h t t p s : / / c s . n y u . e d u / m i s h r a / C O U R S E S / 1 3 . L O G I C / T a r s k i . p d f

Ingest: Truth and Proof (Tarski, Scientific American 1969)

Type: types/ingest-report.md

Classification

This is a popular exposition of logical results by their originator. It has no citations, no proofs and no experiments; Tarski says the technical construction "is too involved to be explained here in detail". It is therefore a conceptual-essay in this KB's vocabulary, not a scientific paper. The official source is Alfred Tarski, "Truth and Proof", Scientific American 220(6), June 1969, pp. 63–77. The captured representation is a PDF scan hosted on an NYU course site (the source URL). Every page carries a "© 1969 SCIENTIFIC AMERICAN, INC" notice. Whether the course site is authorized to host the copy is unverified. The text layer comes from pdftotext. It includes interleaved advertisements: pages 71–74 are ads only, and ad fragments split the text on pages 67, 69 and 77. It also has OCR artefacts such as "logiC", "pOSSibly" and words split by spaces ("lan guage", "componen ts"). The article text runs without gaps from p. 63 to p. 77. Figures or captions, if the printed article had any, were not checked. Sentence (6) depends on being "printed in red" on p. 65, and a text capture cannot show that. Author: Alfred Tarski, who originated the semantic definition of truth and the undefinability theorem. Here he restates his own 1933–1936 results 35 years later, for a general audience.

Summary

The article has three sections. Truth: Tarski takes the classical, Aristotelian conception and states adequacy as implying every instance of "'p' is true if and only if p". For a finite fragment of English with precise syntax and without the word "true", he builds a definition as a finite conjunction of these instances. Whole English resists this for three reasons: grammar does not fix which expressions are sentences, the set of sentences is potentially infinite, and "true" is part of the language. Then comes a "more serious and fundamental reason": the liar (in Łukasiewicz's form). He hedges this step: it "seems to preclude" a definition, and the supposition of adequate use "appears to lead to a contradiction". He traces the liar to the semantically universal character of common language. Such a language names its own expressions and contains semantic terms such as "truth", so for any of its sentences it can say that the sentence is true or false. The languages of chemistry, linguistics and metalogic do not have to be semantically universal. For them a definition succeeds under stated conditions: a full vocabulary, purely formal syntactic rules, meaning that depends only on form (so no "this" or "here"), and a metalanguage that is essentially richer and not translatable into the object-language. Formalized languages "of real interest" are fragments of natural languages, not something "essentially opposed" to them. Proof: a truth definition gives no "workable criterion" for deciding truth. The axiomatic method developed by restricting appeal to intuitive evidence, and proof changed from a psychological activity into formal proof checked by comparing shapes. Relationship: in formalized arithmetic, provability can be defined in the object-language but truth cannot, because otherwise the liar could be reconstructed there. The idea is credited as "closely related" to Gödel's. With sound axioms and rules, some true sentences are unprovable. The result extends to any theory that contains arithmetic. Elementary geometry and the elementary algebra of real numbers are exceptions in which provable and true sentences coincide. Truth is "an ideal limit" that we approximate by adding axioms, and we choose those axioms with truth as the guide. Tarski suggests that truth plays a similar role in empirical knowledge.

Quotes

The common language is universal and is in­ tended to be so. --- kb/sources/.snapshots/tarski-truth-and-proof.md @ sha256:870b717aeb0fa47b4b81b150e1743382fbc7a4bb52287bcd67f3098e0d3c700a — p. 67, universality of the common language

In particular, it is semanti­ cally universal in the following sense. Together with the linguistic objects, such as sentences and terms, which are com­ ponen ts of this language, names of these objects are also included in the lan guage (as we know, names of expressions can be obtained by puttin g the expressions in quotes) ; in addition, the language contains semantic terms such as "truth", "name", "designation", which directly or indirectly refer to the relationship be­ tween linguistic objects and what is expressed by them . Consequently, for every sentence formulated in the com­ mon language, we can form in the same language another sentence to the effect that the first sentence is true or that it is false. --- kb/sources/.snapshots/tarski-truth-and-proof.md @ sha256:870b717aeb0fa47b4b81b150e1743382fbc7a4bb52287bcd67f3098e0d3c700a — p. 67, the sense of "semantically universal"

I do not have in mind any­ thing essentially opposed to natural lan­ guages. On the contrary, the only formal­ ized languages that seem to be of real interest are those which are fragments of natural languages (fragments provid­ ed with complete vocabularies and pre­ cise syn tactical rules) or those which can at least be adequately translated. into natural languages . --- kb/sources/.snapshots/tarski-truth-and-proof.md @ sha256:870b717aeb0fa47b4b81b150e1743382fbc7a4bb52287bcd67f3098e0d3c700a — p. 68, formalized languages as fragments of natural languages

Furthermore, the syntactical rules should be purely formal, that is, they should refer exclusively to the form ( the shape) of expressions ; the function and the meaning of an expres­ sion should depend exclUSively on its form. --- kb/sources/.snapshots/tarski-truth-and-proof.md @ sha256:870b717aeb0fa47b4b81b150e1743382fbc7a4bb52287bcd67f3098e0d3c700a — p. 68, the purely formal syntactical rules condition

Hence we conclude that the metalan­ guage which provides sufficient means for defining truth must be essen tially richer than the object-language ; it can­ not coincide with or be translatable into the latter, since otherwise both lan­ guages would turn out to be seman tical­ ly universal, and the an tinomy of the liar could be reconstructed in both of them. --- kb/sources/.snapshots/tarski-truth-and-proof.md @ sha256:870b717aeb0fa47b4b81b150e1743382fbc7a4bb52287bcd67f3098e0d3c700a — p. 69, untranslatable metalanguage blocks reconstruction of the liar

It compelled us to abandon all attempts at clarifying the notion of truth for natural languages . We had to restrict our endeavors to formalized languages of scientific discourse. As a safeguard against a possible reappearance of the antinomy, we had to complicate con­ siderably the discussion by distinguish­ ing between a language and its meta­ language. --- kb/sources/.snapshots/tarski-truth-and-proof.md @ sha256:870b717aeb0fa47b4b81b150e1743382fbc7a4bb52287bcd67f3098e0d3c700a — p. 77, abandonment of truth for natural languages

Connections Found

The source has two roles. It is a primary-source anchor for Tarski's universality argument in his own words, which the 1944 Tarski ingest records it cannot ground. It is also a limiting source for the impossibility appeal in Prompting changes failure diagnosis with the semantic reference (is-evidence-for). That note said a complete executable semantics for natural language "as a whole" is unattainable 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. This article supports a narrower reading than the note's sentence. It ties the liar to semantic universality and hedges the natural-language conclusion. Its truth ≠ provability argument applies to formalized arithmetic, so that result cannot be what separates natural languages from programming languages. The 1944 ingest is the companion (compares-with, on which Tarski text grounds which point). The 1944 paper states the liar's assumptions, including classical logic, more carefully. The 1969 article supplies the universality wording, the list of conditions that excludes indexicals, the claim that formalized languages are fragments of natural language, and the full truth-and-proof argument.

For Codification the article is classical support (is-evidence-for). Tarski's formalized languages are English fragments whose meaning "should depend exclusively on its form". That fits the definition's view that the clerk-passing fragment is "a programming language in the making". It also fits the definition's caveat that the boundary is not a claim that natural language could never have a formal semantics. Natural-language meaning is coordination among learned approximations remains a rival account (compares-with, on whether natural language has a semantics that formalization recovers). Tarski's obstacles are structural features of the language, not features of learned competence, so the article neither supports nor refutes that hypothesis.

Section 3 is a classical anchor for the KB's proof-gate cluster (is-evidence-for). It gives Gödel machines a reason that "some claims may be unprovable under the axioms" for any formalization that contains arithmetic. It fixes the proof oracle's domain as a proper subset of the truths in Warranted autonomy is bounded by oracle domain, with named exceptions. It bounds the "different axiomatization" remedy in Improvements outside the admitted formal language: axioms can be added repeatedly, truth guides the choice, and truth is never reached. For Causal and proof obligations Tarski says the same thing as the note about judgment moving upstream: formal proof reduces doubt to the axioms and the rules of proof. The history of proof is a worked example for Oracle strength spectrum. Proof went from intuitive convincing to a check made by comparing shapes. Tarski's method for antinomies treats one as a "symptom of disease" and tests a reformed system by trying to rebuild the old antinomy, "akin to carrying out crucial experiments". This is a see-also for the fallibilism in Epistemic stance.

Extractable Value

  1. Occasion claim (1) is supported as worded, with Tarski's definition of the term. [quick-win] On p. 67 Tarski says "The common language is universal and is intended to be so", and that it is "semantically universal" in a stated sense. It contains its own linguistic objects, names of those objects, and semantic terms ("truth", "name", "designation"). "Consequently" it can say, of every one of its sentences, that the sentence is true or false. With an extra "trick" it can form a self-referential sentence, and the liar follows (pp. 67–68). The definition has no separate condition that all T-instances are assertible. That condition appears separately, as the "adequate use" of "true" whose mere supposition leads to contradiction (p. 65). A citer who equates "semantically universal" with the 1944 "semantically closed" should note this difference. Closure in 1944 includes T-assertibility; universality in 1969 does not.

  2. Occasion claim (2) is supported, but the second half needs Tarski's wording and conditions. [quick-win] On p. 68 the first half is explicit. Universal languages "are in general not needed for the purposes of science". The language of chemistry "does not have to be semantically universal". The language of linguistics names the components of the languages it discusses "but not the names of its own components; thus, again, it does not have to be semantically universal". The same applies to metalogic and metamathematics. The second half has three corrections. (a) The fragments are not a second category beside formalized languages. Tarski says "the only formalized languages that seem to be of real interest are those which are fragments of natural languages (fragments provided with complete vocabularies and precise syntactical rules)", or languages translatable into them. (b) He writes "precise syntactical rules", and the conditions are stricter than "complete vocabularies and precise rules". The rules must be "purely formal", meaning must "depend exclusively on its form", sentencehood and assertibility must not vary with context, and "this" and "here" are excluded. (c) Even then he says the definition succeeds "Under certain conditions", which include an essentially richer metalanguage (pp. 68–69). The finite-fragment construction on p. 65 is a separate and weaker example. It covers a finite fragment without the word "true".

  3. Occasion claim (3) is supported in substance, but not as a quotable single sentence. Reword it. [quick-win] The article gives the liar-reconstruction argument in two places. Neither is phrased as "any fragment made semantically universal". (a) For the general case (p. 69): the metalanguage "cannot coincide with or be translatable into" the object-language, "since otherwise both languages would turn out to be semantically universal, and the antinomy of the liar could be reconstructed in both of them". (b) For arithmetic (p. 76): if truth could be translated into the object-language, it "would prove to be in a sense semantically universal". Tarski first calls this a "conjecture" and then confirms it by showing that the liar "could actually be reconstructed in this language". The restriction to formalized languages is explained on p. 77: the liar "compelled us to abandon all attempts at clarifying the notion of truth for natural languages. We had to restrict our endeavors to formalized languages of scientific discourse". In that passage the metalanguage distinction is the "safeguard against a possible reappearance". A faithful paraphrase therefore has two steps. The liar in a semantically universal natural language led him to restrict truth definitions to formalized languages. Within those, he keeps the object-language from becoming semantically universal by defining truth only in a richer metalanguage. The occasion's single causal sentence merges these two steps.

  4. What the article adds to or changes in the 1944 audit of the library appeal (our opinion). [deep-dive] Adds: (a) It grounds the universality framing in Tarski's own words, closing the gap the 1944 ingest records. (b) It makes truth ≠ provability central, rather than putting it in a note. It also says the liar's "basic idea" is what yields the incompleteness result. This supports grouping Tarski with Gödel as one family of liar-based arguments. Tarski calls the main idea "closely related" to Gödel's. (c) It states the scope of that result. The gap between proof and truth holds for any theory that includes arithmetic and has true axioms and infallible rules. It may close for weaker theories such as elementary geometry and real algebra. So the boundary is the theory's expressive strength, not natural versus formal language. This strengthens the 1944 audit's point that undefinability cannot separate natural languages from programming languages. (d) It says formalized languages are not "essentially opposed" to natural languages, which works against a sharp contrast between natural and formal languages. (e) It restates definition versus criterion with an argument: definitions of "atom", "gene" and most mathematical notions also carry no decision criterion. Changes: (f) It drops the explicit classical-logic assumption. The 1969 liar derivation uses classical reasoning without stating it, so the 1944 audit's "cannot draw an assumption-free impossibility" point still stands. The 1944 paper is the text to cite for that dependence. (g) Its statements about natural language are firmer in tone ("compelled us to abandon all attempts") but still hedged where it makes the argument ("seems to preclude", "appears to lead"). It reports abandoning attempts, not proving that a definition cannot exist. Unchanged: the article never mentions computation, machines or halting. Nothing in it supports the "halting-style" part of the appeal, or any claim about executable semantics as distinct from truth.

  5. Classical anchor for the proof-gate limit. [just-a-reference] Provability is definable and sound but strictly narrower than truth in arithmetic-containing theories. Repeated axiom extension guided by truth approaches truth without reaching it. This supports the existing Gödel-machine, warranted-autonomy and unformalized-improvements notes as evidence. It is standard logic, not a new claim for the KB.

  6. Worked case of oracle hardening. [quick-win] Proof went from "an intellectual activity that aimed at convincing" to formal proof. In formal proof each step is recognized "simply by comparing its shape", and remaining doubt is "reduced to possible doubts concerning the truth of the few sentences listed as axioms and the infallibility of the few simple rules of proof". This is a historical example the oracle-strength note could cite, and it shows where residual doubt goes after hardening.

  7. Antinomy as a test case for a revised system. [just-a-reference] Tarski treats an antinomy as a "symptom of disease". The reformed system is tested by trying to reconstruct the old antinomy in it, which he compares to a crucial experiment. This fits the KB's Popperian stance, but it is a methodological remark, not an argument.

Limitations (our opinion)

This is a popular exposition, and it argues less than it reports. It proves none of its formal claims: the recursive satisfaction definition, the reconstruction of the liar in arithmetic ("We shall not go into any further details") and the extension to arithmetic-containing theories are all asserted. It states the liar's dependence on classical logic only implicitly. A citer who needs the assumptions behind the liar should use the 1944 paper or the 1933 monograph. The natural-language claims are informal and hedged, so citing the article for a natural-language impossibility theorem would cite its hedges. There is an internal overstatement. It says that "in no domain of mathematics" is provability a perfect substitute for truth, after naming elementary geometry and real algebra as theories where the two coincide. The broad sentence should be read as limited to arithmetic-containing theories. Using a 1969 retrospective to stand in for the 1933 monograph attributes Tarski's 1969 formulation, not his 1933 one. The terminology may have shifted, and "semantically universal" here is defined by naming and semantic terms, not by T-assertibility. Reading the empirical-knowledge analogy for truth as an "ideal limit" as an argued claim would overread it: Tarski marks it as a "likely" aside "for different reasons". The capture boundary also limits reuse. The copy's authorization is unverified. Ads and OCR artefacts are interleaved, so exact wording must be checked against the snapshot or the printed article. The "printed in red" device behind sentence (6) cannot be verified from text.

Since rewritten: the paragraph was removed rather than narrowed, so this recommendation is closed. Narrow the Tarski appeal in Prompting changes failure diagnosis with the semantic reference to the two-step account in Extractable Value items 2–4: the liar motivates restricting truth definitions to formalized languages, and a richer metalanguage prevents its reconstruction within those languages. Keep the natural-language hedges and the distinction between truth, proof and execution explicit.

Relevant Notes: