Articles by two American mathematicians, E. V. Huntington and Oswald Veblen, are discussed as examples of a movement in foundational research in the period 1900-1930 called American postulate theory. This movement also included E. H. Moore, R. L. Moore, C. H. Langford, H. M. Sheffer, C. J. Keyser, and others. The articles discussed exemplify American postulate theorists' standards for axiomatizations of mathematical theories, and their investigations of such axiomatizations with respect to metatheoretic properties such as independence, completeness, and consistency.

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn S, called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (...) (Łoś, Suszko, Słupecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic. Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section. (shrink)

Carnap’s work was instrumental to the liberalization of empiricism in the 1930s that transformed the logical positivism of the Vienna Circle to what came to be known as logical empiricism. A central feature of this liberalization was the deployment of the Principle of Tolerance, originally introduced in logic, but now invoked in an epistemological context in “Testability and Meaning”. Immediately afterwards, starting with Foundations of Logic and Mathematics, Carnap embraced semantics and turned to interpretation to guide the choice of a (...) theoretical language for science. The first thesis of this paper is that recourse to an intended interpretation led to a partial retrenchment of the conventionalism implied by the Principle of Tolerance. It required that the choice of a language be based on abstraction from a context; this procedure later became a component of the process of explication that was distinctive to Carnap’s mature views. The interpretive origin of formal systems also ensured their likely syntactic consistency, an issue on which Carnap was strongly criticized by figures such as Beth and Gödel. The second thesis of this paper is that this reliance on an intended interpretation enabled constructed formal systems to be relevant to the development of empirical science. (shrink)

Anaphoric deflationism is a kind of prosententialist account of the use of “true.” It holds that “true” is an expressive operator and not a predicate. In particular, “is true” is explained as a “prosentence.” Prosentences are, for sentences, the equivalent of what pronouns are for nouns: As pronouns refer to previously introduced nouns, so prosentences like “that’s true” inherit their semantic content from previously introduced sentences. So, if Jim says, “The candidate is going to win the election,” and Bill replies (...) “that’s true,” the real meaning of Bill’s statement is “It is true that the candidate is going to win the election.” This kind of prosententialist deflationism around the use of “true,” especially in Robert Brandom’s version, is an explanation given in terms of anaphora. The prosentence is an anaphoric dependent of the sentence providing its content. Båve (Philosophical Studies, 145, 297-310. 2009) argued that the anaphoric account is not as general as prosententialists claim, and that the analogy between prosentences and pronouns is explanatorily idle because it does not do any real explanatory work. The two criticisms are connected: The lack of unity within the anaphoric theory can be used to show its poor explanatory value. The plurality of uses of “is true” exceeds the anaphoric account indeed. Therefore, prosententialism is just a superficial re-description and the real work is done by means of more general semantic terms, namely “semantic equivalence and consequence” between “p” and ““p” is true.” I analyze Båve’s arguments and highlight that he fails to acknowledge the importance of a pragmatic and expressive dimension explained by the anaphoric account, a dimension that semantic “equivalence” and “consequence” are not capable of explaining. I then show that the anaphoric account can actually explain semantic equivalence and consequence, and this is crucial because equivalence and consequence do not explain anaphoric dependence. This reverses the allegation of generality: The anaphoric account is more general. Again, the cases typically used to defend prosententialism, if correctly described, show a unitary structure: They are all versions of lazy anaphoric dependence. Therefore, the unifying principle performing the explanation here is lazy anaphora. (shrink)

In this article I give an overview of some recent work in philosophy of science dedicated to analysing the scientific process in terms of (conceptual) mathematical models of theories and the various semantic relations between such models, scientific theories, and aspects of reality. In current philosophy of science, the most interesting questions centre around the ways in which writers distinguish between theories and the mathematical structures that interpret them and in which they are true, i.e. between scientific theories as linguistic (...) systems and their non-linguistic models. In philosophy of science literature there are two main approaches to the structure of scientific theories, the statement or syntactic approach -- advocated by Carnap, Hempel and Nagel -- and the non- statement or semantic approach --advocated, among others, by Suppes, the structuralists, Beth, Van Fraassen, Giere, Wojcicki. In conclusion, I briefly review some of the usual realist inspired questions about the possibility and character of relations between scientific theories and reality as implied by the various approaches I discuss in the course of the article. The models of a scientific theory should indeed be adequate to the phenomena, but if the theory is 'adequate' to (true in) its conceptual (mathematical) models as well, we have a model-theoretic realism that addresses the possible meaning and reference of 'theoretical entities' without relapsing into the metaphysics typical of the usual scientific realist approaches. (shrink)

This essay assesses the account of truth presented in Wiggins's 2002 paper ‘An indefinibilist cum normative view of truth and the marks of truth'. I agree with Wiggins that we should seek, not to define truth, but to elucidate it by unfolding its connections with other basic notions. However, I give reasons for preferring an elucidation based on Ramsey's account of truth to Wiggins's Tarski-inspired approach. I also cast doubt on Wiggins's thesis that convergence is a mark of truth, arguing (...) instead that a claim which is up for assessment as true or false must be one to which different speakers/hearers can attach, and know that they are attaching, the same sense. I use this principle to rule out an account of indicative conditionals, and bring some considerations to bear on the question of whether those conditionals have truth values. An appendix revisits a debate about the determinateness of distinctness. (shrink)

I advance arguments in favour of PKF as an articulation of a central sense of the predicate ‘true’, and show how it illuminates the relationship between that sense and the ‘external’ notion of truth found in such claims as ‘An utterance of the Liar Sentence does not say anything, and so is not true’.

Should we explicate truth in terms of meaning, or meaning in terms of truth? Ramsey, Prior and Strawson all favoured the former approach: a statement is true if and only if things are as the speaker, in making the statement, states them to be; similarly, a belief is true if and only if things are as a thinker with that belief thereby believes them to be. I defend this explication of truth against a range of objections.Ramsey formalized this account of (...) truth as follows: B is true =df ∃P; in §i, I defend this formula against the late Peter Geach's objection that its right‐hand side is ill‐formed. Davidson held that Ramsey and co. had the whole matter back to front: on his view, we should explicate meaning in terms of truth, not vice versa. In §ii, I argue that Ramsey's approach opens the way to a more promising approach to semantic theorizing than Davidson's. Ramsey presents his formula as a definition of truth, apparently contradicting Tarski's theorem that truth is indefinable. In §iii, I show that the contradiction is only apparent: Tarski assumes that the Liar‐like inscription he uses to prove his theorem has a content, but Ramsey can and should reject that assumption. As I explain in §iv, versions of the Liar Paradox may be generated without making any assumptions about truth: paradox arises when the impredicativity that is found when a statement's content depends on the contents of a collection of statements to which it belongs turns pathological. Since they do not succeed in saying anything, such pathological utterances or inscriptions pose no threat to the laws of logic, when these are understood as universal principles about the ways things may be said or thought to be. There is, though, a call for rules by following which we can be sure that any conclusion deduced from true premisses is true, and hence says something. Such rules cannot be purely formal, but in §v I propose a system of them: this opens the way to the construction of deductive theories even in circumstances where producing a well‐formed formula is no guarantee of saying anything. (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)

Revenge arguments purport to show that any proposed solution to the semantic paradoxes generates new paradoxes that prove that solution to be inadequate. In this paper, I focus on revenge arguments that employ the model-theoretic semantics of a target theory and I argue, contra the current revenge-theoretic wisdom, that they can constitute genuine expressive limitations. I consider the anti-revenge strategy elaborated by Field and argue that it does not offer a way out of the revenge problem. More generally, I argue (...) that the difference between ‘standard’ and ‘revenge’ paradoxes is ill-conceived and should be abandoned. This will contribute to show that the theories that provide a uniform account of truth and other semantic notions are the ones best equipped to avoid the paradoxes altogether—‘standard’ and ‘revenge’ alike. (shrink)

The paper is devoted to the discussion of some philosophical and historical problems connected with the theorem on the undefinability of the notion of truth. In particular the problem of the priority of proving this theorem will be considered. It is claimed that Tarski obtained this theorem independently though he made clear his indebtedness to Gödel’s methods. On the other hand, Gödel was aware of the formal undefinability of truth in 1931, but he did not publish this result. Reasons for (...) that are also considered. (shrink)

In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successorS(whereSa=a+ 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation (...) of divisibility | (wherex|ymeansxdividesy). (shrink)

I offer an interpretation of a familiar, but poorly understood portion of Tarskis work on truth – bringing to light a number of unnoticed aspects of Tarskis work. A serious misreading of this part of Tarski to be found in Scott Soames Understanding Truth is treated in detail. Soamesreading vies with the textual evidence, and would make Tarskis position inconsistent in an unsubtle way. I show that Soames does not finally have a coherent interpretation of Tarski. This is unfortunate, since (...) Soames ultimately arrogates to himself a key position that he has denied to Tarski and which is rightfully Tarskis own. (shrink)

Alfred Tarski (1944) wrote that "the condition of the 'essential richness' of the metalanguage proves to be, not only necessary, but also sufficient for the construction of a satisfactory definition of truth." But it has remained unclear what Tarski meant by an 'essentially richer' metalanguage. Moreover, DeVidi and Solomon (1999) have argued in this Journal that there is nothing that Tarski could have meant by that phrase which would make his pronouncement true. We develop an answer to the historical question (...) of what Tarski meant by 'essentially richer' and pinpoint the general result that stands behind his essential richness claim. In defense of Tarski, we then show that each of the several arguments of DeVidi and Solomon are either moot or mistaken. One of the fruits of our investigation is the reclamation of what Tarski took to be his central result on truth. This is a reclamation since: (i) if one does not understand 'essential richness', one does not know what that result is, and (ii) we must unearth a heretofore unrecognized change that occurs in Tarski's view - an alteration of his main thesis in light of a failing he discovered in it. (shrink)

Kaplan (1989a) insists that natural languages do not contain displacing devices that operate on character—such displacing devices are called monsters. This thesis has recently faced various empirical challenges (e.g., Schlenker 2003; Anand and Nevins 2004). In this note, the thesis is challenged on grounds of a more theoretical nature. It is argued that the standard compositional semantics of variable binding employs monstrous operations. As a dramatic first example, Kaplan’s formal language, the Logic of Demonstratives, is shown to contain monsters. For (...) similar reasons, the orthodox lambda-calculus-based semantics for variable binding is argued to be monstrous. This technical point promises to provide some far-reaching implications for our understanding of semantic theory and content. The theoretical upshot of the discussion is at least threefold: (i) the Kaplanian thesis that “directly referential” terms are not shiftable/bindable is unmotivated, (ii) since monsters operate on something distinct from the assertoric content of their operands, we must distinguish ingredient sense from assertoric content (cf. Dummett 1973; Evans 1979; Stanley 1997), and (iii) since the case of variable binding provides a paradigm of semantic shift that differs from the other types, it is plausible to think that indexicals—which are standardly treated by means of the assignment function—might undergo the same kind of shift. (shrink)

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

The present article purports to show that the protocol sentence debate, pursued by some leading members of the Vienna Circle in the mid-1930s, was essentially a controversy over the explanation and the real significance of the concept of truth. It is further shown that the fundamental issue underlying the discussions about the concept of truth was the relationship between form and content, as well as between logic/language and the world. R. Carnap was the philosopher who most explicitly and systematically attempted (...) to come to grips with this problem. It is shown that the form-content distinction pervades the three most important phases of Carnap's philosophical development: the structuralist (in Der logische Aufbau der Welt), the syntactical and the semantical. His final semantical stance is essentially determined by the concept of linguistic frameworks. The article purports to demonstrate that this concept cannot be dispensed with in philosophy, but that Carnap failed to work out its ontological implications. Finally, the concept of an internal ontology is briefly delineated. (shrink)

This paper argues for tlie claims that a) a natural language such as English is semanticaly closed b) semantic closure implies inconsistency. A corollary of these is that the semantics of English must be paraconsistent. The first part of the paper formulates a definition of semantic closure which applies to natural languages and shows that this implies inconsistency. The second section argues that English is semeantically closed. The preceding discussion is predicated on the assumption that there are no truth value (...) gaps. The next section of the paper considers whether the possibility of these makes any difference to the substantive conclusions of the previous sections, and argues that it does not. The crux of the preceding arguments is that none of the consistent semantical accounts that have been offered for solving the semantical paradoxes is a semantic ofEnglish. The final section of the paper produces a general argument as to why this must always be the case. (shrink)

The paper gives a semantics for naive (inconsistent) set theory in terms of substitutional quantification. Soundness is proved in an appendix. In the light of this construction, Several philosophical issues are discussed, Including mathematical necessity and the set theoretic paradoxes. Most importantly, It is argued, These semantics allow for a nominalist account of mathematical truth not committed to the existence of a domain of abstract entities.

The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications.

Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it “the antinomy of the variable”. Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (i) the variables ‘x’ and ‘y’ cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (ii) there is a strong temptation to say that distinct variables ‘x’ (...) and ‘y’ are synonymous, since sentences differing by the total, proper substitution of ‘x’ for ‘y’ always agree in meaning. We offer a precise interpretation of the challenge posed by (i) and (ii). We then develop some neglected passages of Tarski to show that his semantics for variables has the resources to resolve the antinomy without abandoning standard compositional semantics. (shrink)

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel’s first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded (...) by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn’t give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics. (shrink)

This article presents three extracts from the introductory course in mathematical logic that Gödel gave at the University of Notre Dame in 1939. The lectures include a few digressions, which give insight into Gödel's views on logic prior to his philosophical papers of the 1940s. The first extract is Gödel's first lecture. It gives the flavour of Gödel's leisurely style in this course. It also includes a curious definition of logic and a discussion of implication in logic and natural language. (...) The second extract is a discussion on undecidability and on Leibniz. The third extract concerns the paradoxes and Russell's theory of types. (shrink)

Formal semantics and pragmatics as they have developed since the late 1960's have been shaped by fruitful interdisciplinary collaboration among linguists, philosophers, and logicians, among others, and in turn have had noticeable effects on developments in syntax, philosophy of language, computational linguistics, and cognitive science.In this paper I describe the environment in which formal semantics was born and took root, highlighting the differences in ways of thinking about natural language semantics in linguistics and in philosophy and logic. With Montague as (...) a central but not solo player in the story, I reflect on crucial developments in the 1960's and 70's in linguistics and philosophy, and the growth of formal semantics and formal pragmatics from there. I discuss innovations, key players, and leading ideas that shaped the development of formal semantics and its relation to syntax, to pragmatics, and to the philosophy of language in its early years, and some central aspects of its early impact on those fields. (shrink)

Tarski, Carnap and Quine spent the academic year 1940?1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnap's Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking (...) place. In my article, I present a survey of these notes with special emphasis on Tarski's rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarski's finitism/nominalism, and the construction of a finitist language for mathematics and science. (shrink)

Davidson’s 1965 paper, “Theories of Meaning and Learnable Languages”, has invariably been interpreted, by others and by myself, as arguing that natural languages must have a compositional semantics, or at least a systematic semantics, that can be finitely specified. However, in his reply to me in the Żegleń volume, Davidson denies that compositionality is in any need of an argument. How does this add up? In this paper I consider Davidson’s first three meaning theoretic papers from this perspective. I conclude (...) that Davidson was right in his reply to me that he never took compositionality, or systematic semantics, to be in need of justification. What Davidson had been concerned with, clearly in the 1965 paper and in “Truth and Meaning” from 1967, and to some extent in his Carnap critique from 1963, is that we need a general theory of natural language meaning, that such a theory should not be in conflict with the learnability of a language, and that such a theory bring out should how knowledge of a finite number of features of a language suffices for the understanding of all the sentences of that language. (shrink)

Summary This paper is intended to discuss the problems occurring in the relation between the notion of truth and the question of self-reference. To do this, we shall review Tarski's (T) convention and its related terminology. We shall clarify the relation between truth and extension in order to lead into the question of semantic paradoxes appearing in the theoretical models concerned with truth. Subsequently, we shall review the logical system which develops in the reformulation of the modal proposal of the (...) (T) convention. In closing, we shall critically examine Kripke's interpretation from the proposals made by Tarski. (shrink)

Explication is the conceptual cornerstone of Carnap’s approach to the methodology of scientific analysis. From a philosophical point of view, it gives rise to a number of questions that need to be addressed, but which do not seem to have been fully addressed by Carnap himself. This paper reconsiders Carnapian explication by comparing it to a different approach: the ‘formalisms as cognitive tools’ conception. The comparison allows us to discuss a number of aspects of the Carnapian methodology, as well as (...) issues pertaining to formalization in general. We start by introducing Carnap’s conception of explication, arguing that there is a tension between his proposed criteria of fruitfulness and similarity; we also argue that his further desideratum of exactness is less crucial than might appear at first. We then bring in the general idea of formalisms as cognitive tools, mainly by discussing the reliability of so-called statistical prediction rules, i.e. simple algorithms used to make predictions across a range of areas. SPRs allow for a concrete instantiation of Carnap’s fruitfulness desideratum, which is arguably the most important desideratum for him. Finally, we elaborate on what we call the ‘paradox of adequate formalization’, which for the Carnapian corresponds to the tension between similarity and fruitfulness. We conclude by noting that formalization is an inherently paradoxical enterprise in general, but one worth engaging in given the ‘cognitive boost’ it affords as a tool for discovery. (shrink)

This chapter contains sections titled: Highlights of Past Literature Current Work Future Work.

A formula is a contingent logical truth when it is true in every model M but, for some model M , false at some world of M . We argue that there are such truths, given the logic of actuality. Our argument turns on defending Tarski’s definition of truth and logical truth, extended so as to apply to modal languages with an actuality operator. We argue that this extension is the philosophically proper account of validity. We counter recent arguments to (...) the contrary presented in Hanson’s ‘Actuality, Necessity, and Logical Truth’ (Philos Stud 130:437–459, 2006 ). (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

Since Saul Kripke’s influential work in the 1970s, the revisionary approach to semantic paradox—the idea that semantic paradoxes must be solved by weakening classical logic—has been increasingly popular. In this paper, we present a new revenge argument to the effect that the main revisionary approaches breed new paradoxes that they are unable to block.

We use quotation marks when we wish to refer to an expression. We can and do so refer even when this expression is composed of characters that do not occur in our alphabet. That's why Tarski, Quine, and Geach's theories of quotation don't work. The proposals of Davidson, Frege, and C. Washington, however, do not provide a plausible account of quotation either. (Section I). The problem is to construct a Tarskian theory of truth for an object language that contains quotation (...) marks, without appealing to quotation marks in the metalanguage. I propose to supply Tarski's truth definition with one axiom that determines the denotation of all expressions containing quotation marks. According to this axiom, quotation marks create a non-extensional context. Since admitting such contexts does not lead to any difficulties in the recursive truth characterization, we may indeed dispense with extensionalism. (Section II). Finally, I argue that we classify and denote expressions in the very same way that we classify and denote extralinguistic entities. Both tokens and types of written signs can be easily incorporated into the naturalist's worldview. (Section III). (shrink)

Are we perhaps in the "matrix", or anyway, victims of perfect and permanent computer simulation? No. The most convincing—and shortest—version of Putnam's argument against the possibility of our eternal envattment is due to Crispin Wright (1994). It avoids most of the misunderstandings that have been elicited by Putnam's original presentation of the argument in "Reason, Truth and History" (1981). But it is still open to the charge of question-begging. True enough, the premisses of the argument (disquotation and externalism) can be (...) formulated and defended without presupposing external objects whose existence appears doubtful in the light of the very skeptical scenario which Putnam wants to repudiate. However, the argument is only valid if we add an extra premiss as to the existence of some external objects. In order to avoid circularity, we should run the argument with external objects which must exist even if we are brains in a vat, e.g. with computers rather than with trees. As long as the skeptic is engaged in a discussion of the brain-in-a-vat scenario, she should neither deny the existence of computers nor the existence of causal relations; for if she does, she is in fact denying that we are brains in a vat. (shrink)