The Limits of Logic: Paradoxes and The Failure of Formal Logic, #1.

Mr Nemo
7 min readMar 18, 2024

By Joseph Wayne Smith

Alice laughed. “There’s no use trying,” she said: “One ca’n’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, I sometimes believed as many as six impossible things before breakfast!” (Carroll, 1871/1988: pp. 91–92)

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TABLE OF CONTENTS

1. Introduction: A Skeptical Challenge to Formal Logic

2. The Nature of Formal Logic

3. Problems with Logical Validity

4. Logical Skepticism and The Problem of Deduction

5. The Logico-Semantical Paradoxes

6. Paraconsistency

7. The Refutation of Formal Logic

8. Conclusion: Bankruptcy, Non-Formalism, Limits, and Humility

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The essay that follows will be published here in eight installments, two sections per installment; this, the first installment, contains sections 1 and 2.

But you can also download and read or share a .pdf of the complete text of the essay, including the REFERENCES, by scrolling to the bottom of the post and clicking on the Download tab.

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The Limits of Logic: Paradoxes and The Failure of Formal Logic

Indeed, if there is no formalization of logic as a whole, then there is no exact description of what logic is, for it is the very nature of an exact description that it implies a formalization. And if there is no exact description of logic, then there is no sound basis for supposing that there is such a thing as logic. (Church, 1934: p. 360)

Those who claim for themselves to judge the truth are bound to possess a criterion of truth. This criterion, then, either is without a judge’s approval or has been approved. But if it is without approval, whence comes it that it is trustworthy? For no matter of dispute is to be trusted without judging. And, if it has been approved, that which approves it, in turn, either has been approved or has not been approved, and so on ad infinitum. (Sextus Empiricus, 1935, 179)

1. Introduction: A Skeptical Challenge to Formal Logic

In this essay, I provide an overview of the major skeptical challenges to formal or symbolic logic, both so-called classical and non-classical. The idea of formal logic, a logic of “form,” will be outlined the next section, but the conclusion that I will draw is that formal logic fails to constitute some sort of formalization of correct reasoning. Indeed, we will see that the attempts by generations of formal logicians to achieve this have in fact undermined the foundations of the formal-logical endeavor. But this is not to abandon reasoning for silence or literature, for outside of Analytic philosophy, disciplines such as law, and even mathematics, get by fully adequately using informal reasoning methods, without the need for formalization, a needless burden for much of contemporary research. Rationality, and argumentation are much wider and richer fields than formal or symbolic logic per se (Hanna, 2006). And I should add in this introduction, that in striving to cover a wide field, I’m not primarily addressing this essay to the professional formal logicians who would almost certainly never abandon their position, whatever arguments are given, but rather to interested outsiders. Hence technicalities, as much as possible, will be kept to a minimum, and much use will be made of prior papers by others establishing relevant skeptical conclusions.

2. The Nature of Formal Logic

I’ll now give a brief outline of formal or mathematical logic. According to Dale Jacquette:

Logic is formal, and by itself has no content. It applies at most only indirectly to the world, as the formal theory of thoughts about and descriptions of the world. (Jacquette, 2002: p. 3)

Bertrand Russell wrote in his Introduction to Mathematical Philosophy that

logic (or mathematics) is concerned with forms, and is concerned with them only in the way of stating that they are always or sometimes true, with all the permutations of “always” and “sometimes” that may occur. (Russell, 1919: pp. 199–200; see also Chomsky, 1975; May, 1985; Moody, 1986).

More precisely, formal deductive logic is concerned with arguments made in formal languages, which have a precise structure, a syntax, and a semantics or interpretation. The syntax defines the relationship between signs in the language and is comprised of a vocabulary, rules of formation, axioms and rules of inference. Well-formed formulas (WFFs) are specified by giving a set of symbols and rules of formation, which specifies what sequences of symbols are meaningful well-formed formulas. The semantics of a language defines the relationships between expressions in the syntax and non-linguistic objects, which collectively give an interpretation of the language. Formal languages are interpreted by assigning objects (such as numbers, physical entities and so on) to the symbols and/or well-formed formulas. A formula has a model in a language L, if and only if there is an interpretation of the language which makes the formulae true. If X and Y are WFFs (well-formed formulas) of a language L, then Y is a logical consequence of X in the language L, if and only if Y is true in all models of L in which X is true. X is valid in the language if and only if X is true in all models of the language. While the concept of logical consequence is a semantic concept, the concept of proof is syntactical. A proof in the language is a set of well-formed formulas such that each formula is either an axiom of the language or derivable by means of the inferential rules of the system (Hunter, 1971; Kleene, 1967; Manaster, 1975). A formal logical system is consistent (proof-theoretically or syntactically) if and only if there is no well-formed formula X such that both X and not X (written “~X”) are provable in the system. Even at this point there are major philosophical problems, such as with the concept of logical consequence and the definition of validity, but we will pass over this (Etchemendy, 1990; McGee, 1990, 1992; Priest, 1995; Gómez-Torrente).

Languages or formal logical systems are said to be complete if for valid arguments there is a proof in the formal system. The language or formal logical system is sound if no invalid arguments are provable: only proofs of valid arguments can be constructed. Some formal logical systems are incomplete, but this is not a fatal defect in the system. Unsoundness is a fatal flaw because formal deductive logic requires that if the logical form of an argument is valid, then given that its premises are true, then its conclusions must by “logical necessity” be true as well, or so the story goes (Pap, 1962: pp. 94–106). We will see that formal logic, via the paradoxes, fails here.

Most, but far from all, logicians and mathematicians accept “classical logic,” which can be vaguely defined as the logic of Russell and Whitehead’s Principia Mathematica, together with related developments. The logic is not many-valued (having three or more values to be well formed formula, such as “true”, “false” and “indeterminate”), but instead has two and only two values “true” and “false” (all propositions are either true or false). Quantification — quantifiers are operators which indicate whether a statement is general (universal quantifier) or particular (existential quantifier)–occurs only over existent objects, not non-existing “objects” such as “the round square” or the “present King of the USA.” Most importantly, in classical logic it is a necessary and a sufficient condition for an argument to be valid, that in every possible world (or complete interpretation), if the premises of the argument are true, then the conclusion must also be true (Read, 1988: p. 31).

The classical account of validity means that all arguments with a necessarily true conclusion, and all arguments with a necessary false conclusion or (according to classical logic) inconsistent premises, are valid. On this account, a contradiction logically implies anything. Let “&” mean “and”, and “~” mean “not”, then

p & ~p, therefore q

is classically valid and can be proved to be so (Read, 1988). Relevant or relevance logics reject these inferences, because the logicians championing these positions believe that there should be a “logical content” connection between these premises and the conclusion of a deductively valid argument X → Y (where “→” means “implies”), where the “logical content” of the conclusion is contained in the premises (Anderson & Belnap, 1975; Routley et al., 1982; Iseminger, 1980). A movement in modern formal logic associated with relevant or relevance logics are paraconsistent logic (and mathematics) which holds that there are propositions for which X and ~X are both true, true contradictions. If there were true contradictions, and no “logical content” restrictions on logical implications, then p & ~p → q would be counter-modelled, for the premises could be true and the conclusion, an arbitrary proposition q, could be false (Priest, 2006). We will look at the significance of the so-called logico-semantical paradoxes and paraconsistency shortly in this context. It is time now to begin to examine the problems that modern formal logic and mathematics faces (Smith et al., 2023).

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Mr Nemo

Formerly Captain Nemo. A not-so-very-angry, but still unemployed, full-time philosopher-nobody.