Volker Halbach | ||||||

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In this book a develop a deflationist approach to semantics where truth and reference are treated as logico-mathematical notions.
In this book I provide a technical survey of many axiomatic theories of truth as well as a survey of many semantical approaches to the paradoxes.
Axiomatizing Semantic Theories of Truth? (with Martin Fischer, Jönne Speck and Johannes Stern) We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ω-categoricity and discuss its usefulness and limits. Norms for reflexive theories of truth (with Leon Horsten),
in In the past two decades we have witnessed a shift to axiomatic theories of truth. But in this tradition there has been a proliferation of truth theories. In this article we carry out a meta-theoretical reflection on the conditions that we should want axiomatic truth theories to satisfy. Self-reference in Arithmetic I (with Albert Visser), 7 (2014), 671-691
Review of Symbolic LogicA Gödel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence of arithmetic to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed points for the formulae are obtained. This paper is the first of two papers. In the present paper we focus on provability. In part II, we will consider other properties like Rosser provability and partial truth predicates. Self-reference in Arithmetic II (with Albert Visser), , 7 (2014), 692-712
Review of Symbolic LogicIn this sequel to \emph{Self-reference in arithmetic I} we continue our discussion of the question: What does it mean for a sentence of arithmetic to ascribe to itself a property? We investigate how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the expressing formulae are obtained. In this second part we look at some further examples. In particular, we study sentences apparently expressing their Rosser-provability, their own $\Sigma^0_n$-truth or their own $\Pi^0_n$-truth. Finally we offer an assessment of the results of both papers. The Henkin sentence (with Albert Visser), In this paper we discuss Henkin's question concerning a formula that has been described as expressing its own provability. We analyze enkin's formulation of the question and the early repsonses by Kreisel and Löb and sketch how this discussion led to the development of provability Logic. We argue that, in addition to that, the question has philosophical aspects that are still interesting. Reducing Compositional to Disquotational Truth, 2 (2009), 786-798
Review of Symbolic LogicUsually disquotational theories of truth, that is theories of truth based on the T-sentences or similar equivalences as axioms are often thought to be deductively weak. This is correct if the truth predicate is allowed to apply only to sentences not containing the truth predicate. By taking a slightly more liberal approach towards the paradoxes, I obtain a disquotational theory of truth that is proof-theoretically as strong as strong compositional theories such as the Kripke-Feferman theory although it doesn't coincide with these theories. On the Benefits of a Reduction of
Modal Predicates to Modal Operators, in A unary predicate is an expression that can be combined with a singular to give a formula. A unary operator is an expression that yields combined with a formula another formula. I consider whether formalising notions such as necessity, analyticity, and aprioricity should be formalised as unary operators or predicates. On the operator account quantified sentences are not easily formalised. I advocate a reduction of the predicate account to an operator conception of modalities. Benefits include the applicability of possible-worlds semantics to modal operators, avoidance of various paradoxes arising from a predicate conception, and ontological parsimony. Necessities and Necessary Truths:
A Prolegomenon to the Metaphysics of Modality, (with Philip Welch), Necessity is usually conceived as a sentential operator rather than a predicate. An intensional sentential operator does not allow one to express quantified statements such as ‘There are necessary a posteriori propositions’ or ‘All laws of physics are necessary’ in the framework of pure first-order logic. Replacing the operator conception of necessity (and other intensional notions) by the predicate conception causes various problems and forces one to reject many philosophical accounts involving necessity. It is argued that the expressive power of the predicate account might be restored if a truth predicate is added to the language of first-order modal logic; then the predicate ‘is necessary’ can be replaced by ‘is necessarily true’. We prove a result showing that this substitution is technically feasible by providing possible-worlds-semantics for the language with a predicate of necessity and performing the reduction of necessity to necessary truth. On a Side Effect of Solving Fitch's Paradox
by Typing Knowledge, It has been proposed to block Fitch's paradox by disallowing a predicate or sentential operator of knowledge that can be applied to sentences containing the same predicate or operator of knowledge. Furthermore it has been claimed that this move is not ad hoc as there is independent motivation for this restriction, because this restriction provides a solution also to paradoxes arising from self-reference like the paradox of the Knower. Such a solution is only needed if knowledge is treated as a predicate that can be diagonalized. However, if knowledge and possibility are conceived as such predicates with type restrictions, a new paradox arises. Very basic, jointly consistent assumptions on the predicates of knowledge and possibility yield an inconsistency if (a typed version of) the verifiability principle is added. Thus Fitch's paradox cannot be resolved by typing knowledge.
How not to state the T-sentences, In the context of a theory allowing for diagonalisation, the set of the T-sentences T[A]<->A is inconsistent, because of the Liar paradox. Often it is assumed that consistency can be restored by imposing the condition that the sentence A must not contain the truth predicate T. If the underlying language contains also a predicate of necessity (or another modality) and certain natural assumption are made on necessity, then the restricted T-sentences are shown still to yield an inconsistency, although the theory of truth and the theory of necessity taken separately a re perfectly consistent. The Axiomatizing Kripke’s Theory of Truth
(with Leon Horsten), We investigate axiomatizations of Kripke’s theory of truth based on the
Strong Kleene evaluation scheme for treating sentences lacking a truth value.
Feferman’s axiomatization KF formulated in classical logic is an indirect
approach, because it is not sound with respect to Kripke’s semantics in
the straightforward sense; only the sentence provably The Deflationist’s Axioms for Truth
(with Leon Horsten), in From a deflationist point of view, axiomatic theories of truth seem more appropriate than semantic approaches to truth. The question then arises which set of axioms for truth the deflationist ought to adopt. While there probably does not exist a single best axiomatic theory of truth, it is argued that the truth theory FS that was proposed by Friedman and Sheard looks more promising than is commonly appreciated. Computational Structuralism (with Leon Horsten), According to structuralism in philosophy of mathematics, arithmetic is about a certain structure. First-order theories are satisfied by (nonstandard) models that do not exhibit this structure. Proponents of structuralism have put forward various accounts of how we succeed to fix one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum’s theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On our account, the intended models of arithmetic are the notation systems which are computably intertranslatable with our notation system of the Arabic numerals. Bealers Masterargument: ein Lehrstück
zum Verhältnis von Metaphysik und Semantik (with Holger Sturm), In his article We agree with Bealer that the Master Argument applies against certain kinds
of Can We Grasp Consistency?, in In this paper, I will scrutinize a family of arguments which are supposed to show that consistency is not accessible in an epistemically relevant sense. If these arguments were sound, then consistency would be on a par with external facts: neither are directly accessible to us. External factors are not directly accessible because they need to be mediated in some way. Consistency would be inaccessible, according to these arguments, because proving or determining consistency in the relevant cases exceeds our intellectual capabilities. Thus, if consistency were actually inaccessible, then consistency could hardly play a role in an internalist account of epistemology. Traditionally (for instance, in Schlick’s account 1934), however, consistency has been seen as a main ingredient of coherence. Modern epistemologists, like BonJour 1985, have also used consistency as a criterion of coherence, and they have even used consistency in order to define coherence. The arguments against the accessibility of consistency jeopardizes this central role of consistency in any internalist account of epistemic justification. Thus, epistemic justification cannot imply then consistency of the belief system. If consistency is not directly accessible, then it cannot be used as a criterion for consistency on an internalist account. Possible Worlds Semantics for Modal Notions
Conceived as Predicates (with Hannes Leitgeb and Philip Welch), If L (or the box symbol of modal logic) is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where L is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate L, we tackle both problems. Given a frame (W,R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret L at every world in such a way that L[A] holds at a world w in W if and only if A holds at every world v in W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several `paradoxes’ (like Montague’s Theorem, Gödel’s Second Incompleteness Theorem, McGee’s Theorem on the omega-inconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of L at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. War Descartes erkenntnistheoretischer Voluntarist?, According to certain authors like Alvin Goldman and William Alston, normative epistemology presupposes epistemological voluntarism. This is the doctrine that epistemic behaviour like believing, judging, suspension of judgement and so on are voluntary actions. On this account, epistemic norms can have effects on our behaviour because we voluntarily follow those norms. But epistemic voluntarism seems to be in conflict with the observation that our epistemic behaviour is normally not subject to our voluntary control. Descartes has been cited as a typical proponent of epistemic voluntarism by Goldman, Alston and other authors. I argue that Descartes was not an epistemic voluntarist and that his epistemology does not presuppose this brand of voluntarism. In particular, I show that Descartes rejected voluntarism with respect to the two modes of epistemic behaviour most central to Descartes’ epistemology, viz.judgement and suspension of judgement. Finally Descartes’ account of the influence of norms on our epistemic behaviour turns out to be very close to Goldman’s own account. Modalized Disquotationalism, in An axiom to the effect that all instances of the disquotation scheme are necessary is combined with axioms for necessity. The resulting theory overcomes deficiencies of traditional disquotationalist theories of truth. Finally the exact proof-theoretic strength of the theory and some of its variants is determined. Disquotational Truth and Analyticity, The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences, yet not in the scope of negation, the system with the reflection schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to epsilon_0. How Innocent is Deflationism?, pp. 167-194The deflationist pretends to advocate a simple, clear and thin concept of truth by explaining how the phrase ``is true’’ axiomatized in a certain way can serve certain purposes. Unlike the representative of substantial accounts of truth, he does not commit himself to any deep and worrisome doctrines about the world, facts, coherence or correspondence. Maybe the deflationist makes a hash out of the notion of truth, but at least, it seems, he does so without any deep and dark mumbo-jumbo. But is he as innocent as he looks? Recently, several authors among them Horsten 1995, Shapiro 1998 and Ketland 1999 have reproached the deflationist with an alleged weighty consequence of the deflationist’s account. They have suggested that the deflationist has tied himself down to the conservativeness of his axioms and rules for truth. At first sight this reproach seems plausible because deflationism is often
characterized as a position opposed to `substantial’ theories of truth,
such as the correspondence theory. An insubstantial theory ought to have only
insubstantial consequences; that is, the theory should not imply any new non-semantical
consequences. In other words, the theory should not prove any new consequences
not involving truth; technically, it should be If the truth theory is conservative, it does not contribute any substantial insights to what we already know. Also, a proof of conservativeness yields a relative consistency proof. If the theory in which we reason is consistent and allows for the proof of conservativeness of the truth theory, then the truth theory is also consistent. Thus conservativeness is a desirable feature of a truth theory---especially in the eyes of a deflationist who advances a thin concept of truth. Some deflationists may have given the impression that their truth theories yield only trivial consequences. However, I am not aware of any deflationist who has explicitly bound him- or herself to the doctrine that his theory of truth is conservative, before the opponents of deflationism broached the topic. Therefore, the latter were very cautious in their attributions or tried to provide arguments for their charge. Recently, however, one arch-deflationist, Field 1999 has come close to confessing a commitment to conservativeness by saying:
In this paper, I investigate whether the deflationist can agree with Shapiro without being in conflict with some theorems on formal truth theories. Some theories of truth that the deflationists have advanced are not conservative indeed. That is, they do prove new non-semantical sentences and therefore yield, metaphorically speaking, new insights in non-semantical facts. Different formal arguments will be presented that show how the conservativeness requirement is violated by allegedly deflationist truth theories. Furthermore I shall evaluate these formal results with respect to the deflationist position. As will be shown, an exact formulation of the conservativeness requirement is very important. In order to judge the objections against deflationism one has to clarify two intricate and controversial points: what set of axioms and rules is adequate as a theory of truth from a deflationist point of view? And what kind of conservativeness is at issue? In this paper, two axiomatizations will mainly be considered: the traditional disquotation or T-sentences and the inductive defining clauses (a la Tarski) turned into axioms. Both axiom sets have been used by deflationists. However, it seems that many deflationists consider the pure T-sentences insufficient and prefer the inductive clauses. Whether the latter or the T-sentences are used makes a difference, since they behave in different ways with respect to conservativeness. But what kind of conservativeness is at issue at all? Even if the deflationist
is committed to conservativeness, it is unclear what Clearly, conservativeness over logic guarantees the complete neutrality of the truth theory with respect to any non-semantical questions, while conservativeness over a theory of expressions (or their Gödel codes) renders the axioms neutral only with respect to questions that are left undecided by the theory of expressions. Therefore, conservativeness over logic is more desirable but also more likely to be out of the deflationist’s reach. Thus, opponents have to be careful about what kind of conservativeness they ascribe to the deflationist. Both kinds of conservativeness will be discussed in the following. The plan of the paper is as follows. In the next somewhat technical section, I will explain the basic terminology and some basic results concerning conservativeness and make the above remarks more precise. In order to determine whether any kind of conservativeness follows from the deflationist account of truth, I will present the core doctrines of deflationism. I will check what particular axioms are appropriate as deflationist axioms for truth. Furthermore, it will be investigated whether anything about conservativeness follows from typical deflationist claims such as that truth is only a device of generalization or that truth is not a property. In two technical intermezzi I will recall some formal results relevant to the philosophical discussion on conservativeness. In the final sections the results of the findings of section 3 are surveyed. It will turn out that deflationism is wrong if it implies that an adequate
axiomatization of truth is conservative over Disquotationalism Fortified, in A variant of modalized disquotationalism is discussed. A sentence is said to be analytic in the truth predicate if and only if it is a logical consequence of the uniform T-sentences. It is claimed that the disquotationalist is committed to the view that T-analyticity parsed this way is sound, that is, if A is analytic in the truth predicate, then A. The resulting theory of truth turns out to be equivalent to the “Tarskian” theory of truth and is therefore much stronger than the theory of the pure T-sentences often associated with disquotationalism. Moreover, truth in foreign languages is discussed from a disquotationalist point of view. Truth and Reduction, In this paper ontological commitments are attributed to axiomatic systems rather than interpreted languages, when ontological reductions in the abstract sciences are to be carried out. In consequence, persons make ontological commitments to abstract objects by accepting formal theories. This feature allows to investigate ontological commitment and reduction in a naturalized setting. I consider subsystems of analysis, i.e., subsystems of second-order arithmetic extending Peano arithmetic. The ontological commitments of these systems to sets depend on the axioms postulating the existence of second-order objects. Proof-theoretic investigations have shown that some of these subsystems are reducible to certain axiomatic theories of truth formulated in the language of Peano arithmetic augmented by a one-place predicate symbol for truth. Hence certain set existence principles may be superseded by assumptions on the truth predicate. These reductions are considered to be ontological reductions: ontological commitments to sets of natural numbers are replaced by commitments to semantical principles. On the other hand, inverse results are known, where commitments to semantical principles are replaced by ontological commitments. Therefore certain semantical principles are related to certain set existence assumptions. In particular, I relate the commitment to parameterless arithmetical, arithmetical, predicative and inductively definable sets to the semantical principles of the Tarskian equivalences, the recursive definition of truth, compositional semantics and Kripke’s semantics, respectively. Two proof-theoretic remarks on EA+ECT (with
Leon Horsten), In this note two propositions about the epistemic formalization of Church’s Thesis (ECT) are proved. First it is shown that all arithmetical sentences deducible in Shapiro’s system of Epistemic Arithmetic (EA) from ECT are derivable from Peano arithmetic PA + uniform reflection for PA. Second it is shown that the system EA+ECT has the epistemic disjunction property and the epistemic numerical existence property for arithmetical formulas. On Lehrer’s Principle of Trustworthiness, (1999), pp.259-272According to the usual foundationalist picture of knowledge, basic beliefs provide the empirical input on which the whole empirical knowledge of a person should rely. There are convincing arguments showing that this picture is not completely convincing, because the proposed basic beliefs were not sufficient for providing foundations of knowledge or they were in need of further justification and therefore not really basic. Now coherentism claims that the distinction of basic and non-basic beliefs is not sensible at all. There may be basic beliefs and coherentism does not have to deny this, but these possibly existing basic beliefs do not figure prominently in a sound account of epistemic justification. The rejection of basic beliefs as foundations of knowledge, however, seems to deprive of the empirical input. But if knowledge does not rest on basic beliefs providing the empirical input, how can external factors influence our knowledge at all? Coherentism seems prone to a conception where epistemic agents are completely isolated from the world. Coherentists have developed several strategies to evade the isolation objection. Keith Lehrer’s approach to solve the problem became especially important and was thus discussed intensively. In order to give a rough idea how Lehrer accomplishes a plausible picture of how we acquire empirical information, I will present an example. Suppose that I accept that there is a red rose. This assumptions implies in particular that I do not believe that there is a red rose because of wishful thinking etc; rather I came to believe it with the objective of accepting it just in the case it is true; but so far my belief lacks justification and is thus not yet knowledge. I cannot conclude (and thereby justify) from my acceptance alone that there is a red rose indeed. Given some additional information, however, I can and do usually infer this. The additional premises required may include that I am not dreaming, that normal daylight is present, and that I know roses. The common feature of all these premises is that I must be a reliable epistemic agent in the present kind of situation. Exactly this is expressed by the principle of trustworthiness proposed by Lehrer in his “Theory of Knowledge” (1990):
He explains the benefits of this principle in the following way:
Although this account seems to be plausible at first, I will show that the principle of trustworthiness cannot be used as intended by Lehrer. By reformulating the principle of trustworthiness and by an appeal to Löb’s theorem I will show that only trivial instances of the principle are consistent with our basic assumptions on beliefs. Disquotationalism and Infinite Conjunctions, According to the disquotationalist theory of truth, the Tarskian equivalences, conceived as axioms, yield all there is to say about truth. Several authors have claimed that the expression of infinite conjunctions and disjunctions is the only purpose of the disquotationalist truth predicate. The way in which infinite conjunctions can be expressed by an axiomatized truth predicate is explored and it is considered whether the disquotationalist truth predicate is adequate. Conservative theories of classical truth, Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. The truth theories are formulated in the language of PA expanded by an additional unary predicate symbol for truth. PA + “there is a satisfaction class” is the theory with the clauses of Tarski’s inductive definition of truth turned into axioms with only arithmetical induction axioms. The conservativeness over PA of PA + “there is a satisfaction class” and of the theory FS with arithmetical induction only of Halbach (1994) is established by purely finitisitc means. postscript 2015: The conservativeness proof was shwon to be incorrect by Kentaro Fujimoto. Graham Leigh has corrected the proof. Tarskian and Kripkean truth, An analysis of Kripke’s theory of truth with the Strong Kleene evaluation scheme at the least fixed is provided in terms of a Tarskian hierarchy of languages. This hierarchy has been studied in Halbach (1996). In particular, some results on definability in Kripke’s theory are obtained via a translation of the language with an untyped truth predicate into a language with indexed truth predicates. Tarski-hierachies, The hierarchy of truth predicates for all levels smaller than thefirst non-recursive ordinal is equivalent to the ramified analytical hierarchy up to the same level. A system of complete and consistent truth, The theory FS is formulated in the language of arithmetic plus a unary predicate symbol T for truth. FS is given by the axioms of Peano arithmetic plus all induction axiom in the language FS. The special axioms and rules for T are the following: - axioms saying that truth commutes with all connectives and quantifiers. For instance, one axiom says that -A is true iff A is not true, where A is any sentence possibly containing T.
- From A one can conclude T[A], and vice versa. A is here again any sentence.
FS is shown to be equivalent to a system studied by Friedman and Sheard (1987). I follows from a result due to McGee (1985) that the system is omega-inconsistent. FS may be seen as an axiomatization of the finite levels of rule-of-revision
semantics. This insight is used in order to show that FS is proof-theoretically
equivalent to ramified analysis for all finite levels. This implies that FS
is arithmetically sound. If induction is restricted to arithmetical formulas,
the resutling system is proved to be conservative over PA. The theory of satisfaction
classes is employed in this argument ( |