A teaching document I've used in my courses on truth and on incompleteness. Aimed at students who have a good grasp of basic logic, and decent math skills, it attempts to give them the background they need to understand a proper statement of the classic results due to Gödel and Tarski, and sketches their proofs. Topics covered include the notions of language and theory, the basics of formal syntax and arithmetization, formal arithmetic (Q and PA), representability, diagonalization, and the (...) incompleteness and undefinability theorems. (shrink)

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (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 period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

In his seminal work “The Concept of Truth in Formalized Languages”, Alfred Tarski showed how to construct a formally correct and materially adequate definition of true sentence for certain formalized languages. These results have, eventually, been accepted and applauded by philosophers and logicians nearly in unison. Its Postscript, written two years later, however, has given rise to a considerable amount of controversy. There is an ongoing debate on what Tarski really said in the postscript. These discussions often regard Tarski as (...) putatively changing his logical framework from type theory to set theory. In what follows, we will compare the original results with those presented two years later. After a brief outline of Carnap’s program in The Logical Syntax of Language we will determine its significance for Tarski’s final results. (shrink)

Tarski "proved" that there cannot possibly be any correct formalization of the notion of truth entirely on the basis of an insufficiently expressive formal system that was incapable of recognizing and rejecting semantically incorrect expressions of language. -/- The only thing required to eliminate incompleteness, undecidability and inconsistency from formal systems is transforming the formal proofs of symbolic logic to use the sound deductive inference model.

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)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The result can (...) then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established (...) to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. -/- Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. -/- However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. -/- The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion. (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.

The aim of this study is to discuss in what sense one can speak about universal character of logic. The authors argue that the role of logic stands mainly in the generality of its language and its unrestricted applications to any field of knowledge and normal human life. The authors try to precise that universality of logic tends in: (a) general character of inference rules and the possibility of using those rules as a tool of justification of theorems of every (...) science, (b) principles of correct formulating of thoughts and using language, and (c) general deductive method of reasoning, i.e in the study of language syntax and semantics. The authors present two theories of language syntax and semantics as exemplifications based on the Tarski’s famous results in metalogic: (1) a formalization of universal syntax system by Andrzej Grzegorczyk based on a new primitive notion of quotation-marks operator, and (2) an enlarged system of axiomatic theory of syntax of any categorial language of Urszula Wybraniec-Skardowska’s in which the main role plays the relation of concatenation understood as a set-theoretical function. The study concludes in arguing that the importance of the result (2) lies not only in solving some philosophical problems but also in different fields of knowledge and spheres of human life as ex. in discussion and education as well. (shrink)

It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than (...) as a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with -/- ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” -/- I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. -/- Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019). (shrink)

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...) be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. -/- We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. -/- We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. -/- We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences. -/- . (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)

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)

Developing a suggestion of Wittgenstein, I provide an account of truth tables as formulas of a formal language. I define the syntax and semantics of TPL (the language of Tabular Propositional Logic), and develop its proof theory. Single formulas of TPL, and finite groups of formulas with the same top row and TF matrix (depiction of possible valuations), are able to serve as their own proofs with respect to metalogical properties of interest. The situation is different, however, for groups of (...) formulas whose top rows differ. For them I provide (i) a tree-style system of ‘row tree proofs’, which is shown to be sound and complete, and (ii) an alternative, re-writing strategy. (shrink)

It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal (...) logic of linear discrete time with gaps follows. (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)

The starting point of this paper concerns the apparent difference between what we might call absolute truth and truth in a model, following Donald Davidson. The notion of absolute truth is the one familiar from Tarski’s T-schema: ‘Snow is white’ is true if and only if snow is white. Instead of being a property of sentences as absolute truth appears to be, truth in a model, that is relative truth, is evaluated in terms of the relation between sentences and models. (...) I wish to examine the apparent dual nature of logical truth (without dwelling on Davidson), and suggest that we are dealing with a distinction between a metaphysical and a linguistic interpretation of truth. I take my cue from John Etchemendy, who suggests that absolute truth could be considered as being equivalent to truth in the ‘right model’, i.e., the model that corresponds with the world. However, the notion of ‘model’ is not entirely appropriate here as it is closely associated with relative truth. Instead, I propose that the metaphysical interpretation of truth may be illustrated in modal terms, by metaphysical modality in particular. One of the tasks that I will undertake in this paper is to develop this modal interpretation, partly building on my previous work on the metaphysical interpretation of the law of non-contradiction (Tahko 2009). After an explication of the metaphysical interpretation of logical truth, a brief study of how this interpretation connects with some recent important themes in philosophical logic follows. In particular, I discuss logical pluralism and propose an understanding of pluralism from the point of view of the metaphysical interpretation. (shrink)

Gila Sher interviewed by Chen Bo: -/- I. Academic Background and Earlier Research: 1. Sher’s early years. 2. Intellectual influence: Kant, Quine, and Tarski. 3. Origin and main Ideas of The Bounds of Logic. 4. Branching quantifiers and IF logic. 5. Preparation for the next step. -/- II. Foundational Holism and a Post-Quinean Model of Knowledge: 1. General characterization of foundational holism. 2. Circularity, infinite regress, and philosophical arguments. 3. Comparing foundational holism and foundherentism. 4. A post-Quinean model of knowledge. (...) 5. Intellect and figuring out. 6. Comparing foundational holism with Quine’s holism. 7. Evaluation of Quine’s Philosophy -/- III. Substantive Theory of Truth and Relevant Issues: 1. Outline of Sher’s substantive theory of truth. 2. Criticism of deflationism and treatment of the Liar. 3. Comparing Sher’s substantive theory of truth with Tarski’s theory of truth. -/- IV. A New Philosophy of Logic and Comparison with Other Theories: 1. Foundational account of logic. 2. Standard of logicality, set theory and logic. 3. Psychologism, Hanna’s and Maddy’s conceptions of logic. 4. Quine’s theses about the revisability of logic. -/- V. Epilogue. (shrink)

Alfred Tarski was one of the greatest logicians of the twentieth century. His influence comes not merely through his own work but from the legion of students who pursued his projects, both in Poland and Berkeley. This chapter focuses on three key areas of Tarski's research, beginning with his groundbreaking studies of the concept of truth. Tarski's work led to the creation of the area of mathematical logic known as model theory and prefigured semantic approaches in the philosophy of language (...) and philosophical logic, such as Kripke's possible worlds semantics for modal logic. We also examine the paradoxical decomposition of the sphere known as the Banach–Tarski paradox. Finally we examine Tarski's work on decidable and undecidable theories, which he carried out in collaboration with students such as Mostowski, Presburger, Robinson and others. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

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.

The book is a collection of papers and aims to unify the questions of syntax and semantics of language, which are included in logic, philosophy and ontology of language. The leading motif of the presented selection of works is the differentiation between linguistic tokens (material, concrete objects) and linguistic types (ideal, abstract objects) following two philosophical trends: nominalism (concretism) and Platonizing version of realism. The opening article under the title “The Dual Ontological Nature of Language Signs and the Problem of (...) Their Mutual Relations” provides a broad introduction into the problem area connected with this differentiation, while the logic-formal characteristics of the distinction are framed in the work entitled “On the Type-Token Relationships” (Chapter 1). The basic part of the book deals with issues relating to syntax (Chapters 2-4) and semantics of language (Chapters 5-6), as well as pertaining to syntactic-semantic pragmatic questions (Chapters 7-13). Throughout the book, language, categorial language, is characterized syntactically as generated by classical categorial grammar (Chapter 2) and formalized on two opposing levels: as language of expression-tokens (level of tokens) and language of expression-types (level of types). The author’s considerations contained in Chapters 2 and 4 lead to the important philosophical conclusion that in formal-logical syntactic studies on language the assumption that expression-types constitute the primary language layer while expression-tokens make the secondary one, can be neglected; thus, this speaks in favour of the opposing standpoint—the concretistic one—in the ontology of language syntax. In the works “Meaning and Interpretations”, Parts I and II (Chapters 5 and 6), it is underlined, however, that such semantic concepts as: meaning, denotation and interpretation are defined on the types level, yet their formal definitions require making use of notions of the tokens level. The semantic notions introduced in the above-mentioned articles are also used in the following works of the present selection, under the titles: “Three Principles of Compositionality” and “On Metaknowledge and Truth” (Chapters 7 and 8). They formalize two principles of compositionality that are well known in the literature on the subject, deriving from Frege, i.e. those of meaning and of denotation; they are related to the syntactic principle of compositionality which was introduced by the author. All the three principles are, at the same time, three conditions of homomorphism of categorial language algebra into three kinds of non-standard models of language (one syntactic and two semantic ones: intensional and extensional), which allows introducing three definitions of truthfulness into these models. The next two works in the collection, entitled: “On Language Adequacy” and “What is the Sense in Logic and Philosophy of Language” (Chapters 9 and 10) concern adequacy of categorial language syntax along with its dual semantics: intensional and extensional, and categorial compatibility of any of its syntactic categories with two corresponding semantic categories: intensional and extensional, based on the compatibility the syntactic category of each language expression with the ontological category assigned to its denotatum. The well-known problem of categorial compatibility for first-order quantifiers finds its solution in the paper “Categories of First-Order Quantifiers” (Chapter 11). In the work “Logic and Ontology of Language” (Chapter 12), being in a sense a summary of the considerations presented in the preceding chapters of the book, language is treated as an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. Application—like throughout the book—of tools of classical logic and set theory has resulted in emergence of a general formal logical theory of syntax, semantics and pragmatics of language, which takes into account duality in the understanding of linguistic expressions as tokens (concretes) and types (abstract objects). In terms that take into account a functional approach to language itself, there comes out an ontological neutrality of logic with respect to existential assumptions relating to the ontological nature of linguistic expressions and their extra-linguistic ontological counterparts. The issues connected with applying logic while explaining the manner of using linguistic tokens and linguistic types to determine notions of language communication are raised and illustrated in the last chapter of the work, bearing the title “A Logical Conceptualization of Knowledge on the Notion of Language Communication”. (shrink)

The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic value of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide the valid deductive inference. Sound deductive conclusions are the result of these finite string transformation rules.

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. Two of his works appeared posthumously [Hist. Philos. Logic 7 (1986), no. 2, 143--154; MR0868748 (88b:03010); Tarski and Givant, A formalization of set theory without variables, Amer. Math. Soc., Providence, RI, 1987; MR0920815 (89g:03012)]. Tarski's voluminous writings were widely scattered in numerous journals, some quite rare. It has been extremely difficult to study the development of Tarski's thought and to trace the interconnections and interdependence of his various papers. Thanks to the present collection all this has changed, and it is likely that the increased accessibility of Tarski's papers will have the effect of increasing Tarski's already enormous influence. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

We consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth (...) to the compound formulas of PA over N---I_PA(N; SV ) and I_PA(N; SC). We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both I_PA(N; SV ) and I_PA(N; SC). We then show: (a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under I_PA(N; SV ), then this assignment corresponds to the classical non-finitary putative standard interpretation I_PA(N; S) of PA over the domain N; and (b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment I_PA(N; SC), from which we may finitarily conclude that PA is consistent. We further conclude that the appropriate inference to be drawn from Goedel's 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which---under interpretation---are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N. We conclude from this that Lucas' Goedelian argument is validated if the assignment I_PA(N; SV ) can be treated as circumscribing the ambit of human reasoning about `true' arithmetical propositions, and the assignment I_PA(N; SC) as circumscribing the ambit of mechanistic reasoning about `true' arithmetical propositions. (shrink)

There is a long-standing gap in the literature as to whether Gödelian incompleteness constitutes a challenge for Neo-Logicism, and if so how serious it is. In this paper, I articulate and address the challenge in detail. The Neo-Logicist project is to demonstrate the analyticity of arithmetic by deriving all its truths from logical principles and suitable definitions. The specific concern raised by Gödel’s first incompleteness theorem is that no single sound system of logic syntactically implies all arithmetical truths. (...) I set out some responses that initially seem appealing and explain why they are not compelling. The upshot is that Neo-Logicism either offers an epistemic route only to some truths of arithmetic; or that it has to move from a syntactic to a semantic notion of logical consequence, which risks undermining its epistemic goals. I conclude by considering Crispin Wright’s recent attempt to address Gödelian incompleteness, which I argue is not satisfactory. (shrink)

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.

Tarski's Undefinability of Truth Theorem comes in two versions: that no consistent theory which interprets Robinson's Arithmetic (Q) can prove all instances of the T-Scheme and hence define truth; and that no such theory, if sound, can even express truth. In this note, I prove corresponding limitative results for validity. While Peano Arithmetic already has the resources to define a predicate expressing logical validity, as Jeff Ketland has recently pointed out (2012, Validity as a primitive. Analysis 72: 421-30), no theory (...) which interprets Q closed under the standard structural rules can define nor express validity, on pain of triviality. The results put pressure on the widespread view that there is an asymmetry between truth and validity, viz. that while the former cannot be defined within the language, the latter can. I argue that Vann McGee's and Hartry Field's arguments for the asymmetry view are problematic. (shrink)

The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬G(x, y)→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[G(x, y)→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's indefinability theorem (...) in mathematical logic. Gödel numbering is the key for arithmetic modeling applied in this paper. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes (...) multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

Kuhn's alleged taxonomic interpretation of incommensurability is grounded on an ill defined notion of untranslatability and is hence radically incomplete. To supplement it, I reconstruct Kuhn's taxonomic interpretation on the basis of a logical-semantic theory of taxonomy, a semantic theory of truth-value, and a truth-value conditional theory of cross-language communication. According to the reconstruction, two scientific languages are incommensurable when core sentences of one language, which have truth values when considered within its own context, lack truth values when considered within (...) the context of the other due to the unmatchable taxonomic structures underlying them. So constructed, Kuhn's mature interpretation of incommensurability does not depend upon the notion of truth-preserving translatability, but rather depends on the notion of truth-value-status-preserving cross-language communication. The reconstruction makes Kuhn's notion of incommensurability a well grounded, tenable and integrated notion.Author Keywords: Incommensurability; Thomas Kuhn; Taxonomic structures; Lexicons; Truth-value; Untranslatability; Cross-language communication. (shrink)

The generalized conclusion of the Tarski and Gödel proofs: All formal systems of greater expressive power than arithmetic necessarily have undecidable sentences. Is not the immutable truth that Tarski made it out to be it is only based on his starting assumptions. -/- When we reexamine these starting assumptions from the perspective of the philosophy of logic we find that there are alternative ways that formal systems can be defined that make undecidability inexpressible in all of these formal systems.

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, `it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in (...) modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal. (shrink)

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)

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)

This paper presents four key results. Firstly, it distinguishes between _partial_ and _consistent_ assertion of a sentence, and introduces the concept of an _equivocal_ sentence, which is both partially asserted and partially denied. Secondly, it proposes a novel definition of truth, stating that _a true sentence is one that is consistently asserted_. This definition is immune from the Liar paradox, does not restrict classical logic, and can be applied to declarative sentences in the language used by any particular person. Thirdly, (...) the paper introduces an epistemic model of language, known as _assertional language_, which is used to formalize the definition of truth. Finally, it provides an argument for the falsity of so-called Liar sentences. The paper also discusses Tarski's solution to the Liar paradox and argues for the abandonment of the Tarski Scheme: _'p' is true if and only if p_, in the context of everyday language. The proposed definition of truth can be viewed as a formal account of the correspondence theory. The epistemic model is a powerful concept on its own that allows for combining different languages in a meaningful way. This model is uniquely capable of reflecting on epistemic inconsistencies, such as logical paradoxes, in a consistent manner. (shrink)

Tarski's analysis of the concept of truth gives rise to a hierarchy of languages. Does this fragment the concept all the way to philosophical unacceptability? I argue it doesn't, drawing on a modification of Kaplan's theory of indexicals.

We present Logical Description Grammar (LDG), a model ofgrammar and the syntax-semantics interface based on descriptions inelementary logic. A description may simultaneously describe the syntacticstructure and the semantics of a natural language expression, i.e., thedescribing logic talks about the trees and about the truth-conditionsof the language described. Logical Description Grammars offer a naturalway of dealing with underspecification in natural language syntax andsemantics. If a logical description (up to isomorphism) has exactly onetree plus truth-conditions as a model, it completely specifies thatgrammatical (...) object. More common is the situation, corresponding tounderspecification, in which there is more than one model. A situation inwhich there are no models corresponds to an ungrammatical input. (shrink)

In the period after the First World War when the various national-ideological “truths” that led to it were not well resolved which resulted in the Second World War, one of the greatest world crises occurs. In those turbulent times, one philosopher renounces his national identity (changes his religion and name), wanting not to save himself from an evil world that is emerging but to join the creation of a completely new world – the world of modern logic. This man is (...) Alfred Tarski, whom many recognize as “the man who defined truth”, linking him to the formulation of the distinction between object and metalanguage. In the paper, I argue with historians of philosophy who claim that this distinction was first set by the logician and philosopher of the Lvov-Warsaw school Stanislaw Leśniewski (whose Tarski was the only doctoral student) accusing Tarski of violating professional laws without attributing the authorship of this distinction to Leśniewski. I show that this is not true and I cite the places where Tarski clearly emphasizes Leśniewski ‘s contribution. (shrink)

The paper presents Alfred Tarski’s debate with the semantic antinomies: the basic Liar Paradox, and its more sophisticated versions, which are currently discussed in philosophy: Strengthen Liar Paradox, Cyclical Liar Paradox, Contingent Liar Paradox, Correct Liar Paradox, Card Paradox, Yablo’s Paradox and a few others. Since Tarski, himself did not addressed these paradoxes—neither in his famous work published in 1933, nor in later papers in which he developed the Semantic Theory of Truth—therefore, We try to defend his concept of truth (...) against these antinomies. It is demonstrated that Tarskian theory of truth is resistant to the paradoxes and it is still the best solution to avoid the antinomies and remain within a classical logic, that is, accepting the laws of noncontradiction, excluded middle, and the principle of bivalence. Thus, the goal of the paper is double—firstly, to show that none of the versions of the Liar Paradox’s is a serious threat to Tarski’s concept of truth, and secondly, that Semantic Theory of Truth allows to remain within classical logic, and at the same time, avoid antinomies—which makes it the most attractive among classical theories of truth. (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)

►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)

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken (...) together, have substantial logical strength, amounting to a kind of abstract consistency statement. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed in discussions of (...) axiomatic theories of truth. These results are then used to address a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism. -/- This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now. (shrink)