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
3. Problems with Logical Validity
4. Logical Skepticism and The Problem of Deduction
5. The Logico-Semantical Paradoxes
7. The Refutation of Formal Logic
8. Conclusion: Bankruptcy, Non-Formalism, Limits, and Humility
The essay that follows has been published here in four installments, two sections per installment; this, the fourth and final installment, contains sections 7 and 8.
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.
7. The Refutation of Formal Logic
Modern developments in formal logic have resulted in an almost unbounded power to construct exotic counter-examples and counter-models to refute once cherished logical principles. Paraconsistent logician Chris Mortensen has said on this point:
One of the directions of recent logical research has been into semantical conditions under which various propositions hold and fail. One of the upshots has been a growing body of information about how to construct models to refute more and more propositions. It is, for example, no news that countermodels can be constructed to large numbers of theorems of the very natural modal logic S5, on which David Lewis’ modal realism is based. It is also a straightforward matter to construct countermodels to the laws of excluded middle and noncontradiction. Recent work by Errol Martin has even shown how to construct countermodels to every instance of A → A. In light of these kinds of results, it seems to me that it would be a bold claim that there is any proposition that cannot be made to come out false in some structure. (Mortensen, 1981: p. 57, 1989)
Countermodels to every instance of A → A? Mortensen goes on to argue that given Martin’s counter-modelling of A → A (in a weak propositional calculus) one can in principle doubt the seeming logical necessity of statements such as “If Smith is a bachelor then Smith is an unmarried man.” That statement presupposes that “If Smith is a bachelor then Smith is a bachelor” is also necessary, which is of course a substitution instance of A → A (Mortensen, 1989: p. 329). On this basis, Mortensen says, we can conceive how our mathematics could be false:
[It] seems to me that the intuitive solidity of mathematics rests on the same foundation. Short, quite obvious inferences in mathematics often derive, like the previous bachelor case, from some definitional decision to use terms interchangeably applied to A → A, (or to (A&B) → A or A → (A v B)). Mathematical connections established by longer chains of reasonings appealing to more complex deductive principles are to that extent less evidently necessary. I am not suggesting here that it is easy to understand how standard mathematics might have been false. But then we should beware of projecting the limitations of our imaginations onto the world. (Mortensen,1989: p. 329)
Routley and Meyer have constructed relevant logic semantics where any formula of the form x → y may fail (Martin, 1978; Martin & Meyer, 1982; Routley et al., 1982). Priest agrees that the countermodels can be constructed to any arbitrary formula (Priest, 1992). According to Priest:
[The] prime notion of logic is inference; and valid (deductive) inferences are expressed by statements of entailment, α → β, (that α entails that β). Hence in a logically impossible world we should expect statements of this form to take values other than the correct ones. Is there a limit to the value that such a conditional might take? I do not see why. Just as we can imagine a world where the laws of physics are arbitrarily different, indeed, an anomalous world where there are no such laws; so we can imagine worlds where the laws of logic are arbitrarily different, indeed an anomalous world where there are no such laws. (Priest, 1992: pp. 292-293)
Relatedly, the late Richard Sylvan (formerly Richard Routley), developed a theory of items based upon the ideas of logician Alexius Meinong (Routley, 1980). Items are everything that can be the object of thought, and things which cannot, such as: if I is defined as I = that object which is not an item, then it is an item. One can thus speculate about a prime number p between 11 and 13 or even an infinite number of prime numbers between 11 and 13, even though standard Peano arithmetic has no such p (Sylvan, nd). This position as stated has some logical difficulties including a problem with absolute inconsistency (i.e., it allows the derivation that 1=0), which Priest has addressed and which we need not discuss here (Priest, 2005). Given all this logical freedom it is seemingly inevitable that counter-examples to one’s most prized principles and counter-arguments to beloved arguments would multiply “in a way that makes the breeding habits of rabbits look like family planning” (Priest, 1987: 145-146).
8. Conclusion: Bankruptcy, Non-Formalism, Limits, and Humility
The conclusion reached here is the same one I reached 40 years ago: formal logic is bankrupt: there are no “laws of form” (Smith, 1984). The same conclusion was reached by the late Australian philosopher David Stove. Stove said in his book The Rationality of Induction:
There are no logical forms, above a low level of generality … There are few or no logical forms, above a low level of generality, of which every instance is valid: nearly every such supposed form has invalid cases or paradoxical cases. The natural conclusion to draw is that formal logic is a myth and that over validity, as well over invalidity, forms do not rule: cases do. (Stove, 1986: p. 127)
More recently, Hofweber has considered that there are counter-examples to all the inference rules, so the rules are not strictly valid, but are only valid over some range (Hofweber, 2007). The idea that formal logic has its limitations has been expressed before, of course (Rohatyn, 1974; Kekes, 1982; Devlin, 1997), but the full skeptical ramifications have seldom been embraced. Clearly, if the most precise area of human knowledge has numerous “black holes” of reason, we can expect paradoxes a-plenty in every other field, and that is exactly what we find. The existence of these unsolved logical and semantic paradoxes challenges the rationality of science, since science depends upon mathematics, and mathematics, being a so-called deductive science, crucially depends upon logic. But, if the foundations of logic are insecure, so too will be the foundations of mathematics. At this point mathematicians, who were probably Platonists before confronting the paradoxes, are likely to become pragmatists, saying that their concern is with merely making deductions from axioms, without concern about the ultimate truth and justification of them. The skeptic would be pleased to accept this, replying that if this is so, mathematics is not epistemologically different from the rest of human knowledge, where at the end of the day, pragmatism rules.
In general, we have seen that the logico-semantical paradoxes remain unsolved, even by the paraconsistency school which has taken the paradoxical sentences to be “true contradictions.” And, even if the paraconsistent school is right about the limits of classical logic, their own position faces crippling objections, namely that they do not escape all the paradoxes, so that they therefore fail to produce a satisfactory general response to the logical challenge of the paradoxes, which the once radical move of positing “true contradictions” was supposed to solve. For the most precise of all sciences, this is indeed a major epistemological king hit. It is ironic, that increased technical sophistication in formal logic has led to a type of process of self-undermining, where all former “logical truths” and once taken-for-granted principles, such as even modus ponens, face counter-examples (McGee, 1985). An epistemological skeptic would see this as a major objection to the rationality of the discipline itself, and a major epistemological crisis that seemingly is intractable, at least from the perspective of Analytic philosophy.
However, as an alternative, the arguments given here can be taken to show the limits of the Analytical philosophical framework, and the need to move to a non-formalist approach to logic, as has been previously explored by Robert Hanna (Hanna, 2006), and today, by many others in the informal logic schools. Moreover and finally, this situation makes a strong case for philosophical limitationism and epistemic humility, according to which as Rae Langton puts it, “[t]here are inevitable constraints on what we can know, inevitable limits on what we can become acquainted with” (Langton, 1998: p. 2).
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