The Limits of Logic: Paradoxes and The Failure of Formal Logic, #2.

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)

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

The essay that follows will be published here in four installments, two sections per installment; this, the second installment, contains sections 3 and 4.

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.

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)

3. Problems with Logical Validity

According to the classical account of validity, an argument is valid if its conclusion follows from (or: is a logical consequence of, or: is logically entailed by, or: is logically implied by) its premises, and invalid if it is possible for its premises to be true and its conclusion false (or: there is some interpretation in which all its premises are true and its conclusion false). It is a necessary condition of validity that the premises of an argument cannot be true while the conclusion is false, because valid arguments are supposed to go from truth to truth, not truth to falsity (Read, 1979). The logician Stephen Read developed an argument traditionally known as the “Pseudo-Scotus,” which prima facie shows the inconsistency of the concept of validity. Woodbridge and Armour-Garb have said that this paradox shows

not just a problem with the “classical account” of validity … [but] what it shows is that our very concept of validity (and, thus the language we use to express it is inconsistent—at least prima facie. (Woodbridge & Armour-Garb, 2008: p. 64, 2005; Jacquette, 1996; Read, 2001)

Consider the following argument:

A: 1=1

Therefore, argument A is invalid.

To paraphrase Read’s argument: suppose that argument A is valid, then A has a true premise and a false conclusion. By the classical account of validity, A is therefore invalid. Hence, if A is valid, then A is invalid. Therefore (by reductio ad absurdum) A is invalid. However, the premise 1=1 is a necessary truth. It is a principle of modal logic (the logic of notions such as necessity and possibility) that any proposition deduced from a necessarily true proposition, is itself necessarily true. Thus, it is necessarily true that A is invalid, and A has a necessarily true conclusion. However, on the classical account of validity (that is, the necessary truth of the conclusion of an argument is sufficient for the validity of an argument), A is valid. Therefore, A is invalid and valid: a contradiction (Read, 2001).

Another paradox can be generated with

B: This argument is valid, therefore, this argument is invalid. (Read, 1979: p. 267)

Along similar lines it can also be shown that from these two propositions:

(I) P

and

(II) There is no sound deduction of (I) from (I) and (II)

that there is a proof that P is not true, that is, a refutation of any proposition at all! (Windt, 1973).

4. Logical Skepticism and The Problem of Deduction

Some philosophers who have attempted to solve the problem of justifying induction have argued that induction is justified because of its success and that this proposal is not question-begging because deduction itself can only be justified by deduction. Stated very roughly, deductively valid arguments are those arguments where it is logically contradictory to assert the premises and deny the conclusion, that is, it is logically impossible for the conclusion to be false and the premises true. (We have seen that there are problems even with this, the classical account of validity). Susan Haack argued in her paper “The Justification of Deduction” (Haack, 1976), that deduction faces a parallel dilemma to that which Hume raised for induction: inductive justifications of deduction will be too weak, but deductive justifications will be circular. To attempt to show the validity of the rules of inference of a formal logical system in general, would be circular in the sense of using principles of inference for which the conclusion asserts the validity of the argument (Dummett, 1973; Keene, 1975, 1983; Oakley, 1976; Strom, 1977; Bickenbach, 1979; Gallois, 1993; Fox, 1999). As Cellucci puts it: “The trouble with the standard characterization of deductive inferences is that … the proof of the validity of the rules of deductive logic is circular” (Cellucci, 2006: p. 225).

George Couvalis has also argued that we cannot know logical and mathematical truths without using experience and induction. This makes induction epistemologically prior to deduction (Couvalis, 2004). Modernizing an argument found in the work of David Hume, Couvalis says:

To get to know a logical truth we must use an appropriately functioning entity such as a computer or a brain. Past philosophers talked about transparently infallible immediate apprehensions by the soul. But such views rely on dubious ontological assumptions and do not fit well with the fact that we sometimes make mistakes, even in simple cases. To the best of our knowledge, our minds can know logical or mathematical truths only if they at least supervene on a structured material entity, such as a brain or a computer. If it is to be reliable, this entity must function in an appropriate way. Because it is a structured material entity, it is liable to malfunction. Its malfunctions damage the power of the mental processes which it instantiates or which supervene on it. To be fairly sure it is reliable, we need ways of telling that it is functioning in an appropriate way. All such ways use inductive reasoning to reason to the conclusion that someone’s brain or computer is likely to function well from knowledge that that brain or computer seems to have functioned well in the past. This implies that our knowledge that we know that reasoning is logically valid or invalid, or that axioms are true, is dependent on the cogency of inductive reasoning. That is, if no inductive reasoning is cogent, we natural beings [nomologically] cannot know that we know any particular mathematical or logical statement to be true. (Couvalis, 2004: p. 34)

Couvalis goes on to argue that while many logicians and philosophers believe that axioms (statements for which no proof or argument is given) and rules of inference are self-evident, there are problems with this view that were recognized by two of the founding fathers of modern mathematical logic, Bertrand Russell and Alfred North Whitehead. Russell and Whitehead said:

[S]elf-evidence is never more than a part of a reason for accepting an axiom, and is never indispensable. The reason for accepting an axiom, as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by which these propositions could be true and the axiom were false, and nothing which is probably false can be deduced from it. … In formal logic, the element of doubt is less than in most sciences, but it is not absent, as appears from the fact that the paradoxes followed from premises which were not previously known to require limitation. (Russell & Whitehead, 1927: p. 59)

If Couvalis is right—and his arguments strike me as being correct—then there is a dilemma. Either some inductive reasoning must be accepted as valid or we should be skeptical about the justification of our knowledge of logical and mathematical knowledge. Couvalis does not deal with the resolution of this dilemma in the essay I’ve cited. Indications are that he is not a skeptic about logical and mathematical knowledge. But that will require a solution to the problem of justifying induction, which most philosophers grant is unsolved. Hence, deduction requires a justification as much as induction, and this problem is no closer to a solution than the solution of the problem of induction (Cellucci, 2006). However, there are other reasons for supposing that deduction can fail, producing unsoundness, indicted by the logico-semantic paradoxes.

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