Ingest: Liar Paradox (Internet Encyclopedia of Philosophy, Dowden)

Type: types/ingest-report.md

Classification

This is secondary scholarship: an Internet Encyclopedia of Philosophy entry, last revised 2023-03-28. It surveys history, variants and five families of solutions. It also takes positions in its own voice. Examples are its adequacy criteria for a solution (formalism plus philosophical justification), the Strengthened Liar as "the acid test", and judgments on which treatments succeed. That makes it closer to a conceptual essay than to a neutral reference article. Tarski, Kripke, Putnam and others reach the reader through Dowden's selection and paraphrase. The only direct Tarski quotations are from the 1944 paper, not the 1933 monograph. The capture is complete in text, but trafilatura dropped every section heading and the table of contents. Several key paragraphs appear wrapped in square brackets. These include the Tarski-claim paragraph, the classical-logic definition and the opening of the Kripke section. The brackets look like capture markup, not the author's editorial insertions, but the snapshot cannot settle that. Author: Bradley Dowden, a philosopher of logic. He has published on the semantic paradoxes himself, and his bibliography cites his own "Accepting Inconsistencies from the Paradoxes" (1984).

Summary

The entry presents the Liar as an argument from apparently obvious premises to a contradiction, which is ruinous because under classical logic anything follows from a contradiction. It sets out the Classical, Strengthened ("revenge"), contingent, chain, Curry and Yablo variants. It then presents Tarski's Convention T and his formal version of the argument for a classical, interpreted language able to express elementary arithmetic. Here it quotes Tarski 1944's two assumptions: semantic closure (I) and "the ordinary laws of logic" (II). It reports Tarski's conclusion that such a language cannot contain its own global truth predicate, and that truth can be defined for it only from a metalanguage. It sorts solutions into five ways out by the step they reject: ungrammatical, meaningless, truth-value gap, a faulty other step, or dialetheism. It names four features of classical logic, starting with bivalence and explosion, that many solutions revise. Russell's types, Tarski's hierarchy, Kripke's fixed-point theory, Barwise–Etchemendy and Priest's paraconsistency are each assessed for virtues and drawbacks. Kripke gets one language with a partially interpreted truth predicate whose least fixed point leaves the Liar in a truth gap. The entry states that Tarski's undefinability theorem "does not apply" to languages with such gaps. It also reports the Strengthened Liar problem and the objection that Kripke's theory "cannot state its own thesis". It leaves open, explicitly, whether natural language is semantically closed, whether it is inconsistent, and how formal treatments relate to natural language. Read it for a compact map of post-Tarski positions. Do not read it for primary wording.

Quotes

Here is a quotation from (Tarski 1944): --- kb/sources/.snapshots/iep-liar-paradox.md:198-198 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — Tarski section, lead-in to the quoted Tarski 1944 assumptions

I. We have implicitly assumed that the language in which the antinomy is constructed contains, in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term “true” referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called “semantically closed.” --- kb/sources/.snapshots/iep-liar-paradox.md:200-200 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — Tarski section, quoted Tarski 1944 assumption I

Reflection on the Liar Paradox suggests that either informal English (or any other natural language) is not semantically closed or, if it is semantically closed as it appears to be, then it is inconsistent—assuming for the moment that it does make sense to apply the term inconsistent to a natural language with a vague structure. --- kb/sources/.snapshots/iep-liar-paradox.md:388-388 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — opening of the Tarski hierarchy section

An important question regarding the Liar Paradox is: What is the relationship between a solution to the Paradox for (interpreted) formal languages and a solution to the Paradox for natural languages? There is significant disagreement on this issue. --- kb/sources/.snapshots/iep-liar-paradox.md:354-354 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — section on formal versus natural languages

Is our natural language, for example, English, a semantically closed language? --- kb/sources/.snapshots/iep-liar-paradox.md:356-356 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — section on formal versus natural languages

This article raises, but will not resolve, these questions, although some are easier to answer than others. --- kb/sources/.snapshots/iep-liar-paradox.md:358-358 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — section on formal versus natural languages

Many of the most important ways out of the Liar Paradox recommend revising classical formal logic. Classical logic is the formal logic known to introductory logic students as Predicate Logic in which, among other things, (i) all sentences of the formal language have exactly one of two possible truth values (TRUE, FALSE), (ii) the rules of inference allow one to deduce any sentence from an inconsistent set of assumptions, (iii) all predicates are totally defined on the range of the variables, and (iv) classical formal semantics is the one invented by Tarski that provided the first precise definition of truth for a formal language in its metalanguage. --- kb/sources/.snapshots/iep-liar-paradox.md:360-360 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — bracketed paragraph introducing revisions of classical logic

Kripke rejects the hierarchy of languages and retains the intuition that there is a single, semantically coherent and meaningful Liar Sentence, but argues that it is neither true nor false and so falls into a truth value gap. Kripke successfully develops the details using the tools of symbolic logic. Tarski’s Undefinability Theorem does not apply to languages having sentences that are neither true nor false. So, it can be argued that Kripke successfully shows that a semantically coherent formal language can contain its own global truth predicate in the sense that --- kb/sources/.snapshots/iep-liar-paradox.md:428-428 @ sha256:5d1c4f89d687f46f675bba6c174e7f4988c0a93425871089200b2d60122ba03b — opening bracketed paragraph of the Kripke section

Connections Found

For the occasion, the source's role is a bounding secondary source. It can confirm how the field standardly reports Tarski's and Kripke's positions. It cannot stand in for Tarski's own 1933 wording. On claim (2) it does not support the attribution as the occasion phrases it (see Extractable Value item 2).

In the library, the main partner is the Tarski 1944 ingest (compares-with, axis: which assumption of the Liar argument a treatment gives up). Tarski holds classical logic fixed and gives up semantic closure. This entry sets out the later ways out that give up bivalence, explosion or full disquotation instead. That ingest could state those developments only "background knowledge, not from the source". This entry now gives them a citable, if secondary, source.

The entry bears on the impossibility paragraph in Prompting changes failure diagnosis with the semantic reference (is-evidence-for, mostly by narrowing). That note said complete semantics for natural language "as a whole" is unattainable "even in principle" because of "Tarski/Gödel/halting-style impossibility results". Since rewritten: the note, now titled by the link, no longer contains the impossibility paragraph and states that it needs no impossibility claim about formalizing English. The entry treats natural-language closure and inconsistency as open questions. It reports formal languages that contain their own (partial) truth predicate once bivalence is dropped. So the undefinability barrier is conditional on classical assumptions, and the field has not settled it for natural language. This matches the narrowing the Tarski 1944 ingest proposes, and it rests on a survey of the field rather than one author.

The entry also gives rival positions for Natural-language meaning is coordination among learned approximations (compares-with, axis: whether natural language has a determinate semantics that formalization recovers). They are Russell's "underlying sensible part is an ideal formal language", the defence of formal languages as "the best that can be done rigorously", logical monism versus pluralism, and Slater's indexicality objection. None of them addresses learned competence, so the comparison locates the hypothesis but does not test it.

Extractable Value

1. Attribution audit for the occasion's three claims. [quick-win] (1) Semantically closed. The entry gives the definition only by quoting Tarski 1944, assumption I. That is the same passage already held in the Tarski 1944 ingest, so the entry adds no 1933 wording. Its own gloss, inside a bracketed paragraph, says Tarski "claimed" that any semantically closed language "contains or represents its own global truth predicate", and that this self-reference is the "crucial, unacceptable assumption". "Global truth predicate" is the entry's term, not a quotation. (2) Natural language unclosed or inconsistent. The disjunction appears in the entry's own voice, not as a report of Tarski: "Reflection on the Liar Paradox suggests that either informal English … is not semantically closed or, if it is semantically closed as it appears to be, then it is inconsistent". It comes with the entry's own hedge: "assuming for the moment that it does make sense to apply the term inconsistent to a natural language with a vague structure". The next sentence attributes to Tarski a motive and a move, not that conclusion. Because of natural language's vagueness he "quit trying to find the paradox-free structure within natural languages". He developed formal languages that "diverge from natural language as little as possible". The level-subscripted hierarchy is presented as "Tarski's way out". The one passage that has Tarski holding ordinary language "semantically closed and hence inconsistent" is a quoted Putnam (1990) sentence. That is Putnam's attribution. A later Putnam quotation says Tarski "was wrong (as I believe he was) in supposing that ordinary language is a theory". The entry also quotes Tarski 1944 that scientific languages "do not have to be semantically closed". A citation can use this entry for the disjunction as a standard reading. It cannot cite it as evidence that Tarski asserted the inconsistency claim. (3) Kripke. The entry describes one formal language expressing elementary number theory. Its truth predicate T has an extension, an anti-extension and "undecideds". The interpretation is built up to the "unique lowest fixed point", where Convention T holds and the Liar "falls into the truth gap". The sentence the occasion needs is in the entry's own voice: "Tarski's Undefinability Theorem does not apply to languages having sentences that are neither true nor false". The entry hedges the success claim: "it can be argued that Kripke successfully shows …", and "Kripke's controversial claim that T is a truth predicate at that point". It reports three objections: the Strengthened Liar, the charge that the predicate is only "a clever restriction" and not global, and that the theory "cannot state its own thesis", because the judgment that the Liar is not true is made from a metalanguage. That last objection bounds the occasion's "its own truth predicate": the hierarchy of languages returns at the level of stating the theory.

2. Assumptions the entry names. [quick-win] Tarski's theorem is stated for "classically interpreted languages capable of expressing elementary arithmetic". It rests on Convention T plus assumption II, "the ordinary laws of logic hold". The entry defines classical logic by four features: "(i) all sentences … have exactly one of two possible truth values" (bivalence); "(ii) … deduce any sentence from an inconsistent set of assumptions" (explosion); "(iii) all predicates are totally defined"; and "(iv)" Tarski's own semantics. Kripke's partial T gives up (i) and (iii). Priest's paraconsistency gives up (ii) and lets sentences be both true and false. The entry adds that all five ways out presuppose logical monism. These named conditions support stating the undefinability barrier as conditional rather than absolute.

3. Map of post-Tarski alternatives with failure tests. [just-a-reference] The five-ways classification and the Strengthened Liar "acid test" give a compact vocabulary for saying which assumption a treatment drops and what it pays. The Tarski hierarchy cannot say "every language has true sentences". Kripke cannot state his own thesis. Barwise–Etchemendy propositions cannot be about the whole world. In Priest's theory, contradictions propagate into the metalanguage. The common pattern, that each treatment loses some self-description, is useful when the KB discusses limits of self-describing formal systems. The entry does not argue the pattern is necessary. It quotes Putnam's pessimism and offers ZFC as the optimistic analogue.

4. Change-of-subject caution for explicating "true". [just-a-reference] The entry warns that an explication of true must stay "true to some core of ordinary meaning" or become "a change of subject". It also says a formalism that blocks the Liar may only show "the inadequacy of the formalism". This parallels the Tarski 1944 ingest's precedent on renaming terms, and it is not new to the KB.

Limitations (our opinion)

The entry is secondary and partly argumentative. On the occasion's claims it bounds attribution to Tarski; it does not supply primary-source corroboration. It is in the same position as the KB's Church–Turing thesis and operationalism SEP ingests. It quotes no 1933 text, so it cannot replace the monograph for the universality or colloquial-language framing. Its one direct definition comes from the 1944 paper the KB already holds. Its dating is loose. It names the undefinability theorem "of 1936" while citing the 1933 article, and the snapshot cannot show which dating the entry intends. Its statement that undefinability "does not apply" to gappy languages is compressed. Kripke's theory still needs a metalanguage to say that the Liar is not true. The entry reports this only as an objection ("cannot state its own thesis"), beside the sentence and not inside it. A reader could take the sentence as a stronger theorem than the entry's own hedges support. The entry's Kripke account is given as exposition; it reproduces no construction, so formal details rest on Kripke 1975. The bracketed paragraphs and missing headings come from the capture. Locators for any retained quote will need line ranges or nearby wording rather than section titles. IEP entries are revised. This observation is the 2023-03-28 version, and later revisions are distinct observations.

Recommended Next Action

Since rewritten: the paragraph was removed rather than narrowed, so this recommendation is closed. Qualify the impossibility appeal in Prompting changes failure diagnosis with the semantic reference with the entry's account of partial truth predicates and its open questions about natural language. Attribute the unclosed-or-inconsistent disjunction to the entry's reading, not to Tarski's assertion, and keep both the vagueness hedge and the objections to Kripke's account beside the claims they limit.