The liar paradox is perhaps the most perplexing antimony to ever confront reason. Suppose there is a sentence that says “this sentence is not true”. If the sentence is true, then what it asserts must be true and since it asserts that it is false, it must be false. Therefore, if it is true, it is false. But, since it is false, then what it says must be false and what it says is that it is false. So, it must be false that it is false which means it must be true. Therefore, if it is false, then it is true. This passage of reasoning entails that the liar sentence is true if and only if (iff) it is false. This conclusion is an absurd contradiction. Tarski and Kripke each propose their own solutions to the liar which I shall briefly outline. I argue that one of Kripke’s criticisms of Tarski’s approach is successful because it shows Tarski’s solution suffers from a subsequent pair of liar sentences. However, whilst Kripke’s solution manages to avoid this problem, it succeeds only at the peril of suffering its own revenge paradox, namely, the strengthened liar sentence.
Tarski shows that a truth-predicate cannot exist in a semantically closed language. He considers the necessary components to generate a liar sentence which we shall here consider. We begin with a formal language “𝐿0” whose structure is exactly specified. (1) This means that we have characterised all sentences that are considered meaningful and the conditions under which we may assert them. (2) We can only assert the theorems of 𝐿0 because they constitute its true sentences. (3) We want to express the truth of these theorems so we need to introduce a truth-predicate to 𝐿0, let us call it “𝑇𝑟1”. New rules are needed so that the structure of 𝐿0 remains exactly specified and so that 𝑇𝑟1 behaves properly. Tarski appeals to a basic conception of truth to describe how 𝑇𝑟1 should work. Consider the sentence “snow is white”; the basic conception of truth seems to imply that the sentence is true iff snow is white. (4) We generalise this as: 𝑋 is true iff 𝑝, where “𝑝” is an arbitrary sentence and “𝑋” is the name of that sentence. (5) This is known as the T-schema (6) and it governs the behaviour of 𝑇𝑟1. Moreover, we now have an instance of a sentence in 𝐿0 that says of itself that it is not 𝑇𝑟1. We have constructed the liar sentence and will get a contradiction. Something has gone wrong so we must examine our initial assumptions. Tarski observes that we have made three assumptions (7) but I consider only two as the third is inconsequential for our purpose:
(1) 𝐿0is semantically closed (is expressively rich enough to talk about its own semantics without
leaving the language) and contains a truth-predicate.
(2) The laws of classical logic hold.
He considers (2) non-negotiable so (1) must go. (8) The contradiction tells us that 𝑇𝑟1 cannot be in 𝐿0. (9) So, any sentence with 𝑇𝑟1 is not in 𝐿0. (10) The paradox is blocked because 𝐿0 does not contain 𝑇𝑟1, so the liar sentence cannot be expressed in 𝐿0 (11) because the sentence contains the negation of 𝑇𝑟1. The liar sentence turns out semantically defective and expresses nothing. Therefore, we cannot use semantically closed languages to define truth and cannot solve the liar in such a language. We need another language – call it 𝐿1 – that contains 𝑇𝑟1 and allows us to talk about sentences in 𝐿0. Tarski refers to 𝐿0 as the object language and 𝐿1 as the meta-language. (12) The definition of truth is formulated in the metalanguage so all sentences in the object language exist in the meta-language. (13) If we consider the Tschema, we see that 𝑋 exists in the meta-language and 𝑝 in the object-language. (14) Although it is counterintuitive to postulate a meta-language, we created it when we tried to introduce 𝑇𝑟1 to 𝐿0 since the resulting language is no longer 𝐿0 it is 𝐿0 + 𝑇𝑟1 which we called 𝐿1. Whilst Tarski’s proposal blocks the liar paradox, Kripke shows that it suffers some problems.
Kripke argues that Tarski’s solution suffers from a liar sentence of a different kind. Suppose Nixon and Dean assert the following two sentences: (15)
(1) Dean: All of Nixon’s utterances about Watergate are false.
(2) Nixon: Everything Dean says about Watergate is false.
Neither of these is paradoxical in isolation but together we can get a version of the liar paradox. Tarski’s hierarchy of languages which assign levels to truth-predicates applicable to sentences of lower languages cannot handle this case. (16) Neither sentence can predicate truth over the other because we do not know what levels to assign them. (17) If we assign a level to (1) we can say that it is expressed in 𝐿1 so (2) must be a sentence in the object language 𝐿0. However, (2) contains a truth-predicate that applies to (1) so its level must be higher than (2) but 0 < 1 and we are in trouble again. Despite the potential for paradox, Kripke provides sufficient conditions for which we could assign truth-values to each if the empirical facts allow. (18) If at least one of Dean’s statements about Watergate is true then (2) is false, if everything Nixon said about Watergate is false then (1) is true. (19) Kripke’s complaint is that Tarski’s approach cannot account for these intuitive results. Kripke’s solution to the liar makes use of only one truth-predicate applicable to sentences that contain the truth-predicate itself and which allows truth value gaps. (20) He then claims that paradoxical sentences are neither true or false because they suffer from gaps. (21) I will first describe the nature of Kripke’s solution with an analogy which makes some of its formal aspects a little easier to describe.
The analogy captures the idea that the truth of a sentence depends on something outside of itself which is called the notion of grounding. (22) Suppose you want to explain the concept of truth to Saulfred who is unfamiliar with its meaning. You tell Saulfred that he should only call “true” those sentences he is willing to assert and “false” those sentences he is willing to deny. (23) First, you show him sentences that he should call true in accordance with our rule but none of them can contain the word “true” (24) because 3 he does not yet know what it means. Once he understands a sentence to be true, you present him with sentences that do contain the word “true”. Since he knows that “true” applies to “p”, he also knows that “true” applies to “p is true”. (25) Now imagine Saulfred climbing a ladder where at the ground there are the base sentences that do not mention “true” but at each higher step are sentences that invoke truth about those immediately below (and so forth, all the way to the ground). (26) The analogy shows that we can only learn to apply the concept of truth if there exists a set of sentences on the ground that does not express the term “true”. The truth of sentences depends on whether they are grounded. Now imagine Saulfred starts somewhere near the top and wants to climb down; he cannot declare any of the sentences true because he does not know whether their corresponding sentences at the ground are true.
(27) Suppose the last four steps are (28):
(4) Step (3) is true
(3) Step (2) is true
(2) Step (1) is true (Saulfred is about to learn whether any of this is grounded)
(1) Step (2) is true
Truth, in this case, wanders in a circle (29) between the last two steps. The sentences are neither true or false (30) so we can think of them as falling into a truth-value gap. Saulfred never reaches the ground and neither does truth.
Kripke’s main criticism of Tarski’s definition of truth is that it does not guarantee grounding. We can now survey some formal aspects of Kripke’s solution that attempt to rectify this. First, he needs a precise formal language. A sentence in this language is semantically well-formed if there are sufficient conditions that can determine its truth. (31) Next, he needs to deal with partially defined predicates because he employs a partially defined truth-predicate. A monadic predicate 𝑃(𝑥) is interpreted by a pair (𝑆1, 𝑆2) of disjoint subsets of a domain. (32) S1 is said to be the extension of the predicate, and S2 the antiextension. (33) This means 𝑃(𝑥) is true for all elements of 𝑆1, false for all elements of 𝑆2 and otherwise undefined (34) (hence it is a partially defined predicate). Only predicates that end up with a truth-value are grounded and otherwise ungrounded. 35 Next we take, a language 𝐿0, assume it can make self-reference, and that the interpretation of the predicates in 𝐿0 remain fixed. 36 We extend 𝐿0 with the addition of a monadic truth-predicate 𝑇(𝑥) to make 𝐿1. 37 𝑇(𝑥) is interpreted by the partial set (𝑆1, 𝑆2) and 𝑇(𝑥) is undefined for all objects that are not an element of the union of 𝑆1 and 𝑆2. (38) Kripke proves – by means which I must omit – that 𝑇(𝑥) is true of 𝐴 iff 𝐴 is true, and false of 𝐴 iff 𝐴 is false. (39) Any pair (𝑆1, 𝑆2) that satisfies these conditions Kripke calls a fixed point, he then shows formally how a fixed point may be constructed by considering a hierarchy of languages. (40) This starts to look similar to Tarski’s hierarchy but then we see a surprising difference. 𝑇(𝑥) does not only define truth for 𝐿0, it extends its interpretation at each successive step in the hierarchy. (41) At each succession, more undefined sentences will receive truth-values. He then proves that once a sentence gets a truth-value at a fixed point, the 4 value never changes (42) and this seems to fit our intuitions about grounding. The rest of the technicalities must be omitted here but eventually one cannot ascend any further because we run out of sentences at some level 𝐿𝑛 which is also a fixed point that contains its own truth-predicate. (43) We can now see that the liar sentence never gets a truth-value and remains ungrounded within a gap throughout the entirety of the ascension because its truth wonders in a circle like at the end of Saulfred’s journey down the ladder.
Kripke’s criticism of Tarski’s proposal seems to undermine its candidacy as a solution and Kripke’s own does not suffer from the same problem. It is not obvious how the T-schema could assign an appropriate level of metalanguage to each of the statements uttered by Nixon and Dean. Since Kripke requires only one truth-predicate in formal language, the potential for paradox between the two statements is ameliorated because they suffer a truth-value gap. Additionally, a liar sentence such as “This sentence is false” receives the same treatment and is no problem for Kripke. However, his solution suffers from a problem that presents no such trouble for Tarski. Note that the liar sentence “This
sentence is false” is not the paradoxical sentence that Tarski solved (although his solution handles this sentence all the same). Tarski solved the paradoxical liar sentence “This sentence is not true” which is known as the strengthened liar. (44) Let the former liar sentence be “𝜔0” and the strengthened liar be “𝜔1”; since 𝜔0 will suffer from a truth-value gap per Kripke, 𝜔0 is neither true or false. But then this entails that 𝜔0 is not false, that is, the negation of 𝜔0, so not 𝜔0 is true. (45) This entails that 𝜔0 is false because
any sentence with a true negation is false so the notion of a truth-value gap entails a contradiction, namely that 𝜔0 is not false and 𝜔0 is false. (46) However, if 𝜔0 is not false, this does not mean it is true because it lies in a gap so we can at least say it is not true and if 𝜔0 is false we can also say it is not true. This generates 𝜔1 which is the sentence Tarski initially constructed in his resolution of the liar paradox. Therefore, whilst Kripke’s criticism is well placed his own solution, metaphorically speaking, throws the baby out with the bathwater. Kripke’s solution avoids the issue of assigning levels to sentences that are not able to receive any without generating a paradox however suffers from precisely the paradox that Tarski resolves.
I have attempted to give a brief outline of both Tarski’s and Kripke’s solution to the liar paradox. Whilst the criticism Kripke makes of one interpretation of Tarski’s solution successfully exposes it to a pair of liar sentences which Tarski’s hierarchy of languages is unable to handle, Kripke’s own solution succeeds in avoiding the same problem but suffers the revenge of the strengthened liar paradox due to Kripke’s inclusion of truth-value gaps in a formally precise language.
1 Tarski, Alfred. 1944. “The Semantic Conception of Truth: and the Foundations of Semantics.” Philosophy and Phenomenological Research 346.
4 Ibid, p.344.
5 Ibid, p.346.
6 Ibid, p.344.
7 Ibid, p.348.
8 Ibid, p.349.
9 Sainsbury, R M. 2009. “Chapter 6: Classes and truth.” In Paradoxes, by R M Sainsbury, 134. Cambridge: Cambridge University Press.
12 Tarski, Alfred. 1944. “The Semantic Conception of Truth: and the Foundations of Semantics.” Philosophy and
Phenomenological Research 350.
15 Kripke, Saul. 1975. “Outline of a Theory of Truth.” The Journal of Philosophy 697.
20 Ibid, p.698.
22 Sainsbury, R M. 2009. “Chapter 6: Classes and truth.” In Paradoxes, by R M Sainsbury, 129. Cambridge:
Cambridge University Press.
23 Ibid, p.134.
24 Ibid, p.129.
25 Ibid, p.134.
30 Ibid, p.131.
31 Kripke, Saul. 1975. “Outline of a Theory of Truth.” The Journal of Philosophy 699-700.
32 Ibid, p.700.
35 Ibid, p.701.
36 Ibid, p.702.
41 Ibid, p.703-704.
42 Ibid, p.704.
44 Sainsbury, R M. 2009. “Chapter 6: Classes and truth.” In Paradoxes, by R M Sainsbury, 132. Cambridge:
Cambridge University Press.
Kripke, Saul. 1975. “Outline of a Theory of Truth.” The Journal of Philosophy 690-716.
Sainsbury, R M. 2009. “Chapter 6: Classes and truth.” In Paradoxes, by R M Sainsbury, 123-145.
Cambridge: Cambridge University Press.
Tarski, Alfred. 1944. “The Semantic Conception of Truth: and the Foundations of Semantics.”
Philosophy and Phenomenological Research 341-376.