We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘\-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules are all valid in fixed-point models; the second by ‘\-like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. We show that (...) to any natural system in one cluster—corresponding to natural variations on induction schemata—there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth. (shrink)

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, (...) we obtain results on the quantifier complexity, finite axiomatizability and relative strength of schemes for Δn+1-formulas. (shrink)

A compositional nihilist believes that the only objects that exist are simples. However, a non-nihilist believes in the existence of composite objects and challenges the nihilist to explain why there are true sentences about chairs, tables, etc., if composite objects do not exist. Different nihilist views have been suggested to explain this (the paraphrase strategy and the truthmaker theory), but I believe that they are unsuccessful (either they do not successfully paraphrase every sentence apparently about composite objects, or (...) they are not precise about the truthmakers of those sentences). I will suggest that a new truthmaker theory, which uses resources from Cameron’s truthmaker theory and van Inwagen’s paraphrase strategy, can give such an explanation and does not face the problems of other theories. Therefore, we have a good reason to accept this truthmaker theory as the best nihilist way to accommodate sentences apparently about composite objects. (shrink)

This paper considers one of the most significant and controversial attempts to account for the meaning of pejoratives as lexical items, namely Hom and May’s. After outlining the theory, we pinpoint sets of pejorative sentences that come out true on their account and for which the question as to whether they are compatible with the view advocated by them (so-called Moral and Semantic Innocence) remains open. Helping ourselves to the standard model-theoretical framework Hom and May (presumably) work in, (...) we prove they are compatible with the view. Given that the issues of both the moral import of pejoratives and the practical effects of their utterance are not settled by the proof, we then highlight unwelcome moral and pragmatic implications for some of the pejorative sentences under scrutiny, thereby showing that the view, broadly understood, is not as morally and semantically innocuous as it is meant to be. (shrink)

RESUMO O presente artigo contempla uma das tentativas mais significativas e controversas de explicar o significado de pejorativos como itens lexicais, a saber, a de Hom e May. Após apresentarmos em linhas gerais a teoria, identificamos conjuntos de sentenças pejorativas que saem verdadeiras nessa teoria e para as quais a questão da sua compatibilidade com a visão por eles defendida (a chamada Inocência Moral e Semântica) permanece em aberto. Explorando o arcabouço teórico padrão da teoria dos modelos em que Hom (...) e May (presumivelmente) trabalham, damos uma prova de sua compatibilidade com a visão. Dado que as questões relativas tanto à importância moral de pejorativos quanto aos efeitos práticos de seu proferimento não são resolvidas pela prova, destacamos, em seguida, implicações morais e pragmáticas indesejáveis para algumas das sentenças pejorativas sob escrutínio, mostrando assim que a visão de Hom e May, interpretada de maneira ampla, não é tão moral e semanticamente inócua como deveria ser. (shrink)

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...) a broadly Dummettian framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true-as the logicist contends-then there is tension between this conservation requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility, of arithmetic truth. (shrink)

This chapter contains sections titled: Preamble The case of proper names The difference principle The ‘within a world’ version of the argument using the difference principle Sentences containing “actual” and “actually” Demonstrative adjectives Natural kind terms A passing comment on centering Where to from here?

Some earlier remarks Michael Dummett made on Gödel’s theorem have recently inspired attempts to formulate an alternative to the standard demonstration of the truth of the Gödel sentence. The idea underlying the non-standard approach is to treat the Gödel sentence as an ordinary arithmetical one. But the Gödel sentence is of a very specific nature. Consequently, the non-standard arguments are conceptually mistaken. In this paper, both the faulty arguments themselves and the general reasons underlying their failure are analysed. The analysis (...) reveals the true nature of the epistemological relation between the Gödel sentence and its numerical instances. (shrink)

Various authors of logic texts are cited who either suggest or explicitly state that the Gödel incompleteness result shows that some unprovable sentence of arithmetic is true. Against this, the paper argues that the matter is one of philosophical controversy, that it is not a mathematical or logical issue.

Proofs of Gödel's First Incompleteness Theorem are often accompanied by claims such as that the gödel sentence constructed in the course of the proof says of itself that it is unprovable and that it is true. The validity of such claims depends closely on how the sentence is constructed. Only by tightly constraining the means of construction can one obtain gödel sentences of which it is correct, without further ado, to say that they say of themselves that they (...) are unprovable and that they are true; otherwise a false theory can yield false gödel sentences. (shrink)

Paraconsistent and dialetheist approaches to a theory of truth are faced with a problem: the expressive resources of the logic do not suffice to express that a sentence is just true—i.e., true and not also false—or to express that a sentence is consistent. In his recent book, Spandrels of Truth, Jc Beall proposes a ‘just true’-operator to identify sentences that are true and not also false. Beall suggests seven principles that a ‘just true’-operator must (...) fulfill, and proves that his operator indeed fulfills all of them. He concludes that just true has been expressed in the language. I argue that, while the seven conditions may be necessary for an operator to express just true, they are not jointly sufficient. Specifically, first, I prove that a further plausible desideratum for necessary conditions on ‘just true’ is not fulfilled by Beall's proposal, namely that ‘just true’ ascriptions should themselves be just true, and not also false (or equivalently, that the ‘just true’-operator iterates). Second, I show that Beall's operator does not adequately express just true, but that it merely captures an arbitrary proper subset of the just true sentences. Further, there is no prospect of extending the proposal in order to encompass a more reasonable subset of the just true sentences without presupposing that we have antecedent means to characterize the class of just true sentences. (shrink)

In Tractatus, Wittgenstein held that there are null sentences – prominently including logical truths and the truths of mathematics. He says that such sentences are without sense (sinnlos), that they say nothing; he also denies that they are nonsensical (unsinning). Surely it is what a sentence says which is true or false. So if a sentence says nothing, how can it be true or false? The paper discusses the issue.

Frege’s claim that sentences are names of truth-values, I argue, was drawn to fit the formal project, but it respects our pre-theoretical intuitions and does not undermine the sentence’s central semantic role. I do a minimal work both on the expression and on its referent, connecting the sentence and the definite description, suggesting an intuitive referent for a true sentence, suggesting a motive for Frege’s choice of the truth-values as referents, and finally suggesting an understanding of the False (...) as a referent. (shrink)

In In Contradiction, Graham Priest shows, as clearly as anything like this can be shown, that it is coherent to maintain that some sentences can be both true and false at the same time. As a consequence, some contradictions are true, and an appreciation of this possibility advances our understanding of the nature of logic and language.

There are sentences that express the same temporally fully specified proposition at all contexts--call them 'context-insensitive, temporally specific sentences.' Sentence (1) 'Obama was born in 1961' is a case in point: at all contexts, it expresses the proposition ascribing to the year 1961 the property of being a time in which Obama was born. Suppose that someone uttered (1) in a context located on Christmas 2000 in our world. In this context, (1) is a true sentence about (...) the past. Moreover, it seems impossible that (1) will be false in a successive context (one located, say, on Christmas 2020 in our world). More generally, one might be tempted to endorse the following principle: if a context-insensitive, temporally specific sentence is uttered in a context in which it is about the past and takes a certain truth value in this context, it cannot be the case that it takes a different truth value in a successive context located in the same world. In this paper, we present linguistic evidence that shows that this principle fails. On this basis, we draw an apparently crazy conclusion: the past can change. We then explain why this conclusion is not that crazy, after all. (shrink)

We call a semantic theory 'classical' if it includes the assertions that (I) a function V assigning semantic value maps object language proper names into some set D, (ii) V maps object language atomic sentences into some set F, and (iii) the extension of any object language unary predicate is a member of the power set of D. Two theorems can be proven which assert that any classical theory which includes certain other assumptions assigns the same member of F (...) to every true object language sentence. Many accept the following argument: (1) every plausible semantic theory is classical and contains the assertions named in the theorems, (2) if the semantic value of declarative sentences is a representation or representational then, some different true sentences differ in what they represent, hence, declarative sentences are not representational. I show how to avoid the conclusion by arguing for the falsity of (1). (shrink)

We argue that, under the usual assumptions for sufficiently strong arithmetical theories that are subject to Gödel’s First Incompleteness Theorem, one cannot, without impropriety, talk about *the* Gödel sentence of the theory. The reason is that, without violating the requirements of Gödel’s theorem, there could be a true sentence and a false one each of which is provably equivalent to its own unprovability in the theory if the theory is unsound.

The liar paradox can be shown semantically defective if we distinguish the /sentence/ ''snow is white' is true' from the /string/ that constitutes it. This paper develops the String-to-Sentence Theory of Truth---for short, String Theory---according to which, while the /string/ contains the string 'true', the /sentence/ is merely 'snow is white', which contains no such occurrence: more generally, a string like 'S is true' constitutes, relative to an assessor, the sentence which, to the assessor, means the same (...) as S. So suppose we attempt to define a singular term 'L' referring to the sentence 'not: L is true'. Relative to an assessor, 'L' refers to the sentence negating the assessor's sentence meaning the same as the referent of 'L'. So the referent of 'L' means the same as its negation. But no sentence means the same as its negation, so 'L' does not refer. The act of naming with which the liar paradox commences is semantically defective; so there can be no liar paradox. (shrink)

Does truth attach to sentences, or to what sentences express? If to sentences, then certainly not to type sentences, such as ‘I am going to London tomorrow’, but only to token sentences, that is, sentences considered as uttered by a particular speaker at a particular time. It would, however, be inconvenient to restrict truth to utterances that are actually made; we may therefore adopt the device and terminology of Davidson, and speak of a ‘statement’ (...) constituted by a triple [s, i, t] of a type sentence s, an individual i and a time t, the existence of which does not depend on whether i in fact uttered s at t. I shall presume that the identification of a type sentence depends on identifying the language to which it belongs. A familiar, irritating obstacle prevents our explaining that, when i did not utter s at t, the ‘statement’ is to be said to be true just in case he would have said something true if he had done so; the obstacle consists in sentences like, ‘I am not now speaking.’ The difficulty is not serious. What more is needed in order to obtain, from a type sentence s, something apt to be characterised as true or false is an assignment of references to indexical and demonstrative expressions occurring in s; we may say that s ‘comes out’ true or false under such an assignment. We may therefore take a ‘statement’ [s, i, t] to be true or false according as s comes out true or false under the assignment of t to the word ‘now’, i to the words ‘I’, and ‘me’, the place where i is at t to the word ‘here’, and so on. There is no reason to be disconcerted by the fact that this will yield as true a ‘statement’ involving the sentence, ‘I have been dead for a hundred years.’ Demonstrative expressions like ‘that house’, ‘this country’, etc., are less easily dealt with. Many should be regarded as devoid of reference unless, at t, i actually makes a pointing gesture or the equivalent; but the question need not be pursued here. For convenience, however, we may continue to speak of the ascription of truth to sentences, as long as we bear in mind that it is to ‘statements’, in our special sense, to which it is really to be ascribed. (shrink)

We prove that if is a model of size at most [kappa], λ[kappa] = λ, and a game sentence of length 2λ is true in a 2λ-saturated model ≡ , then player has a winning strategy for a related game in some ultrapower ΠD of . The moves in the new game are taken in the cartesian power λA, and the ultrafilter D over λ must be chosen after the game is played. By taking advantage of the expressive power (...) of game sentences, we obtain several applications showing the existence of ultrapowers with certain properties. In each case, it was previously known that such ultrapowers exist under the assumption of the GCH, and we get them without the GCH. Article O. (shrink)

Abstract.According to the normative theory of legal science, juristic ought‐sentences describe rules, since legal science just deals with rules, and rules cannot be described but by means of ought‐sentences. The author challenges this view. Two different constructions of “describing rules” are proposed: Namely, either interpreting or stating the validity of rules. “Interpreting rules,” in its turn, can be understood in three different senses: listing all the possible meanings of rule‐formulations, reporting the different interpretations a rule‐formulation has in fact (...) received by courts, or ascribing meaning to rule‐formulations. However, the author argues that ought‐sentences are not the proper tools to accomplish such tasks. At the same time, juristic ought‐sentences cannot be understood as validity statements, since they neither mention any rule whatsoever, nor include the term “valid.” Further, if validity‐statements were ought‐sentences, their logical behaviour would reflect the logic of rules themselves. However, as the late Kelsen argued, things do not run this way, since two inconsistent ought‐sentences, if understood as validity statements, can paradoxically both be true, as well as both false. Hence, validity‐statements cannot be reduced to ought‐sentences iterating the rules which they (are supposed to) refer to. (shrink)

Proposition and sentence are two separate entities indicating their specific purposes, definitions and problems. A proposition is a logical entity. A proposition asserts that something is or not the case, any proposition may be affirmed or denied, all proportions are either true (1’s) or false (0’s). All proportions are sentences but all sentences are not propositions. Propositions are factual contains three terms: subject, predicate and copula and are always in indicative or declarative mood. While sentence is a (...) grammatical entity, a unit of language that expresses a complete thought; a sentence may express a proposition, but is distinct from the proposition it may be used to express: categories, declarative sentences, exclamatory, imperative and interrogative sentences. Not all sentences are propositions, propositions express sentence. Sentence is a proposition only in condition when it bears truth values i.e. true or false. We use English sentences governed by imprecise rule to state the precise rules of proposition. In logic we use sentence as logical entity having propositional function but grammatical sentences are different from logical sentences while the former are having only two divisions namely subject and predicate and may express wishes, orders, surprise or facts and also have multiple subjects and predicates and the latter must be in a propositional form which states quantity of the subject and the quality of the proposition and multiple subjects and multiple predicate make the proposition multiple. (shrink)

This is a paper about the constituents of arguments. It argues that several different kinds of truth-bearer may be taken to compose arguments, but that none of the obvious candidates—sentences, propositions, sentence/truth-value pairs etc.—make sense of logic as it is actually practiced. The paper goes on to argue that by answering the question in different ways, we can generate different logics, thus ensuring a kind of logical pluralism that is different from that of J. Beall and Greg Restall.

Many competent speakers initially judge that (i) is true and (ii) isfalse, though they know that (iii) is true. (i) Superman leaps more tallbuildings than Clark Kent. (ii) Superman leaps more tall buildings thanSuperman. (iii) Superman is identical with Clark Kent. Semanticexplanations of these intuitions say that (i) and (ii) really can differin truth-value. Pragmatic explanations deny this, and say that theintuitions are due to misleading implicatures. This paper argues thatboth explanations are incorrect. (i) and (ii) cannot differ (...) intruth-value, yet the intuitions are not due to implicatures, but ratherto mistakes in evaluating (i) and (ii). (shrink)

I give several characterizations of the set V₀ proposed in [3] as the set of meaningful and true sentences of first order arthimetic, and show that in Peano arithmetic the Σ₂ completeness of V₀ is provable.

We study here equiformity, the standard identity criterion for sentences. This notion was put forward by Lesniewski, mentioned by Tarski and defined explicitly by Presburger. At the practical level this criterion seems workable but if the notion of sentence is taken as a fundamental basis for logic and mathematics, it seems that this principle cannot be maintained without vicious circle. It seems also that equiformity has some semantical features ; maybe this is not so clear for individual signs but (...) sentences are often considered as meaningful combinations of signs. If meaning has to play a role, we are thus maybe in no better position than when dealing with identity criterion for propositions. In formal logic, one speaks rather about well-formed formulas, but closed formulas are called sentences because they are meaningful in the sense that they can be true or false. Formulas look better like mathematical objects than material inscriptions and equiformity does not seem to apply to them. Various congruencies can be considered as identities between formulas and in particular "to have the same logical form". One can say that the objects of study of logic are rather logical forms than sentences conceived as material inscriptions. (shrink)

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been (...) pressed in the literature by Hilary Putnam and Peter Koellner. In this paper I sketch a conventionalist theory of mathematics, show that this conventionalist theory can meet the challenge just raised , and clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them. (shrink)

The present event‐related potential (ERP) study used picture–sentence verification to investigate the neurolinguistic correlates of the online processing of compositional‐semantic information. To this end, we examined context effects on sentences involving temporal adverbial quantification likeJana war jeden Morgen schwimmen an den Arbeitstagen (“Jana went for a swim every morning during the working week”). We tested whether the conceptual complexity associated with quantifying over time intervals leads to delayed predictions regarding the upcoming words in a sentence. The present study replicated (...) previous results relating to quantification over individuals, which are conceptually less complex than time intervals. Analogous to previous studies, false vs. true sentences elicited an N400 whenever contextual cues did not permit a potential revision of a locally assigned truth value. The present results are compatible with an approach under which contextual cues are immediately considered for predicting how a sentence continues. The fact that the contextual complexity did not lead to processing delays indicates that the processing system quickly abstracts away from the conceptual complexity associated with the linguistic input if such an abstraction is possible. (shrink)

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.

I give several characterizations of the set V₀ proposed in [3] as the set of meaningful and true sentences of first order arthimetic, and show that in Peano arithmetic the Σ₂ completeness of V₀ is provable.

Three-valued logics are standardly used to formalize gappy languages, i.e., interpreted languages in which sentences can be true, false or neither. A three-valued logic that assigns the same truth value to all gappy sentences is, in our view, insufficient to capture important semantic differences between them. In this paper we will argue that there are two different kinds of pathologies that should be treated separately and we defend the usefulness of a four-valued logic to represent adequately these (...) two types of gappy sentences. Our purpose is to begin the formal exploration of the four-valued logics that could be used to represent the phenomena in question and to show that these phenomena are present in natural language, at least according to some semantic theories of natural language. (shrink)

The theory of mental models postulates that meaning and knowledge can modulate the interpretation of conditionals. The theory's computer implementation implied that certain conditionals should be true or false without the need for evidence. Three experiments corroborated this prediction. In Experiment 1, nearly 500 participants evaluated 24 conditionals as true or false, and they justified their judgments by completing sentences of the form, It is impossible that A and ___ appropriately. In Experiment 2, participants evaluated 16 conditionals (...) and provided their own justifications, which tended to be explanations rather than logical justifications. In Experiment 3, the participants also evaluated as possible or impossible each of the four cases in the partitions of 16 conditionals: A and C, A and not-C, not-A and C, not-A and not-C. These evaluations corroborated the model theory. We consider the implications of these results for theories of reasoning based on logic, probabilistic logic, and suppositions. (shrink)

This paper defends scientific realism from the pessimistic meta-induction from past reference failure. It allows that a descriptive theory of reference implies that scientific terms characteristically fail of determinate reference. But it argues that a descriptive theory of reference also implies an equivalence between scientific theories and quantificational claims in the style of Ramsey. Since these quantificational claims do not use any of the referentially suspect scientific terms, they can be approximately true even when those terms fail to refer (...) determinately.Keywords: Science; Realism; Ramsey sentences; Approximate truth; Reference. (shrink)