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The Incompleteness of Formal Logic

A Philosophical Criticism of Symbolic Calculus

Daniel J. Castellano (2009-12)

I. Introduction

In the English-speaking world, philosophy tends to be dominated by a paradigm that assumes philosophical logic is identical with symbolic or mathematical logic. Many philosophy instructors have a background in logic as taught in mathematics or computer science, and it is this impoverished logic that passes for philosophic or true logic. In fact, mathematical logic is really a calculus, independent of the content of the entities it relates. Since it is based solely on the form of an argument or proof, independent of the content of its premises or variables, it is properly called formal logic. Formal logic is necessarily incomplete and incapable of generating a system of epistemology or metaphysics. For all of our progress in developing formal logic, it remains an impoverished and inadequate philosophical logic, as it does not take into account the categorical content of its premises. This explains the chaotic state of modern philosophy, which has been unable to demonstrate any generally accepted theses of metaphysics, since formal logic can only prove barren tautologies.

It must be remembered that the enterprise of reducing philosophical logic to formal logic assumed that the latter would implicitly contain all the richness of the former. Gottlob Frege hoped to prove that, since all logical operations follow a set of rules to be applied to propositions regardless of their specific content, this logic might be described as a calculus. Unfortunately, attempts to formalize philosophical logic did not adequately address the reality that logic does depend on the categorical content of its variables, leaving many gaps, and entire classes of metaphysical theses that, while certainly true, were not provable by formal, symbolic logic. From these considerations alone, it should have been obvious long before Gödel that formal logic was “incomplete.”

Critics of formal logic, most notably Bertrand Russell, often assumed the same indifference to categorical or semantic content. Russell’s famous self-referential paradox, for example, exploits formal logic’s failure to adequately treat substituends in relation to their antecedent. Philosophically, Russell’s paradox is no true paradox, since anyone with an elementary understanding of how to parse meaning in English can resolve the flaw. The paradox is no true paradox because Frege’s logic is no true logic, but a deficient subset of the latter. It is a subset of traditional logic in the sense of the type of information it can prove. In another sense, traditional logic might be considered a subset of formal logic, the latter being overly broad and without reference to ontological reality. Classical logic is grounded in reality, and as its name suggests, is based on the logos, the “word” which is assumed to be linked to the reality it describes. Without this link to reality (actual or virtual), there is no reason to regard words as anything but arbitrary labels, and it is similarly arbitrary to define rules about the relations between these labels (words or propositions).

This fact has not been lost on modern logicians, who have felt at liberty to construct non-Boolean “logics,” with mathematically or practically (e.g., computer science) useful results. These successes prove no more than that these other formal logics are useful calculuses, but some of their exponents have rashly concluded that the choice of logic is arbitrary, and even the principles of non-contradiction and the “excluded middle” are negotiable. This failure to distinguish symbolic logic from a real logic of being (in which it is self-evident that being and non-being are exclusive, when predicated univocally) results in a complete failure to arrive at even the most elementary philosophical truths, since all principles are negotiable or arbitrary. In fact, non-Aristotelian logics (including Boolean logic) are merely calculuses that are implicitly predicated on an Aristotelian logic of being. If this were not the case, it would hardly be possible to conduct mathematical proofs about these logics (meta-proofs), showing that they are internally consistent. In such a scenario, every logic student could argue he deserves a perfect grade, since the rules of mathematical proof are arbitrary. To this day, all mathematical proofs, including those about non-Aristotelian “logics” (calculuses), assume Aristotelian principles of non-contradiction and the excluded middle.

The modern enterprise of formalizing logic did not set out to move beyond Aristotelian logic, but rather to restrict its scope, purging it of metaphysics. This aim was expressly stated by Bertrand Russell (among others), who sought to reduce all of philosophy to the philosophy of natural science. This desire to expunge metaphysics from philosophy was itself grounded in a metaphysical assumption: philosophical naturalism. In this view, there is nothing that can be known beyond natural science, hence there could be no meta-science (metaphysics or metaphysical logic) to be the standard of truth. Consequently, the only logic is the logic of science, whose predicates are the objects of natural science. The reasoning here is manifestly circular. We assume there is nothing knowable beyond natural science, and define our logic to only treat objects of natural science. A stunted epistemology, where physical empiricism is the only valid way of knowing, is built into our logic, which is then used to “show” that there is no logical basis for metaphysics. This is a common feature of philosophical naturalism: metaphysics is evaded by disavowing any logical or epistemological means of arriving at it. It would be like gouging out our eyes so we can deny the existence of color. Similarly, the analytic philosophers and other naturalists practice what Nicholas Capaldi has rightly called a “truncated Aristotelianism,” denying metaphysics by simply ignoring it. In fact, all they have done is build their epistemic and metaphysical assumptions into their logic, rather than allow logic to stand in judgment of these assumptions, which will not withstand such scrutiny.

If it were really true that nothing is knowable beyond natural science and philosophy were but the logic of science, there would be no real need for philosophy as a distinct discipline. If philosophy does not stand in judgment of other sciences, but is utterly derivative of the natural sciences, what need do scientists have of philosophers? Surely the practitioners of each science understand the logic of their disciplines, including the concepts and precepts they use, which supposedly give the truth about everything. If naturalism is taken seriously, logic should not add any knowledge to the natural sciences beyond what can be determined within the sciences. Indeed, in their zeal to lionize “science” (as if there were a single epistemology for all fields of study), mainly in order to cast aspersions upon systems of metaphysics and theology, the analytic philosophers and other naturalists render themselves irrelevant. It is far better to ask a physicist than a philosopher for the logic of physics, if this is nothing more than physics. We would be better off turning to real scientists rather than their hangers-on, who are neither philosophers nor scientists.

Had logicism succeeded, it might have established the intellectual autonomy of those sciences that are grounded in mathematics. Symbolic logic, however, has several serious shortcomings. Since formal logic abstracts from conceptual content, dealing only with the form of statements, it is limited in what it can prove, and often fails to distinguish coherent from incoherent statements, because it is not grounded in sound a priori concepts. Even as a formal logic, symbolic logic fails, yielding paradoxes such as Russell’s antinomy, which result from a neglect of a priori ontological concepts, such as substance. Further, it assumes property-object ontology, so it is incapable of handling universals.

Lastly, even within the domain of mathematics, symbolic logic is incomplete, unable to yield all possible deductions. The art of mathematical proof cannot be reduced to an algorithm, as Kurt Gödel famously proved with his “incompleteness” theorems, which showed the insufficiency of symbolic logic even for purely mathematical proofs. Even in its own domain of mathematics, we need something stronger than symbolic logic.

Formal logic assumes the content of statements does not matter, but only regards their form, for the purpose of valid deduction. Thus it only encompassses those deductions that follow from form. Yet even these deductions contain implicit assumptions about universals and particulars, and what it means to be an instance of a universal, as in “for every” (∀) and “there exists” (∃). Thus even formal logic has conceptual content. In mathematical logic, we interpret “for every” (∀) and “there exists” (∃) in a set theoretic sense, not in terms of universals and particulars. In Aristotelianism, when a universal is in a particular, its essence is in the particular, so you may say that the particular, in a sense, is the universal. This instantiation relation is different from attribution, where you ascribe some accident to a subject. A true logic must be capable of treating such relations, rather than dismiss them out of hand. Only then can logic be truly a priori, rather than a post hoc rationalization of existing sciences.

Without sound ontological concepts, logic is just an arbitrary set of rules, and we could easily define a different formalism. The project of logicism failed because it tried to discard ontology and metaphysics, all the while smuggling in its own peculiar ontological and metaphysical assumptions, many of which were not sound. We must make our ontology explicit, and allow logic to encompass anything the mind can conceive, not confined to the ever-changing paradigms of contemporary science. A logic must transcend theory, not be confined to a theory. Logicism was heavily theory-laden, and blind to the fallacy of the Victorian idea that “science” could be “objective.” When logic is freed from the empiricist straitjacket, it can subject epistemologies and theories of physics and metaphysics to scrutiny under the laws of thought.

Any attempt to recover true logic from the morass of modern formalism must begin with an understanding of the meaning of symbols, taking care to distinguish semantics from syntax, while emphasizing the importance of both. We have done this to some extent already in Logic and Language, but we will recapitulate our results a bit later. Next we will examine the historical transition from classical term logic to modern predicate logic, and its intellectual motivations. Then we will analyze first order predicate logic in considerable detail, taking care to bring out its latent ontological assumptions. In particular, we will deal with the problematic concept of material implication, as well as the quantifiers ∀ and ∃ mentioned above, showing how these confuse universals and particulars. Next, we will move on to higher order logic and rules of inference or proof. This will naturally lead to a discussion of the claims about what symbolic logic could prove, and the definitive refutation of those claims by Gödel.

We will examine Gödel’s undecidability and incompleteness theorems in some detatil, before proceeding to a discussion of formal set theory, and the Zermelo-Fraenkel axiomization in particular. The limitations of Russell’s logic have made necessary several existential axioms, especially the awkward substitution axiom, designed to circumvent Russell's paradox. If logic were established on a more solid semantic foundation, such makeshift remedies would be unnecessary. Set theory does not adequately deal with universals, any more than predicate logic. A major correction is in order.

This discussion will culminate in the development of a formal logic that retains much of the semantic richness of classical logic, while retaining the precision of symbolic logic. We will begin with John Trotter’s proposed predicate calculus that makes use of ordinary language handling of substituends, and then check for ontological robustness. Lastly we will take up Trotter’s invitation to construct a new axiomization of set theory, without existential axioms. We will also clarify the mysterious concepts of elementhood and sethood.

The failure of logicism is eighty years old, yet few have made the obvious inference that a more robust logic is needed, recovering some of the capabilities of classical logic. Many still carry on as if symbolic logic were the only logic, and see Gödel as an indictment of logic itself. Instead, most analytic philosophers have descended deeper into the bunker of scientism, turning to linguistics or neuropsychology for the basis of logic. We will briefly survey this school of thought, and point out its epistemic incoherence. It is ironic that the analytic philosophers, who were realists countering the Kantian turn in philosophy, have now resorted to overt psychologism, albeit an ostensibly “scientific” materialist psychologism. A more thorough refutation of their deification of empiricism shall await a work on epistemology, but we can at least show, using our robust logical system, that there is no rational obstacle to reality beyond natural science. As for the supposed intellectual autonomy of the sciences, we will see in due course that these each have their own characteristic ontological assumptions, which are subject to logical scrutiny. Such philosophical criticism is not intended to refute the sciences, so much as to clarify the ontological and metaphysical content of their claims, in correction of the grandiose philosophical claims made by popularizers of science. Those who deny metaphysics cannot avoid it, but end up doing it poorly, blind to their assumptions, which we shall make explicit throughout this work.

Continue to Part II


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

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