logic have historically divided into two camps: formalists and intuitionists. Formalism, as distinguished from intuitionism, is not to be confused with a general sympathy toward formalization. In the context of mathematical logic, formalism means that mathematics can be completely formalized by proving there is no contradiction among the various theories of mathematics and their systems of symbolic logic. For intuitionists such as Heyting, we need to resort to intuitive concepts as the basis of mathematics. Logicism espoused by Frege and Russell constitute a third theory of the philosophy of mathematics, which would reduce mathematics to a symbolic logic. Their logicism was really a kind of formalism, since their logic was not really a logic, but a logistic. We are not primarily concerned with the philosophy of mathematics, except as it inadvertently relates to our analysis of formal or symbolic logic, which as we shall see, is not truly logic.

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Logic, Logistic and Calculus

Logic is the science of how we understand the world a priori. It contains laws of thought, yet it is not confined to thought. Rather it restricts thought to what is a priori possible. We are fully capable of thinking illogically, so logic is not a rule of our own psychology, but rather the imposition of some a priori relationship among concepts upon our thinking. Concepts, to be sure, can only be known by an intellect, yet they have an objective structure that is not dependent on any particular intellect or its psychology.

A logistic is an interpreted set of symbols, purportedly representing logical statements and inferences. Logistics include what English-speaking philosophers call “symbolic logic,” which we have seen does not encompass a priori logic. The symbols themselves constitute a calculus, which may be parsed or manipulated according to syntactic rules even by a computer. A logistic does not necessarily represent logic, since the system might not be coherent, or it may fail to describe every conceivable a priori reality.

Logistic, or “symbolic logic,” as it is now called, deals with forms of statements as though these could be abstracted from their conceptual content. Classical logic, by contrast, recognizes that the formal relationship among terms depends on their categorical content. Symbolic logic pretends to purge ontology from logic, yet it has its own ontological assumptions embedded, whereas classical logic explicitly expounds a priori ontological possibilities. While we have allowed a full four-category ontology since there are no a priori grounds for discarding any of these, symbolic logic presumes a property-object ontology and metaphysical nominalism, imposing a particular metaphysics upon the world without firing a shot. Symbolic logic is no less dependent than classical logic on an ontological interpretation of the subject-predicate relationship. Symbolic logic pretends only one such relationship is possible, while classical logic embraces the breadth of ontologically possible predications. (See Logic and Language, Part II.)

Some modern philosophers, such as Tarski, recognized that classical logic was not dependent on a particular metaphysics, but failed to recognize its applicability to mathematics and other sciences. Those who have studied the Mertonian school and other specialized forms of Scholasticism know this is not the case. Our purpose here is not to detail the achievements of the Scholastics, nor the fruitfulness of symbolic logic in mathematics and computer science, which we take as given. What we wish to do is challenge the notion, as Mora puts it, that symbolic logic has rules of inference abstracted from any general philosophy or metaphysics. Abstracting logic from philosophy is a modern will-o’-the-wisp (derived from an obsession with ‘objectivity’ and contempt for metaphysics) that is attainable only by reducing logic to a calculus. We will expose the philosophical assumptions of symbolic logic — the interpreted calculus — and examine which of these are sound.

Since symbolic logic is an interpreted calculus, there are as many different logics as calculuses, including those that defy the law of the excluded middle and non-contradiction (e.g., Heyting logic). Yet all of these “logics” (really logistics) are reasoned out according to principles of classical logic, which serves as behind-the-scenes meta-logic. It remains to be seen why one calculus and its logistics should be preferred to others. First, we will examine the constituents of a calculus.

The constituents of a calculus are all symbols or syntactic relations between symbols. A calculus consists of (1) primitive signs, (2) expressions or formulas, (3) well-formed expressions, and (4) theorems. Expressions or formulas are combinations of primitive signs, which may correspond to some semantic expression, either part of a judgment or a complete judgment, or it might not be coherent at all. We need rules of formulation for our syntax, enabling us to define syntactically a “well-formed expression,” which presumably represents a logical judgment. If we further establish rules of inference (again defined syntactically), then we can claim our formula or expression represents what is logically implied by the judgment signified by some other formula. The terminus of a chain of inference is called a theorem. In order to have theorems, we need to define axioms, rules of inference, and our notion of proof. Again, these are all defined purely syntactically.

Clearly, we are playing a game when we say our calculus is purely syntactic, for in reality all of the syntax is inspired by our semantic need to represent logic and its rules. A calculus is the result of a formalization of some determinate part of logic, though ideally we would like it to represent all of logic, as the analytic philosophers claim. Each calculus, with its signs, formulas, well-formed expressions, and theorems, along with the implied rules of formulation, axioms, rules of inference, and definition of proof, can be tested for certain objective qualities we call consistency, completeness, decidablility, and independence. Here lies the real power of symbolic logic, as it allows us to test a calculus (and thus the logic it represents) for the ability to prove all theorems consistent with it, or the absence of any contradiction, et cetera. It remains to be seen, however, if all logic could really be conveyed by a syntax. We will show, at any rate, that the calculuses proposed to date as representing real logic fall short of their goal. We will test each calculus syntactically, using the concepts mentioned above.

Of particular interest to us is sentential calculus, whose signs are letters, connectives, and parentheses. When the letters represent propositions, it serves as a propositional calculus, replacing classical logic. However, we will examine in detail why this does not in fact encompass all of Aristotelian logic.

Sentential calculus can be augmented by the quantifiers ∀ and ∃, giving us quantificational calculus (interpreted as quantificational logic). Quantificational logic consists of sentential letters, predicate letters, argument letters, connectives, parentheses and quantifiers. Predicates are modeled as a function of some argument which is the subject.

There are two kinds of quantificational logic or calculus. We may restrict quantification only to arguments or variables rather than predicates. This is elemental quantification, also known as functional logic or lower predicate calculus. Alternatively, we may further allow quantifiers to be applied directly to predicates. This higher quantificational calculus is known as higher functional logic or higher predicate calculus. Lower predicate calculus, also known as first order logic, is a part of the higher quantificational calculus which fully contains it.

We may augment quantificational calculus with signs of identity, to produce a calculus or language of identity. A calculus of classes uses sentential connectives, Boolean symbols and class symbols. We will find that this calculus does not adequately represent universals. Lastly, we can have a calculus of relations, which uses connective symbols, Boolean symbols, and symbols of relations. Relations are often represented as predicates with more than one argument. We will examine all of these calculuses and determine their limitations compared with classical logic.

Continue to Part III

Notes

[1] Here the word 'term' is used in the common grammatical sense, not in the sense of 'term logic,' where 'term' specifically signifies the subject or predicate of a premise.

[2] If we acknowledged that a term can refer to imaginary things, we might regard the referent as a thought-object considered extensively. This requires distinguishing a formal object from a material object, which the analytic philosophers generally did not do, much as they failed to distinguish objects from concepts.


© 2009-2012 Daniel J. Castellano. All rights reserved. http://www.arcaneknowledge.org

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