The Limits of Logic: Paradoxes and The Failure of Formal Logic, #3.

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 third installment, contains sections 5 and 6.

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.

5. The Logico-Semantical Paradoxes

Logico-semantical paradoxes are almost as old as Western philosophy (Martin, 1970). The liar paradox of Epimenides the Cretan arose from the statement “I am lying,” which is true if it is false and false if it is true. A modern variant to consider is:

(L) This sentence is false.

There are “strengthened paradoxes,” a sentence that says of itself that it is not true and variants of this, such as a sentence that says of itself that it is not definitely true (Mackie, 1973; Goldstein & Goddard, 1980; Parsons, 1984; McGee, 1991; Heck, 1993; Goldstein, 1994; Mills, 1995; Sorensen, 1998; Soamer, 1991; Priest, 2000a, 2000b; Greenough, 2001; Bueno & Colyvan, 2003).

The 20^th century also saw the presentation of a number of other surprising paradoxes. Löb’s paradox involves considering a sentence A which is true if and only if it implies B:

(L1) A ↔ (A→B).

Assume then:

(L2) A,

then

(L3) A → B,

and

(L4) B. Withdraw A,

so:

(L5) A → B,

i.e.,

(L6) A,

therefore,

(L7) B. (Löb, 1955; Van Benthem, 1978: p. 50).

Closely related to this paradox is Curry’s paradox which also proves an arbitrary proposition by generally accepted (that is, until the paradox was uncovered), logical principles (Curry, 1942; Irvine, 1992). This paradox does not involve negation, and can be formulated in set theoretic, property, semantic, and validity versions (Shapiro, 2013; Shapiro & Beall, 2018). An informal argument is as follows (Shapiro & Beall, 2018). Consider a sentence, “If S is true, then F.” Then:

(C1) Given the assumption that S is true, then if S is true, then F.

And as well:

(C2) Given the assumption that S is true, then it is the case that S is true.

Now supposing that S is true, using modus ponens on the above conditional and antecedent gives:

(C3) Given the assumption that S is true, then F.

By conditional proof, the conditional can be affirmed, and assuming the antecedent yields:

(C4) If S is true, then F.

Since (C4) is S, then:

(C5) S is true.

By modus ponens from (C4) and (C5), then:

(C6) F, an arbitrary statement.

Better known than these paradoxes are the paradoxes of set theory such as Russell’s paradox. Consider the set of all sets which are not members of themselves. Is this set a member of itself? If it is, then it is not. If it is not, then it is (Moorecroft, 1993). There are a number of other set-theoretical paradoxes such as the Burali-Forti paradox and Cantor’s paradox, which need not be discussed here (Weir, 1998). Typically the set theoretical paradoxes have been dealt with by modifying our naïve conception of a set through various formal set theories. Ingenious as these theories have been it would appear from a survey of the critical literature that a final resolution of these difficulties has not been accomplished (Weir, 1998).

For example, Grim argued for some time that the set of all truths, or, all true statements, is in conflict with Cantor’s power set theorem (Grim, 1984). The power set is the set of all subsets of a given set, and if a set S has n elements, then the Power set PS has 2ⁿ elements (Suppes, 1960: p. 46-48). If we take the intuitive idea of a “set” to be a “collection of entities of any sort” (Suppes, 1960: p. 1; Wang, 1974: p. 181), then we should be able to deal meaningfully with both the set of all apples, and the set of all true statements. Of course, in the light of the set-theoretical paradoxes, logicians have restricted the objects of sets containing special set constituents, as in the set of all sets, and have made a distinction between sets and classes. But set theory should not yield paradoxes merely from considering elements such as sentences, which are ontologically distinct from sets. However, for the set of all truths, for each subset of this set, there will be a truth, and thus a corresponding statement, so there will be at least as many truths as there are elements of the power set, contrary to the power set theorem—or in some systems, axiom (Suppes, 1960: p. 46). Thus, a counter-example is presented to a provable theorem. Reflecting on this result and other paradoxes of totalities, Rescher and Grim state:

Set theory was born in paradox, was shaped by paradox, and continues to carry the threat of paradox into its current adolescence. Properly understood … the threat of contradiction is not merely formal and is not to be evaded by merely formal techniques. The fact that there can be no set of all non-self-membered sets might be shrugged aside as a minor logical surprise. Beyond Russell’s paradoxical set, however, there are serious philosophical difficulties of coherently conceptualising a set of all things, the realm of unrestricted quantification (or even the sense of restricted quantification), the totality of all events, all facts, all propositions, or all that is true. Sets are structurally incapable of handling any of these. (Rescher & Grim, 2011: p. 6)

Relatedly, it can be noted that the logician Wilfred Hodges once published a paper, “An Editor Recalls Some Hopeless Papers” (Hodges, 1998). This related to “crank” critiques of Cantor’s diagonal argument. He wrote that almost all the “cranks” attacked the matrix representation of the sequence of decimal real numbers, but “none of the authors showed any knowledge of Cantor’s theorem about the cardinalities of power sets” (Hodges, 1998: p. 2). Well, now they have the Rescher-Grim paradox.

From all this, Benson Mates concluded that

although each possible point of contact is identified by someone as the source of the difficulty, each is also exonerated by the great majority; and consequently, no purported solution ever comes close to general acceptance. (Mates, 1981: p. 5)

Mates believed that our fundamental concepts such as set, truth etc. may be radically defective “in the sense, that, the clearer we get about them, the clearer it becomes that they lead to contradiction and must be repaired, if possible, or, failing that, replaced” (Mates, 1981, 57).Heck has also concluded that:

there can be no consistent resolution of the semantic paradoxes that does not involve abandoning truth-theoretic principles that should be every bit dear to our hearts as the T-schema once was. And that leads me … to be tempted to conclude that there can be no truly satisfying, consistent resolution of the Liar paradox. (Heck, 2012, 39)

A subject dear to the hearts of popular science writers in this field is that of Gödel’s incompleteness theorem (Franzen, 2005). The proof of this theorem was published by the logician Kurt Gödel in 1931. The proof showed the existence of formally undecidable propositions in certain formal systems of arithmetic. One such system of arithmetic is Peano arithmetic which has as its axioms: (1) 0 is a number; (2) the successor of any number is a number; (3) no two numbers have the same successor; (4) 0 is not the successor of any number and (5) if a predicate P is true of 0 (i.e. P(0) is true), and if it is true that P (n)→P(n+1), then P is true of all numbers. The formal theory of Peano arithmetic PA is open to Gödel’s first incompleteness theorem. This states that in the formal theory PA there is a sentence G of PA such that if PA is consistent, neither G nor ~G can be proved in the formal theory. There are various ways that this theorem can be proved, with associated logical and philosophical issues (Butrick, 1965). One method involves use of a “diagonal” argument arguably similar to the liar paradox (Martin, 1977; Humphries, 1979; Johnstone, 1981). Let the Gödel sentence be the sentence:

(G) This sentence is not provable from the axioms of Peano arithmetic.

Thus, G is true, but is unprovable in PA. Suppose then, that G is not true. Then given the statement of G’s contents, then G must be provable in PA. Assume that the axioms of PA are true and that the system is logically sound. Then statements provable in PA must be true. G is provable in PA. Therefore, G is true. However, the statement that G is true contradicts our initial assumption that G is not true. Therefore, G is true. If G is true, then by the Tarski principle, True (P) → P, what G says, holds, G is not provable (Barwise & Etchemendy, 2007).

There are a number of interesting papers (and chapters in books), most of them appropriately peer-reviewed, reporting some challenging ramifications of Gödel’s incompleteness theorem, and its method of proof. For example, Martin (Martin, 1977) shows that a diagonal statement: (TS) “Nothing in the discourse D bears a relation R to exactly the things in the discourse D that don’t bear R to themselves,” underlies a number of syntactical and semantical paradoxes, as well as some important results in metalogic such as Gödel’s theorem, Cantor’s power-set theorem, Tarski’s theorem, and every instance of the diagonal argument (Martin, 1977: p. 455). The intriguing philosophical question is how to distinguish between “good” (i.e. non-paradoxical) and “bad” (i.e. paradoxical) uses of (TS). Logical skeptics maintain that there is no method of distinguishing the “good” from the “bad” uses of (TS), so all uses are therefore regarded as problematic (Smith, 1988).

Ketland has proved that there is a sentence K, which “says of itself that it is not a true sentence” (Ketland, 2000: p. 1), such that K is provable in the system PA(S). PA is standard first-order Peano arithmetic in a formal language L, the first-order language of arithmetic. PA(S) is a semantical extension of PA resulting from adding a primitive satisfaction predicate Sat_L(x, y). By way of explanation: an object or sequence of objects satisfies a predicate if the predicate “holds” (is “true”) of the object or sequence of objects. For example, the object “electron” satisfies “does not simultaneously have definite position and momentum values,” because according to mainstream quantum theory the electron does not simultaneously have definite position and momentum values. In formal semantics, the satisfaction concept is used to define a formal concept of systems-relative truth (Kaye, 1991; Ketland, 1999). Therefore, adding a primitive satisfaction predicate to PA is unobjectionable. However, Ketland shows that K, the strengthened liar formula that says of itself that it is not true, is provable in PA(S).

Graham Priest has also produced an argument demonstrating an alleged surprising consequence of Gödel’s theorem (Priest, 2006a; Berto, 2009). He states Gödel’s theorem as follows: let T be a theory which can represent all recursive functions and where the proof relation of T is recursive. To explain: recursive functions, are functions that can be defined from the constant, successor, and projection functions by composition of functions and recursive definitions. A recursive definition applies to the first term of a series and then for a successor term, through the predecessor of that term. To require that the proof theory of T be recursive is to require a proof be effectively recognizable, a reasonable requirement. Priest rightly observes that it is essential to the very concept of a proof, that a proof should be effectively recognizable, for the very point of a proof is to give us a way of determining whether something is true or not. Given all this, Priest states Gödel’s (first) incompleteness theorem as follows: if T is consistent then there is a formula ø, Gödel’s, such that (1) ø is not provable in T and (2) if the axioms and rules of inference of T are intuitively correct, then ø can be shown to be true by an intuitively correct argument. An “intuitively correct argument” refers to the type of non-formalized arguments used by mathematicians in their daily work. These methods of informal proof are generally accepted to be capable of formalization. Thus, the naïve notion of proof satisfies the conditions of Gödel’s theorem.

Priest shows that the assumption of the consistency of the naïve notion of proof leads to contradiction. Let T be the formalization of the naïve theory of proof. T satisfies the conditions of Gödel’s theorem. Thus, if T is consistent then there is a sentence ø which is not provable in T, but which can be shown to be true in T by a naïve proof. But the naïve notion of proof is just T, so ø is provable in T after all! (Priest, 2006 a, 39). Priest then concludes that the

only way out of the problem, other than to accept the contradiction, and thus dialetheism [i.e., the idea that there are true contradictions] anyway, is to accept the inconsistency of naïve proof. (Priest, 2006a: p. 41)

Priest’s argument was first published in a peer-reviewed journal (Priest, 1979b) and has been criticized, but defended by him (Priest, 1984). As Priest notes, the Gödel sentence is a paradoxical sentence. Informally, it is “This sentence is not provably true.” Assume that the sentence is false. Then the sentence is provably true. Therefore, it is true. By reductio ad absurdum it is therefore true. This, however, is a proof (informally). Thus, the Gödel sentence is provably true. But if the Gödel sentence is provably true, then it is not provably true, which is contradictory! Priest speculates at this point that naïve proof procedures may therefore be essentially inconsistent because the theory is capable of giving its own semantics (semantic closure) so that the semantical paradoxes will be provable in the theory. Priest concludes that this vindicates the Kant/Hegel thesis that Reason is inherently inconsistent (Kallestrup, 2007).

Priest could be correct about Reason being inherently inconsistent. He himself does not draw a skeptical conclusion from this because he believes that paraconsistent logic can control the contradictions. The problematic contradictions are not provable falsehoods or necessarily false propositions, but true contradictions. So, Reason, after all is saved. But is it? Consider Priest’s argument from Gödel’s theorem to start with. Gödel’s theorem shows that T, the formalized theory of naïve proof (intuitive mathematical proof) is inconsistent. But note that the proof of Gödel’s theorem given earlier, and quoted from Priest’s own presentation, presupposed that T is consistent. But by Priest’s theorem, T is inconsistent, that is, it is not the case that T is consistent. Therefore, it is not the case that Gödel’s theorem is correct. If Gödel’s theorem is incorrect then Priest’s theorem fails because it presupposes the correctness of Gödel’s theorem, so that the theorem seems to be self-undermining. This does not rehabilitate classical logic, because it was classical consistency assumptions which generated this logical spiral in the first place. There is thus something intrinsically problematic with the Gödel sentence. At this point we need to look more closely at the paraconsistent approach to the logico-semantical paradoxes.

6. Paraconsistency

The paraconsistent criticism of classical logic has led to some interesting developments in metaphysics (Priest, 1999, 2000a, 2006 b; Beall, 2000; Beall & Colyvan, 2001; Kabay, 2006; Baten et al., 2000; Carnielli et al., 2002) and formally useful work in paraconsistent logic and mathematics (Priest, 1997; Mortensen, 1987, 1995), especially for automated reasoning and information processing with computer systems in which a data base contains inconsistent data (Besnard & Hunter, 1998). Needless to say, many of these useful formal developments would still be possible without accepting that there are true contradictions: all that is needed at a minimum for automated reasoning with inconsistent data is to prevent triviality occurring. So why then believe that there are true contradictions?

Priest and others generally believe that the logico-semantical paradoxes present the best case for dialetheism. The classical solutions to the paradoxes all face difficulties and something of a logician’s task of Hercules:

For every single argument they must locate a premise that is untrue, or a step that is invalid. Of course, choosing a point at which to break each argument is not difficult: we can just choose one at random. The problem is to justify the choice. It is my contention that no choice has been satisfactorily justified, and moreover, that no choice can be. (Priest, 1987: p. 11)

Presumably these remarks are made about the logico-semantical paradoxes and not all “logical/metaphysical” paradoxes dear to the hearts of philosophical logicians. Consider, for example, the ancient Sorites paradox, or paradox of the heap, associated with Eubulides of Miletus. This paradox can be stated as follows:

One thousand stones, suitably arranged, might form a heap. If we remove a single stone from a heap of stones we still have a heap; at no point will the removal of just one stone make sufficient difference to transform a heap into something which is not a heap. But, if this is so, we still have a heap, even when we have removed the last stone composing our original structure. (Burgess, 1990: p. 417)

The argument need not use the concept of a “heap” but can still be restated with any number of vague predicates. Thus, 0 is a small number. If n is a small number, then n+1 is a small number. Therefore, by the principle of mathematical induction, all numbers are small (Priest, 1979a: pp. 74-75). As Priest has said: “Mathematical induction is shown to be an invalid form of argument when fuzzy properties are involved” (Priest, 1979a: p. 75). The Sorites paradox can be generated by finitely many applications of modus ponens (if p then q, p, therefore q) or by use of the substitutivity of identicals. So, according to Priest’s claim about mathematical induction, these logical principles too are invalid when fuzzy/vague properties are considered (Dummett, 1975; Lakoff, 1973; Hanfling, 2001; Keefe & Smith, 1999). If not, why not?

The mathematician Florentin Smarandache has produced a number of quantum mechanics sorites paradoxes (Smarandache, 2005). For example, there is not a clear dichotomy between matter which on the large-scale behaves deterministically, and matter which is subject to Heisenberg’s indeterminacy principle (variables specifying the position and momentum of subatomic particles cannot simultaneously both take definitive values). In general, philosophers have paid insufficient attention to the Smarandache paradoxes. No matter: vagueness has already “become a philosopher’s nightmare” (Napoli, 1985: p. 115; Russell, 1923; Schwartz & Throop, 1991). In a survey of solutions to the sorites paradox Richard De Witt says that “all the proposals offered to date as ways of blocking the paradox are seriously deficient” (DeWitt, 1992: pp. 93-94). Priest is also of the opinion that “no extant solution to the Sorites paradox works” (Priest, 1991: p. 296)—and that presumably includes a paraconsistent solution. If one is to postulate that situations of vagueness involve true contradictions, then much of the observable world would be contradictory, a position which Priest does not embrace (Priest, 2006a).

The thesis that taking the paradoxes as being sound arguments delivering a true conclusion (a true contradiction) constitutes a unified and non-ad hoc solution to the logico-semantical paradoxes, is also contestable (Everett, 1993, 1994; Mares, 2000; Beall, 2001; Bromand, 2002). Curry’s paradox and Löb’s paradox, for example, do not have a “true contradiction” as a conclusion, but rather an arbitrary proposition. Correspondingly, some have argued that paraconsistent logics still face the Curry/Löb paradoxes (Everett, 1996). Many paraconsistent logics are reduced to triviality from Curry-style paradoxes (Slaney, 1989; Restall, 2007). One response to this has been to reject the principle of absorption:

(AB) (A → (A → B)) → (A → B),

read as “If A implies A implies B, then A implies B.” This has involved the alleged construction of a countermodel to (AB). Even so, Geach has shown how a sentence A such that A → (A → B), where B is an arbitrary statement can be constructed (Geach, 1955).

Priest’s response to Curry’s paradox, in his system LP, is to reject the general validity of modus ponens (Carrara & Martino, 2011: p. 200). Modus ponens fails, Priest proposes, when dialetheic sentences that are both true and false, occur. It has been argued in reply that this denial is ad hoc, and as well, Curry’s paradox can be derived without modus ponens, using only the naïve notion of deducibility, which Priest accepts (Carrara & Martino, 2011: p. 203-205). Hence Priest’s dialetheism does not avoid trivialism (Carrara et al., 2010).

Armour-Garb and Woodbridge have constructed pathological sentences that defy classical and paraconsistent responses, they alleged (Armour-Garb & Woodbridge, 2006). An example is:

(C1) (C2) is true→ ‘Everything is true’

(C2) (C1) is true→ ‘Everything is true’

The “open pair “ (in the above case “Curried open pair”) has a simpler form:

(1) (2) is false

(2) (1) is false,

which generates a pathological oscillation. Amour-Garb and Woodbridge argue, convincingly in my opinion, that both consistent solutions and paraconsistent solutions to the “open pair” paradox fail. Debate continues on this issue.

It is to be expected that the ultimate result of all this logical research would be ruin. To begin, the dialetheist position, that some contradictions are true, represented as “D is true” where D is a dialetheia (a true contradiction) turns out on Priest’s account of paraconsistency, LP, to be a dialetheia itself, that is, true and false. Thus, the very statement of the position of strong paraconsistency (dialetheism) is contradictory. It has been shown that the principle of non-contradiction in Priest’s system, LP, is both valid and invalid (Heald, 2016). Priest accepts this result (Priest, 1979b). One could argue, as Manuel Bremer does in his excellent Lectures on Paraconsistent Logic, that it is a minimum condition for the assertability of a thesis that it should be true only (Bremer, 2004: p. 205); no doubt dialetheists would counter this by arguing that it begs the question against them because after all they have asserted their thesis, it is open to criticism (e.g. the production of “hypercontradictions” or triviality) and so on. However, arguably, if dialetheism is true and false, then this position is to be rejected in favor of a position which offers a non-ad hoc unified solution to the logico-semantical paradoxes and is arguably true only. As we have seen, paraconsistency fails to provide a simple unified solution to the paradoxes in any case. Classical logic is not such position because of the arguments outlined by its paraconsistent critics. Their critique of classical logic holds, even if, which we believe is the case, paraconsistent logic has its own internally destructive problems.

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