A new solution to the liar paradox that takes it seriously, and does not dismiss it as a confusion that no disciplined thinker could fall into, is always welcome, even if it is an old solution. It is the aim of the volume under review to present, and examine critically, such a new-old solution.
If every statement, in addition to what it asserts explicitly, asserts implicitly its own truth, then the traditional liar statement, ‘This statement is not true’, asserts a simple contradiction, and is unobjectionably false. This is the solution to the liar paradox customarily attributed to Jean Buridan (c.1295–1358). Unity, Truth and the Liar draws attention to the subtler, and comparatively unknown, solution advanced somewhat earlier by his contemporary Thomas Bradwardine (c.1290–1349), and to some other medieval solutions. Part I consists of a zestful presentation by Stephen Read of Bradwardine's theory in modern dress, eleven critical commentaries (including one, by Klima, that is mostly about Buridan), and a firm, though not always convincing, response from Read. Part II, about which I shall say nothing more, consists of two historical papers on other medieval treatments of insolubilia, and includes a critical Latin edition (but no translation) of three anonymous texts reportedly inspired by William Heytesbury (a.1313–c.1373).
Bradwardine's suggestion, informally stated, is that a sentence is true if and only if everything that it says (or signifies) obtains; or ‘[t]hings must be not only how x says they are, but however x says they are, in order that x be true’ (pp. 8f.). Writing ‘:’ for the quasi-relation of saying that or signification, and ‘⇔’ for strict (logical) equivalence, Read symbolizes Bradwardine's definition of truth as
where the possible substituends for ‘x’ are names of (meaningful, univocal) sentences, and the possible substituends for ‘p’ are propositions, not names of propositions. Let L be any sentence that says of itself that it is untrue; that is, L:¬TL. If L says something more, let that something more be summed up in the proposition q. Then the bi-implication TL⇔¬TL∧q follows from (A), and hence ¬TL and ¬q also follow from (A). In short, the untruth of L can be proved, which means that L can be proved. But we cannot thereby prove TL, that L is true. (To see this, interpret x:p to hold for every sentence x and every proposition p, and Tx to hold for no x.) The sentence L is not paradoxical. With the additional postulate that signification is closed under logical implication (logical equivalence would actually suffice), Bradwardine goes on to conclude that L: TL too. Buridan assumed that x:Tx for every sentence x. For self-denying sentences such as L, Bradwardine derives the same result from his definition (A) and his closure postulate.
According to Read, Bradwardine's definition (A) improves on Tarski's T-scheme; that is,
where ‘x’ is replaced by a structural-descriptive name of a sentence, and ‘p’ is replaced by the sentence whose name replaces ‘x’. (A) is ‘a logical equivalence, whereas (T) was for Tarski merely a material equivalence’ (p. 7); and it is ‘an analysis of truth’, though not ‘the final analysis of truth’ (since signification remains unanalysed), whilst ‘(T) is merely, for Tarski, a condition of material adequacy’ (p. 7; p. 210). For elementary languages Tarski did, however, propose a method of defining truth, and the definitions that it yields are logical (not merely material) equivalences. Since a different definition is needed for each language, this method does not yield ‘a philosophical analysis, or explication’ of truth (p. 7), but nothing in the book explains why we need such an analysis, or why Tarski's definitions are not enough. It should be emphasized too that in the context in which Tarski proposed to use (T), the implicit logical consequences of a sentence are as necessary for its truth as is the sentence itself. If a sentence signifies all and only its consequences, then Tarski can easily endorse the spirit (though hardly the letter) of Bradwardine's definition (A). Read's complaint, near the end of Part I, that ‘[w]hat the T-scheme fails to recognise is that our utterances may have implicit consequences which are not fully explicit’ (p. 222), is expatiated on by Dutilh Novaes, but it is neither accurate nor pertinent. What is essential to Bradwardine's solution to the liar paradox is that what a sentence says, and what its consequences are, are not always the same. Only Restall's contribution undertakes to say something informative about this difference.
Read nowhere states the principle (which is singled out by Terence Parsons on p. 139, and by Serény on pp. 162 and 171) that every sentence signifies what it states explicitly:
where ‘x’ and ‘p’ are replaced in the same way as in (T). On occasion (pp. 6, 9), however, he endorses instances of (E), and he recognizes that some assumption is needed to guarantee that signification is not an empty relation, and to make a connection with more traditional forms of the liar paradox. Given the worries expressed about applying (T) to sentences containing indexicals (Read, p. 4; Goldstein, p. 79), the principle (E) may, however, not suffice to show that there exists a sentence L that signifies its own untruth. It is rather extraordinary that, except for one passing reference by Armour-Garb (p. 37), the more robust liar sentences obtained by adapting Gödel's diagonalization procedure to everyday language have no place in this book. Quine's version, ‘“yields a falsehood when appended to its own quotation” yields a falsehood when appended to its own quotation’, may be cumbersome, but (E) and (T) are easily applied to it and, if (E) is accepted, it therefore constitutes an uncontroversial example of a sentence L for which L:¬TL. Goldstein's assertion that ‘Tarski … ignored indexicals, and, insofar as he was interested in providing a solution to the Liar Paradox, he did so at his peril, since indexicality is so intimately implicated in Liar paradoxicality’ (p. 67) is almost the reverse of the truth. For a serious treatment of the liar paradox, those who ignore diagonalization do so at their peril.
Quine's liar calls into question several opinions that are advocated: the doctrine that Goldstein (p. 67) shares with Mills that ‘[t]he Liar paradox gets off the ground only when the unit of assessment is a particular use of a particular sentence-token’ (p. 125); Slater's doctrine that ‘there is no such thing as syntactic self-reference’ (p. 191); and the ‘cassationist position of claiming that [the] Liar says nothing’ (Goldstein, p. 73). It casts doubt too on the soundness of Read's response to Serény's awkward objections concerning bivalence (pp. 179–183), and to the problem of ‘indirect revenge’ (Armour–Garb (pp. 27–37), that ‘[i]t is crucial to Bradwardine's solution to the Liar and other paradoxes that we distinguish the false and self-referential occurrences of ‘¬TL’ from the true and non-self-referential ones’ (p. 220).
It must be added that neither Read nor any of his opponents (even Woleński, who is Tarski's main defender here) gives an adequate report of how the liar sentences fare in Tarski's hands. The usual view, that the ban on semantic closure makes ‘the semantic paradoxes inexpressible’ (Read, p. 9) or that ‘Tarski's orthodoxy takes a Liar string to be ungrammatical expressing no proposition’ (Sagüillo, p. 156), makes little sense if Tarski's aim was to provide (within a fragment of set theory) an explicit definition of the term ‘true’. Explicitly defined terms, after all, are always eliminable.
Read is at pains to distance Bradwardine's theory not only from Tarski's, but also from the assumption of Buridan (and of Albert of Saxony) that every sentence signifies its own truth (p. 14). The reason is that if x:Tx holds universally then every significant sentence is a simple truth-teller; that is, a sentence V that signifies TV and nothing more (except the consequences of TV). For if q is any proposition signified by x, then by (A), Tx ⇒ q. This implies that x signifies nothing more than Tx and its consequences. I doubt that this conclusion is easily avoided within Bradwardine's theory, but Read's anxiety that ‘John and Albert … have no adequate theory of truth at all’ (p. 16) is hyperbolic, as Parsons observes (pp. 132f.). Although ‘no sentence can be shown using (A), … to be true’, it does not follow that it cannot be shown of some sentences that they are untrue (which implies, if bivalence can be assumed, that their negations are true). Let x signify some ordinary factual proposition such as the proposition that snow is white. Then although x says nothing more than Tx, it follows from Tx that snow is white; and hence, if snow is not white, x is not true.
Disaster or not, this result illustrates the risks inherent in talking about ‘the Truth-teller’, as if there were only one sentence V such that V:TV, and in claiming that it says nothing (Read, p. 15, amended in note 14 on p. 213; Goldstein, p. 68). A sentence that says of itself that it is true does not say nothing; a sentence that does not say of itself that it is true does say nothing.
