The focus in this essay will be upon the paradoxes, and foremostly in set theory. A central result is that the librationist set theory £ extension \Pfund $\mathscr{HR}(\mathbf{D})$ of \pounds \ accounts for \textbf{Neumann-Bernays-Gödel} set theory with the \textbf{Axiom of Choice} and \textbf{Tarski's Axiom}. Moreover, \Pfund \ succeeds with defining an impredicative manifestation set $\mathbf{W}$, \emph{die Welt}, so that \Pfund$\mathscr{H}(\mathbf{W})$ %is a model accounts for Quine's \textbf{New Foundations}. Nevertheless, the points of view developed support the view (...) that the truth-paradoxes and the set-paradoxes have common origins, so that the librationist resolutions of the set theoretic paradoxes are at the same time resolutions of the truth theoretic paradoxes. Both the librationist resolutions of the set theoretic paradoxes and the truth theoretic paradoxes have non-trivial philosophical implications: librationist set theories have the consequence that there are no absolutely uncountable sets, and librationist truth theories allow the use of syntactical modalities in ways which circumvent limitations as those of \parencite{Montague1963}, and a truth predicate which is useful for more precise philosophical discourse. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, (...) and finite-universe logic. We review the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and (...) to computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to (...) Bourbaki’s system the axiom of universes for the purpose of considering the theory of categories. In this regard, we make some historical and epistemological remarks that could explain the conservative attitude of the Group. (shrink)

By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as defined (...) by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given. (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)

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be (...) used to reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ${\mathrm {ZFC}}$ to enable the development of interesting mathematics. We propose an axiomatic system ${\mathrm {BZFC}}$, obtained by analysing the ${\mathrm {ZFC}}$ -axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set. Our theory is naturally bi-interpretable with ${\mathrm {ZFC}}$, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1]. Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory. (shrink)

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take (...) care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski. In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical (...) terms, since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)

We study the class of all strongly constructivizable models having \(\omega \) -stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean \(\Sigma ^1_1\) -algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of all Boolean \(\Sigma ^1_1\) -algebras. This gives a characterization to the (...) Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with \(\omega \) -stable theories. (shrink)

Is mathematics 'entangled' with its various formalisations? Or are the central concepts of mathematics largely insensitive to formalisation, or 'formalism free'? What is the semantic point of view and how is it implemented in foundational practice? Does a given semantic framework always have an implicit syntax? Inspired by what she calls the 'natural language moves' of Gödel and Tarski, Juliette Kennedy considers what roles the concepts of 'entanglement' and 'formalism freeness' play in a range of logical settings, from computability (...) and set theory to model theory and second order logic, to logicality, developing an entirely original philosophy of mathematics along the way. The treatment is historically, logically and set-theoretically rich, and topics such as naturalism and foundations receive their due, but now with a new twist. (shrink)

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a (...) subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem (...) and Barwise compactness. (shrink)

Tarski defined a way of assigning to each Boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from ℕ, such that two Boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a Boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In this (...) paper we analyze the complexity of the question "Does B have invariant x?" For each x ∈ In we define a complexity class Γx that could be either Σⁿ, Πⁿ, Σⁿ ∧ Πⁿ, or Πω+1 depending on x, and we prove that the set of indices for computable Boolean algebras with invariant x is complete for the class Γx. Analogs of many of these results for computably enumerable Boolean algebras were proven in earlier works by Selivanov. In a more recent work, he showed that similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new similar results for the complexity of various isomorphism problems for dense Boolean algebras. (shrink)

In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.

The nominalistic ontology of Kotarbinski, Slupecki and Tarski does not provide any direct interpretations of the sets of higher types which play important roles in type theory and in set theory. For this and other reasons I will interpret those theories as descriptions of some finite structures which are actually constructed in human imaginations and stored in their memories. Those structures will be described in this lecture. They are hinted by the idea of Skolem functions and Hilbert's (...) -symbols, and they constitute a finitistic modification of Tarski's concept of a model. They suggest also a form of the evolutionary process which leads to the development of human intelligence and language. (shrink)

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a (...) subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings. (shrink)

The work on set theory made by A. Tarski in the years 1924-1950 is very interesting, but little know.We develope partial questions in set theory in the moment that A. Tarski intervenes and his contributionsand also influences.The principals aims in this development are:1. The axiom of choice [A.C.] and his equivalents;2. the general continuum hypothesis [G.C.H.] and the A.C.;3. the dual trichotomy principle;4. the inaccessible cardinals and his relation with the A.C. and the G.C.H.;5. the notion (...) of finite set and his relation with the A.C and the G.C.H.;6. the almost disjoint set and their relation with the G.C.H.;and finally,7. the results in Berkeley. (shrink)

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given (...) sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

In this paper the authors argue that if Tarski’s definition of truth for the calculus of classes is correct, then set theories which assert the existence of proper classes (classes which are not the member of anything) are incorrect.

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend (...) on the context in which they are embedded. The context here will be the theory CZF− of constructive Zermelo–Fraenkel set theory but without -Induction . Let INAC be the statement that for every set there is an inaccessible set containing it. CZF−+INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF−+INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF−+INAC is a small ordinal known as the Feferman–Schütte ordinal Γ0. (shrink)

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work and (...) with an earlier paper written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper. (shrink)

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic 7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work (...) and with an earlier paper written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper. (shrink)

We recall one Tarski's result about the existence of a non-zero additive measure defined on power set of an infinite set vanishing on finite subsets. This rather surprising result goes back to 1930 and it allows to introduce a non-trivial linear functional invariant under changes of finitely many inputs.

Algebraic Set Theory, a reconsideration of Zermelo-Fraenkel set theory in category-theoretic terms, has been built up in the mid-nineties by André Joyal and Ieke Moerdijk. Since then, it has developed into a whole research program. This paper gets back to the original formulation by Joyal and Moerdijk, and more specifically to its first three axioms. It explains in detail that these axioms set up a framework directly linked to descent theory, a theory having to do with (...) the shift from local data to a global item in modern algebraic geometry. Fibered categories, introduced by Grothendieck, provide a powerful framework for descent theory: They formalize in a very general way the consideration of local data of different kinds over the objects of some base category. The paper shows that fibered categories fit Joyal and Moerdijk’s axiomatization in a natural way, since the latter actually aims to secure a descent condition. As a result, Algebraic Set Theory is shown to accomplish, not only an original and fruitful combination of set theory with category theory, but the genuine graft of a deeply geometric idea onto the usual setting of zfc. (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.

Tarski's general conception of logic placed it at the center of all rational thought, and he took its aim to be the creation of a unified conceptual apparatus. In pursuit of this conviction, from his base at the University of California in Berkeley in the post-war years he campaigned vigorously on behalf of logic, locally, nationally and internationally. Though Tarski was ecumenical in his efforts to establish the importance of logic in these various ways, in his own work—even (...) that part of it which is considered to be philosophically significant—he was first and foremost a mathematician, and, even more so, a set-theoretical one. The paper traces this particular conception of logic back to the influence of his teachers in set theory and topology in Warsaw, and then details how it affected his explication of the notions of satisfaction, truth, logical consequence and logical term. (shrink)

The purpose of this article is to highlight a selected few of Alfred Tarski's career achievements. The choice of these achievements is subjective. Section 1 is a general sketch of his life and work, emphasizing his role as researcher, teacher, organizer and founder of a scientific school. Section 2 discusses his contributions to set theory. Section 3 discusses his contributions to the foundations of geometry and to measure theory. Section 4 looks at his metamathematical work, and especially (...) the decision problem for formalized theories. Section 5 is a selected bibliography to illustrate Sects. 1–4. (shrink)

This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.

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)

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)

A hallmark of correspondence theories of truth is the principle that sentences are made true by some truth-makers. A well-known objection to treating Tarski’s definition of truth as a correspondence theory has been put forward by Donald Davidson. He argued that Tarski’s approach does not relate sentences to any entities (like facts) to which true sentences might correspond. From the historical viewpoint, it is interesting to observe that Tarski’s philosophical teacher Tadeusz Kotarbinski advocated an ontological doctrine (...) of reism which accepted only concrete individuals and rejected all such abstract entities as facts, states of affairs, properties, and sets. Kotarbinski’s physicalism influenced Tarski who also avoided concepts like “fact” and “property” in his theory of truth, but—unlike Kotarbinski—he used freely set-theoretical terminology. In his mature work in model theory in the 1950s, Tarski used systematically the notion of a relational system (i.e., a domain of objects with designated elements, subsets, and relations). Wilfrid Hodges has argued that the notions of “structure” and “truth in a structure” appeared in Tarski’s work only in 1950. In my view, one can find the main ingredients of the model-theoretic account of truth already in the 1930s. These considerations suggest, against Davidson, that Tarski’s definition presupposes that material truth is always related to some kind of truth-maker. Further, facts as truth-makers can be reconstructed by employing the resources of model theory. (shrink)

Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical (...) logic -- Deflationism -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)