Tarski has shown how the argumentation of the liar paradox can be used to prove a theorem about truth in formalized languages. In this paper, it is shown how the paradox concerning the least undefinable ordinal can be used to prove a no go-theorem concerning the notion of satisfaction in formalized languages. Also, the connection of this theorem with the absolute notion of definability is discussed.
Given any finite graph, which transitive graphs approximate it most closely and how fast can we find them? The answer to this question depends on the concept of “closest approximation” involved. In [8,9] a qualitative concept of best approximation is formulated. Roughly, a qualitatively best transitive approximation of a graph is a transitive graph which cannot be “improved” without also going against the original graph. A quantitative concept of best approximation goes back at least to [10]. A quantitatively best transitive (...) approximation is a transitive graph that makes the minimal number of mistakes against the original graph. In other words, the sum of the edges that are removed from and are added to the original graph is minimal. (shrink)
We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by (...) a different type of infinite additivity. (shrink)
I investigate what it means to have an interpretation of our language, how we manage to bestow a determinate interpretation to our utterances, and to which extent our interpretation of the world is determinate. All this is done in dialogue with van Fraassen’s insightful discussion of Putnam’s model-theoretic argument and of scientific structuralism.
This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.
Until now, antirealists have offered sketches of a theory of truth, at best. In this paper, we present a probabilist account of antirealist truth in some formal detail, and we assess its ability to deal with the problems that are standardly taken to beset antirealism.
In this paper, a general perspective on criteria of identity of kinds of objects is developed. The question of the admissibility of impredicative or circular identitycriteria is investigated in the light of the view that is articulated. It is argued that in and of itself impredicativity docs not constitute sufficient grounds for rejecting aputative identity criterion. The view that is presented is applied to Davidson's criterion of identity for events and to the structuralist criterion of identity of placesin a structure.
In this article, the prospects of deflationism about the concept of truth are investigated. A new version of deflationism, called inferential deflationism, is articulated and defended. It is argued that it avoids the pitfalls of earlier deflationist views such as Horwich’s minimalist theory of truth and Field’s version of deflationism.
If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)
In this article, we reflect on the use of formal methods in the philosophy of science. These are taken to comprise not just methods from logic broadly conceived, but also from other formal disciplines such as probability theory, game theory, and graph theory. We explain how formal modelling in the philosophy of science can shed light on difficult problems in this domain.
We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and (...) can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus. (shrink)
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 sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...) KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀. (shrink)
Jonathan Lowe has argued that a particular variation on C.I. Lewis' notion of strict implication avoids the paradoxes of strict implication. We show that Lowe's notion of implication does not achieve this aim, and offer a general argument to demonstrate that no other variation on Lewis' notion of constantly strict implication describes the logical behaviour of natural-language conditionals in a satisfactory way.
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 sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...) KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀. (shrink)
Criteria of identity should mirror the identity relation in being reflexive, symmetrical, and transitive. However, this logical requirement is only rarely met by the criteria that we are most inclined to propose as candidates. The present paper addresses the question how such obvious candidates are best approximated by means of relations that have all of the aforementioned features, i.e., which are equivalence relations. This question divides into two more basic questions. First, what is to be considered a ‘best’ approximation. And (...) second, how can these best approximations be found? In answering these questions, we both rely on and constructively criticize ground-breaking work done by Timothy Williamson. Guiding ideas of our approach are that we allow approximations by means of overlapping equivalence-relations, and that closeness of approximation is measured in terms of the number of mistakes made by the approximation when compared to the obvious candidate criterion. (shrink)
This paper outlines a framework for the abstract investigation of the concept of canonicity of names and of naming systems. Degrees of canonicity of names and of naming systems are distinguished. The structure of the degrees is investigated, and a notion of relative canonicity is defined. The notions of canonicity are formally expressed within a Carnapian system of second-order modal logic.
. Two simple generalized conversational implicatures are investigated :(1) the quantitative scalar implicature associated with ‘or’, and (2) the ‘not-and’-implicature, which is the dual to (1). It is argued that it is more fruitful to consider these implicatures as rules of interpretation and to model them in an algebraic fashion than to consider them as nonmonotonic rules of inference and to model them in a proof-theoretic way.
This paper is a contribution to the program of constructing formal representations of pragmatic aspects of human reasoning. We propose a formalization within the framework of Adaptive Logics of the exclusivity implicature governing the connective ‘or’.Keywords: exclusivity implicature, Adaptive Logics.
A series of unnoticeably small changes in an observable property may add up to a noticeable change. Crispin Wright has used this fact to prove that perceptual indiscriminability is a non-transitive relation. Delia Graff has recently argued that there is a 'tension' between Wright's assumptions. But Graff has misunderstood one of these, that 'phenomenal continua' are possible; and the other, that our powers of discrimination are finite, is sound. If the first assumption is properly understood, it is not in tension (...) with but is actually implied by the second, given a plausible physical assumption. (shrink)
A series of unnoticeably small changes in an observable property may add up to a noticeable change. Crispin Wright has used this fact to prove that perceptual indiscriminability is a non-transitive relation. Delia Graff has recently argued that there is a 'tension' between Wright's assumptions. But Graff has misunderstood one of these, that 'phenomenal continua' are possible; and the other, that our powers of discrimination are finite, is sound. If the first assumption is properly understood, it is not in tension (...) with but is actually implied by the second, given a plausible physical assumption. (shrink)
The difficulties with formalizing the intensional notions necessity, knowability and omniscience, and rational belief are well-known. If these notions are formalized as predicates applying to (codes of) sentences, then from apparently weak and uncontroversial logical principles governing these notions, outright contradictions can be derived. Tense logic is one of the best understood and most extensively developed branches of intensional logic. In tense logic, the temporal notions future and past are formalized as sentential operators rather than as predicates. The question therefore (...) arises whether the notions that are investigated in tense logic can be consistently formalized as predicates. In this paper it is shown that the answer to this question is negative. The logical treatment of the notions of future and past as predicates gives rise to paradoxes due the specific interplay between both notions. For this reason, the tense paradoxes that will be presented are not identical to the paradoxes referred to above. (shrink)
Earman (1993) distinguishes three notions of empirical indistinguishability and offers a rigorous framework to investigate how each of these notions relates to the problem of underdetermination of theory choice. He uses some of the results obtained in this framework to argue for a version of scientific anti- realism. In the present paper we first criticize Earman's arguments for that position. Secondly, we propose and motivate a modification of Earman's framework and establish several results concerning some of the notions of indistinguishability (...) in this modified framework. Finally, we interpret these results in the light of the realism/anti- realism debate. (shrink)
Earman (1993) distinguishes three notions of empirical indistinguishability and offers a rigorous framework to investigate how each of these notions relates to the problem of underdetermination of theory choice. He uses some of the results obtained in this framework to argue for a version of scientific anti-realism. In the present paper we first criticize Earman's arguments for that position. Secondly, we propose and motivate a modification of Earman's framework and establish several results concerning some of the notions of indistinguishability in (...) this modified framework. Finally, we interpret these results in the light of the realism/anti-realism debate. (shrink)
This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.
New epistemic principles are formulated in the language of Shapiros system of Epistemic Arithmetic. It is argued that some plausibility can be attributed to these principles. The relations between these principles and variants of controversial constructivistic principles are investigated. Special attention is given to variants of the intuitionistic version of Churchs thesis and to variants of Markovs principle.
An epistemic formalization of arithmetic is constructed in which certain non-trivial metatheoretical inferences about the system itself can be made. These inferences involve the notion of provability in principle, and cannot be made in any consistent extensions of Stewart Shapiro's system of epistemic arithmetic. The system constructed in the paper can be given a modal-structural interpretation.
We critically discuss Cleland''s analysis of effective procedures as mundane effective procedures. She argues that Turing machines cannot carry out mundane procedures, since Turing machines are abstract entities and therefore cannot generate the causal processes that are generated by mundane procedures. We argue that if Turing machines cannot enter the physical world, then it is hard to see how Cleland''s mundane procedures can enter the world of numbers. Hence her arguments against versions of the Church-Turing thesis for number theoretic functions (...) miss the mark. (shrink)
