Tarskian Theories of Truth

We can simply define Gödel 1931 Incompleteness away by redefining the meaning of the standard definition of Incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). This definition construes the existence of self-contradictory expressions in a formal system as proof that this formal system is incomplete because self-contradictory expressions are neither provable nor disprovable in this formal system. Since self-contradictory expressions are neither provable nor disprovable (...) only because they are self-contradictory we could define them as unsound instead of defining the formal system as incomplete. (shrink)

Both Tarski and Gödel “prove” that provability can diverge from Truth. When we boil their claim down to its simplest possible essence it is really claiming that valid inference from true premises might not always derive a true consequence. This is obviously impossible.

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning (...) of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus. (shrink)

►JOHN CORCORAN AND IDRIS SAMAWI HAMID, Two-method errors: having it both ways. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] Philosophy, Colorado State University, Fort Collins, CO 80523-1781 USA E-mail: [email protected] Where two methods produce similar results, mixing the two sometimes creates errors we call two-method errors, TMEs: in style, syntax, semantics, pragmatics, implicature, logic, or action. This lecture analyzes examples found in technical and in non-technical contexts. One can say “Abe knows whether Ben draws” in two other (...) ways: ‘Abe knows whether or not Ben draws’ or ‘Abe knows whether Ben draws or not’. But a stylistic TME occurs in ‘Abe knows whether or not Ben draws or not’. One can say “Abe knows how Ben looks” using ‘Abe knows what Ben looks like’. But syntactical TMEs are in ‘Abe knows what Ben looks’ and in ‘Abe knows how Ben looks like’. One can deny that Abe knows Ben by prefixing ‘It isn’t that’ or by interpolating ‘doesn’t’. But a pragmatic TME occurs in trying to deny that Abe knows Ben by using ‘It isn’t that Abe doesn’t know Ben’. There are several standard ways of defining truth using sequences. Quine’s discussions in the 1970 first printing of Philosophy of logic [3] and in previous lectures were vitiated by mixing two [1, p. 98]. The logical TME in [3], which eluded Quine’s colleagues, was corrected in the 1978 sixth printing [2]. But Quine never explicitly acknowledged, described, or even mentioned the error. This lecture presents and analyses two-method errors in the logic literature. [1] JOHN CORCORAN, Review of Quine’s 1970 Philosophy of Logic. In Philosophy of Science, vol. 39 (1972), pp. 97–99. [2] JOHN CORCORAN, Review of sixth printing of Quine’s 1970 Philosophy of Logic. In Mathematical Reviews MR0469684 (1979): 57 #9465. [3] WILLARD VAN ORMAN QUINE, Philosophy of logic, Harvard, 1970/1986. (shrink)

Every countable language which conforms to classical logic is shown to have an extension which has a consistent definitional theory of truth. That extension has a consistent semantical theory of truth, if every sentence of the object language is valuated by its meaning either as true or as false. These theories contain both a truth predicate and a non-truth predicate. Theories are equivalent when sentences of the object lqanguage are valuated by their meanings.

Truth pluralists say that there are many ways to be true. Aaron Cotnoir (“Pluralism and Paradox” in: Pedersen and Wright (eds) Truth and pluralism: current debates, Oxford University Press, Oxford, 2013) has suggested a “uniquely pluralist response to the liar”. The basic idea is to maintain that, if a sentence says of itself that it is not true in a certain way, then that sentence is not apt to be true in that way, but is instead apt to be true (...) in a different way. While this is consistent with the basic tenets of truth pluralism, it is an open question whether or not it is amenable to any actual pluralist theory. The primary goal of this paper is to argue that Cotnoir's proposal is amenable to form-based pluralism, rather than domain-based pluralism. In particular, in Section 1, I argue that there are several serious obstacles in the way of the domain-based pluralist who wishes to endorse Cotnoir's proposal; in Section 2, I show how the form-based pluralist can overcome these difficulties. The secondary goal of the paper is to argue that most, if not all, substantivists about truth should find form-based pluralism independently attractive. As such, the possibility of a form-based pluralist solution to the liar is not merely a technical curiosity, but something in which substantivists about truth have a vested interest. (shrink)

There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...) We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. Indeed, we argue that developing a fully compositional semantics in the tradition of Frege leads to a typographic variant of the most radical of Tarskian views: variabilism, the view that names should be modeled as Tarskian variables. We conclude with the consequences of this result for Frege's distinction between sense and reference. (shrink)

The concept of truth and competing philosophical theories on what truth amounts to have an important place in contemporary philosophy. The aim of this chapter is to give a synopsis of different theories of truth and the particular philosophical issues related to the concept of truth. The literature on this topic is vast, and we must necessarily be rather selective and very brief about complex questions of interpretation of various philosophers. The focus of the chapter is mainly on selected systematic (...) issues and the most influential and well-established philosophical theories and key concepts. (shrink)

The purpose of this paper is to communicate - as a proposal - a general method of assigning a 'truthmaker' to any 1st order sentence in each of its models. The respective construct is derived from the standard model theoretic (recursive) satisfaction definition for 1st order languages and is a conservative extension thereof. The heuristics of the proposal (which has been somewhat idiosyncratic from the current point of view) and some more technical detail of the construction may be found in (...) my article on part I of Spinoza's 'ethica, ordine geometrico demonstrata', which is the context within which I elaborated the assignment. But this context need not be repeated here, the presentation of the truthmaker assignment will be comprehensible to anybody with solid basic knowledge in 1st order model theory, anything used is standard and no advanced techniques are required. (shrink)

Confronting the Liar Paradox is commonly viewed as a prerequisite for developing a theory of truth. In this paper I turn the tables on this traditional conception of the relation between the two. The theorist of truth need not constrain his search for a “material” theory of truth, i.e., a theory of the philosophical nature of truth, by committing himself to one solution or another to the Liar Paradox. If he focuses on the nature of truth (leaving issues of formal (...) consistency for a later stage), he can arrive at material principles that prevent the Liar Paradox from arising in the first place. I argue for this point both on general methodological grounds and by example. The example is based on a substantivist theory of truth that emphasizes the role of truth in human cognition. The key point is that truth requires a certain complementarity of “immanence” and “transcendence”, and this means that some hierarchical structure is inherent in truth. Approaching the Liar Paradox from this perspective throws new light on its existent solutions: their differences and commonalities, their purported ad-hocness, and the relevance of natural language and bivalence to truth and the Liar. (shrink)

This paper describes Tarski’s project of rehabilitating the notion of truth, previously considered dubious by many philosophers. The project was realized by providing a formal truth definition, which does not employ any problematic concept.

I here argue for a particular formulation of truth-deflationism, namely, the propositionally quantified formula, (Q) “For all p, <p> is true iff p”. The main argument consists of an enumeration of the other (five) possible formulations and criticisms thereof. Notably, Horwich’s Minimal Theory is found objectionable in that it cannot be accepted by finite beings. Other formulations err in not providing non-questionbegging, sufficiently direct derivations of the T-schema instances. I end by defending (Q) against various objections. In particular, I argue (...) that certain circularity charges rest on mistaken assumptions about logic that lead to Carroll’s regress. I show how the propositional quantifier can be seen as on a par with first-order quantifiers and so equally acceptable to use. While the proposed parallelism between these quantifiers is controversial in general, deflationists have special reasons to affirm it. I further argue that the main three types of approach the truth-paradoxes are open to an adherent of (Q), and that the derivation of general facts about truth can be explained on its basis. (shrink)

نگاهی بر استدلال‌های دونالد دیویدسون در باب ضرورت زبان برای اندیشه .

We discuss misinformation about “the liar antinomy” with special reference to Tarski’s 1933 truth-definition paper [1]. Lies are speech-acts, not merely sentences or propositions. Roughly, lies are statements of propositions not believed by their speakers. Speakers who state their false beliefs are often not lying. And speakers who state true propositions that they don’t believe are often lying—regardless of whether the non-belief is disbelief. Persons who state propositions on which they have no opinion are lying as much as those who (...) state propositions they believe to be false. Not all lies are statements of false propositions—some lies are true; some have no truth-value. People who only occasionally lie are not liars: roughly, liars repeatedly and habitually lie. Some half-truths are statements intended to mislead even though the speakers “interpret” the sentences used as expressing true propositions. Others are statements of propositions believed by the speakers to be questionable but without revealing their supposed problematic nature. The two “formulations” of “the antinomy of the liar” in [1], pp.157–8 and 161–2, have nothing to do with lying or liars. The first focuses on an “expression” Tarski calls ‘c’, namely the following. -/- c is not a true sentence -/- The second focuses on another “expression”, also called ‘c’, namely the following. -/- for all p, if c is identical with the sentence ‘p’, then not p -/- Without argumentation or even discussion, Tarski implies that these strange “expressions” are English sentences. [1] Alfred Tarski, The concept of truth in formalized languages, pp. 152–278, Logic, Semantics, Metamathematics, papers from 1923 to 1938, ed. John Corcoran, Hackett, Indianapolis 1983. -/- https://www.academia.edu/12525833/Sentence_Proposition_Judgment_Statement_and_Fact_Speaking_about_th e_Written_English_Used_in_Logic. (shrink)

This papers discuss the place, if any, of Convention T (the condition of material adequacy of the proper definition of truth formulated by Tarski) in the truth-makers account offered by Kevin Mulligan, Peter Simons and Barry Smith. It is argued that although Tarski’s requirement seems entirely acceptable in the frameworks of truth-makers theories for the first-sight, several doubts arise under a closer inspection. In particular, T-biconditionals have no clear meaning as sentences about truth-makers. Thus, truth-makers theory cannot be considered as (...) the semantic theory of truth enriched by metaphysical (ontological) data. The problem of truth-makers for sentences about future events is discussed at the end of the paper. (shrink)

The paper discusses what kind of truth bearer, or truth-ascription, a deflationist should take as primary. I first present number of arguments against a sententialist view. I then present a deflationary theory which takes propositions as primary, and try to show that it deals neatly with a wide range of linguistic data. Next, I consider both the view that there is no primary truth bearer, and the most common account of sentence truth given by deflationists who take propositions as primary, (...) and argue that they both attribute an implausible type of ambiguity to “true”. This can be avoided, however, if truth-ascriptions to sentences are taken as a certain form of pragmatic ellipses. I end by showing how this hypothesis accommodates a number of intuitions involving truth-ascriptions to sentences. (shrink)

Philosopher’s judgements on the philosophical value of Tarski’s contributions to the theory of truth have varied. For example Karl Popper, Rudolf Carnap, and Donald Davidson have, in their different ways, celebrated Tarski’s achievements and have been enthusiastic about their philosophical relevance. Hilary Putnam, on the other hand, pronounces that “[a]s a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail.” Putnam has several alleged reasons for his dissatisfaction,1 but one of them, (...) the one I call the modal objection (cf. Raatikainen 2003), has been particularly influential. In fact, very similar objections have been presented over and over again in the literature. Already in 1954, Arthur Pap had criticized Tarski’s account with a similar argument (Pap 1954). Moreover, both Scott Soames (1984) and John Etchemendy (1988) use, with an explicit reference to Putnam, similar modal arguments in relation to Tarski. Richard Heck (1997), too, shows some sympathy for such considerations. Simon Blackburn (1984, Ch. 8) has put forward a related argument against Tarski. Recently, Marian David has criticized Tarski’s truth definition with an analogous argument as well (David 2004, p. 389-390).2 This line of argument is thus apparently one of the most influential critiques of Tarski. It is certainly worthy of serious attention. Nevertheless, I shall argue that, given closer scrutiny, it does not present such an acute problem for the Tarskian approach to truth as many philosophers think. But I also believe that it is important to understand clearly why this is so. Moreover, I think that a careful consideration of the issue illuminates certain important but somewhat neglected aspects of the Tarskian approach. (shrink)

According to Field’s influential incompleteness objection, Tarski’s semantic theory of truth is unsatisfactory since the definition that forms its basis is incomplete in two distinct senses: (1) it is physicalistically inadequate, and for this reason, (2) it is conceptually deficient. In this paper, I defend the semantic theory of truth against the incompleteness objection by conceding (1) but rejecting (2). After arguing that Davidson and McDowell’s reply to the incompleteness objection fails to pass muster, I argue that, within the constraints (...) of a non-reductive physicalism and a holism concerning the concepts of truth, reference and meaning, conceding Field’s physicalistic inadequacy conclusion while rejecting his conceptual deficiency conclusion is a promising reply to the incompleteness objection. (shrink)

In the early 20th century, scepticism was common among philosophers about the very meaningfulness of the notion of truth – and of the related notions of denotation, definition etc. (i.e., what Tarski called semantical concepts). Awareness was growing of the various logical paradoxes and anomalies arising from these concepts. In addition, more philosophical reasons were being given for this aversion.1 The atmosphere changed dramatically with Alfred Tarski’s path-breaking contribution. What Tarski did was to show that, assuming that the syntax of (...) the object language is specified exactly enough, and that the metatheory has a certain amount of set theoretic power,2 one can explicitly define truth in the object language. And what can be explicitly defined can be eliminated. It follows that the defined concept cannot give rise to any inconsistencies (that is, paradoxes). This gave new respectability to the concept of truth and related notions. Nevertheless, philosophers’ judgements on the nature and philosophical relevance of Tarski’s work have varied. It is my aim here to review and evaluate some threads in this debate. (shrink)

The thesis deals with the concept of truth and the paradoxes of truth. Philosophical theories usually consider the concept of truth from a wider perspective. They are concerned with questions such as - Is there any connection between the truth and the world? And, if there is - What is the nature of the connection? Contrary to these theories, this analysis is of a logical nature. It deals with the internal semantic structure of language, the mutual semantic connection of sentences, (...) above all the connection of sentences that speak about the truth of other sentences and sentences whose truth they speak about. Truth paradoxes show that there is a problem in our basic understanding of the language meaning and they are a test for any proposed solution. It is important to make a distinction between the normative and analytical aspect of the solution. The former tries to ensure that paradoxes will not emerge. The latter tries to explain paradoxes. Of course, the practical aspect of the solution is also important. It tries to ensure a good framework for logical foundations of knowledge, for related problems in Artificial Intelligence and for the analysis of the natural language. Tarski’s analysis emphasized the T-scheme as the basic intuitive principle for the concept of truth, but it also showed its inconsistency with the classical logic. Tarski’s solution is to preserve the classical logic and to restrict the scheme: we can talk about the truth of sentences of a language only inside another essentially richer metalanguage. This solution is in harmony with the idea of reflexivity of thinking and it has become very fertile for mathematics and science in general. But it has normative nature | truth paradoxes are avoided in a way that in such frame we cannot even express paradoxical sentences. It is also too restrictive because, for the same reason we cannot express a situation in which there is a circular reference of some sentences to other sentences, no matter how common and harmless such a situation may be. Kripke showed that there is no natural restriction to the T-scheme and we have to accept it. But then we must also accept the riskiness of sentences | the possibility that under some circumstances a sentence does not have the classical truth value but it is undetermined. This leads to languages with three-valued semantics. Kripke did not give any definite model, but he gave a theoretical frame for investigations of various models | each fixed point in each three-valued semantics can be a model for the concept of truth. The solutions also have normative nature | we can express the paradoxical sentences, but we escape a contradiction by declaring them undetermined. Such a solution could become an analytical solution only if we provide the analysis that would show in a substantial way that it is the solution that models the concept of truth. Kripke took some steps in the direction of finding an analytical solution. He preferred the strong Kleene three-valued semantics for which he wrote it was "appropriate" but did not explain why it was appropriate. One reason for such a choice is probably that Kripke finds paradoxical sentences meaningful. This eliminates the weak Kleene three valued semantics which corresponds to the idea that paradoxical sentences are meaningless, and thus indeterminate. Another reason could be that the strong Kleene three valued semantics has the so-called investigative interpretation. According to this interpretation, this semantics corresponds to the classical determination of truth, whereby all sentences that do not have an already determined value are temporarily considered indeterminate. When we determine the truth value of these sentences, then we can also determine the truth value of the sentences that are composed of them. Kripke supplemented this investigative interpretation with an intuition about learning the concept of truth. That intuition deals with how we can teach someone who is a competent user of an initial language (without the predicate of truth T) to use sentences that contain the predicate T. That person knows which sentences of the initial language are true and which are not. We give her the rule to assign the T attribute to the former and deny that attribute to the latter. In that way, some new sentences that contain the predicate of truth, and which were indeterminate until then, become determinate. So the person gets a new set of true and false sentences with which he continues the procedure. This intuition leads directly to the smallest fixed point of strong Kleene semantics as an analytically acceptable model for the logical notion of truth. However, since this process is usually saturated only on some transfinite ordinal, this intuition, by climbing on ordinals, increasingly becomes a metaphor. This thesis is an attempt to give an analytical solution to truth paradoxes. It gives an analysis of why and how some sentences lack the classical truth value. The starting point is basic intuition according to which paradoxical sentences are meaningful (because we understand what they are talking about well, moreover we use it for determining their truth values), but they witness the failure of the classical procedure of determining their truth value in some "extreme" circumstances. Paradoxes emerge because the classical procedure of the truth value determination does not always give a classically supposed (and expected) answer. The analysis shows that such an assumption is an unjustified generalization from common situations to all situations. We can accept the classical procedure of the truth value determination and consequently the internal semantic structure of the language, but we must reject the universality of the exterior assumption of a successful ending of the procedure. The consciousness of this transforms paradoxes to normal situations inherent to the classical procedure. Some sentences, although meaningful, when we evaluate them according to the classical truth conditions, the classical conditions do not assign them a unique value. We can assign to them the third value, \undetermined", as a sign of definitive failure of the classical procedure. An analysis of the propagation of the failure in the structure of sentences gives exactly the strong Kleene three-valued semantics, not as an investigative procedure, as it occurs in Kripke, but as the classical truth determination procedure accompanied by the propagation of its own failure. An analysis of the circularities in the determination of the classical truth value gives the criterion of when the classical procedure succeeds and when it fails, when the sentences will have the classical truth value and when they will not. It turns out that the truth values of sentences thus obtained give exactly the largest intrinsic fixed point of the strong Kleene three-valued semantics. In that way, the argumentation is given for that choice among all fixed points of all monotone three-valued semantics for the model of the logical concept of truth. An immediate mathematical description of the fixed point is given, too. It has also been shown how this language can be semantically completed to the classical language which in many respects appears a natural completion of the process of thinking about the truth values of the sentences of a given language. Thus the final model is a language that has one interpretation and two systems of sentence truth evaluation, primary and final evaluation. The language through the symbol T speaks of its primary truth valuation, which is precisely the largest intrinsic fixed point of the strong Kleene three valued semantics. Its final truth valuation is the semantic completion of the first in such a way that all sentences that are not true in the primary valuation are false in the final valuation. (shrink)

Many commentators on Alfred Tarski have, following Hartry Field, claimed that Tarski's truth-definition was motivated by physicalism—the doctrine that all facts, including semantic facts, must be reducible to physical facts. I claim, instead, that Tarski did not aim to reduce semantic facts to physical ones. Thus, Field's criticism that Tarski's truth-definition fails to fulfill physicalist ambitions does not reveal Tarski to be inconsistent, since Tarski's goal is not to vindicate physicalism. I argue that Tarski's only published remarks that speak approvingly (...) of physicalism were written in unusual circumstances: Tarski was likely attempting to appease an audience of physicalists that he viewed as hostile to his ideas. In later sections I develop positive accounts of: (1) Tarski's reduction of semantic concepts; (2) Tarski's motivation to develop formal semantics in the particular way he does; and (3) the role physicalism plays in Tarski's thought. (shrink)

In this paper, I argue that the cleavage between the theory of reference and the theory of meaning, which under the influence of Quine has dominated a large part of the philosophy of language of the last fifty years, is based on a misrepresentation of Tarski's achievement and on an overestimation of the scope and value of disquotation. In particular, I show that, if we accept Davidson's critique of disquotation, the same kind of reasons that Quine offered in opposition to (...) the Carnapian theory of meaning also apply, mutatis mutandis, to the Tarskian theory of reference. (shrink)

Hilary Putnam's famous arguments criticizing Tarski's theory of truth are evaluated. It is argued that they do not succeed to undermine Tarski's approach. One of the arguments is based on the problematic idea of a false instance of T-schema. The other ignores various issues essential for Tarski's setting such as language-relativity of truth definition.

In this paper I present what I call the 'Metaphysical Problem of Truth', which consists in explaining in virtue of what all true sentences are true, and argue that a version of the Correspondence Theory of Truth is the most plausible solution to this problem.

Since the publication of Hartry Field’s influential paper “Tarski’s Theory of Truth” there has been an ongoing discussion about the philosophical import of Tarski’s definition. Most of the arguments have aimed to play down that import, starting with that of Field himself. He interpreted Tarski as trying to provide a physicalistic reduction of semantic concepts like truth, and concluded that Tarski had partially failed. Robert Stalnaker and Scott Soames claimed then that Field should have obtained a stronger conclusion, namely that (...) Tarski’s failure in his allegedly intended physicalistic reduction was total. From another front, Hilary Putnam argued for an even more sweeping thesis: “[a]s a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail.” Scott Soames followed suit also on this count, endorsing Putnam’s argument in section iii of the aforementioned paper; and John Etchemendy used a very similar argument to contend that “a Tarskian definition of truth [...] cannot possibly illuminate the semantic properties of the object language.” The paper argues that these contentions are based on several misunderstandings about the nature of a definition like Tarski’s. Regarding the Putnam-Soames-Etchemendy argument, we must apply a distinction Frege found natural to draw—and is natural to draw anyway—between what he called “constructive” definitions, definitions properly so called, which are mere stipulations, and what he described as “analytic” definitions, definitions which aim to make explicit the meaning of a term already in use. Regarding the Field-Soames-Stalnaker criticism, the paper contends that the true aims of a Tarskian truth definition are perfectly well fulfilled without Field’s amendments. (shrink)

To what in reality do the logically simple sentences with empirical content correspond? Two extreme positions can be distinguished in this regard: 'Great Fact' theories, such as are defended by Davidson; and trope-theories, which see such sentences being made the simply by those events or states to which the relevant main verbs correspond. A position midway between these two extremes is defended, one according to which sentences of the given sort are made tme by what are called 'dependence structures', or (...) in other words by certain complex concrete portions of reality between the parts of which relations of dependence are defined. Principles governing such dependence-structures are laid down, principles of an ontologically motivated sort which serve as basis for a topological semantics conceived as an altemative to standard set-theoretic approaches to semantics of the Tarskian sort. These principles are then used to resolve certain puzzles generated by the (semantically motivated) theory of events put forward by Davidson. (shrink)

CORCORAN REVIEWS THE 4 VOLUMES OF TARSKI’S COLLECTED PAPERS Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most (...) of the Aristotelian corpus is not about logic, whereas virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. (shrink)

To eliminate incompleteness, undecidability and inconsistency from formal systems we only need to convert the formal proofs to theorem consequences of symbolic logic to conform to the sound deductive inference model. -/- Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) thus unlike the formal proofs of symbolic logic provability cannot diverge from truth.

Within the (Haskell Curry) notion of a formal system we complete Tarski's formal correctness: ∀x True(x) ↔ ⊢ x and use this finally formalized notion of Truth to refute his own Undefinability Theorem (based on the Liar Paradox), the Liar Paradox, and the (Panu Raatikainen) essence of the conclusion of the 1931 Incompleteness Theorem.