There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...) We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. Indeed, we argue that developing a fully compositional semantics in the tradition of Frege leads to a typographic variant of the most radical of Tarskian views: variabilism, the view that names should be modeled as Tarskian variables. We conclude with the consequences of this result for Frege's distinction between sense and reference. (shrink)

Noun phrases with overt determiners, such as <i>some apples</i> or <i>a quantity of milk</i>, differ from bare noun phrases like <i>apples</i> or <i>milk</i> in their contribution to aspectual composition. While this has been attributed to syntactic or algebraic properties of these noun phrases, such accounts have explanatory shortcomings. We suggest instead that the relevant property that distinguishes between the two classes of noun phrases derives from two modes of existential quantification, one of which holds the values of a variable fixed (...) throughout a quantificational context while the other allows them to vary. Inspired by Dynamic Plural Logic and Dependence Logic, we propose Plural Predicate Logic as an extension of Predicate Logic to formalize this difference. We suggest that temporal <i>for</i>-adverbials are sensitive to aspect because of the way they manipulate quantificational contexts, and that analogous manipulations occur with spatial <i>for</i>-adverbials, habituals, and the quantifier <i>all</i>. (shrink)

The result of combining classical quantificational logic with modal logic proves necessitism – the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃x t=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted (...) that in order to specify the truth conditions of certain sentences involving constants or variables that don’t denote, one has to apparently quantify over things that are not identical to anything. In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia. (shrink)

In a series of recent works, Kit Fine, 605–631, 2003, 2007) has sketched a novel solution to Frege’s puzzle. Radically departing from previous solutions, Fine argues that Frege’s puzzle forces us to reject compositionality. In this paper we first provide an explicit formalization of the relational semantics for first-order logic suggested, but only briefly sketched, by Fine. We then show why the relational semantics alone is technically inadequate, forcing Fine to enrich the syntax with a coordination schema. Given this (...) enrichment, we argue, that that the semantics is compositional. We then examine the deep consequences of this result for Fine’s proposed solution to Frege’s puzzle. We argue that Fine has mis-diagnosed his own solution–his attempted solution does not deny compositionality. The correct characterization of Fine’s solution fits him more comfortably among familiar solutions to the puzzle. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes (...) multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

MILLIANISM and DESCRIPTIVISM are without question the two most prominent views with respect to the semantics of proper names. However, debates between MILLIANS and DESCRIPTIVISTS have tended to focus on a fairly narrow set of linguistic data and an equally narrow set of problems, mainly how to solve with Frege's puzzle and how to guarantee rigidity. In this article, the author focuses on a set of data that has been given less attention in these debates—namely, so-called predicative uses, bound uses, (...) and shifted uses of names. The author first shows that these data points seem to favor a DESCRIPTIVIST view over a MILLIAN view, but the author then introduces an alternative view of names that not only provides a simple and elegant way of dealing with the data, but also retains rigidity without becoming subject to the problems raised by Frege's puzzle. This is the view that names are variables, also called VARIABILISM. (shrink)

Variables is a project at the intersection of the philosophies of language and logic. Frege, in the Begriffsschrift, crystalized the modern notion of formal logic through the first fully successful characterization of the behaviour of quantifiers. In Variables, I suggest that the logical tradition we have inherited from Frege is importantly flawed, and that Frege's move from treating quantifiers as noun phrases bearing word-world connection to sentential operators in the guise of second-order predicates leaves us both philosophically and (...) technically wanting. (shrink)

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)

We argue that the extant evidence for Stoic logic provides all the elements required for a variable-free theory of multiple generality, including a number of remarkably modern features that straddle logic and semantics, such as the understanding of one- and two-place predicates as functions, the canonical formulation of universals as quantified conditionals, a straightforward relation between elements of propositional and first-order logic, and the roles of anaphora and rigid order in the regimented sentences that express multiply general (...) propositions. We consider and reinterpret some ancient texts that have been neglected in the context of Stoic universal and existential propositions and offer new explanations of some puzzling features in Stoic logic. Our results confirm that Stoic logic surpasses Aristotle’s with regard to multiple generality, and are a reminder that focusing on multiple generality through the lens of Frege-inspired variable-binding quantifier theory may hamper our understanding and appreciation of pre-Fregean theories of multiple generality. (shrink)

Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most of the Aristotelian corpus is not about (...) class='Hi'>logic, whereas virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. Two of his works appeared posthumously [Hist. Philos. Logic 7 (1986), no. 2, 143--154; MR0868748 (88b:03010); Tarski and Givant, A formalization of set theory without variables, Amer. Math. Soc., Providence, RI, 1987; MR0920815 (89g:03012)]. Tarski's voluminous writings were widely scattered in numerous journals, some quite rare. It has been extremely difficult to study the development of Tarski's thought and to trace the interconnections and interdependence of his various papers. Thanks to the present collection all this has changed, and it is likely that the increased accessibility of Tarski's papers will have the effect of increasing Tarski's already enormous influence. (shrink)

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. -/- There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is (...) countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. -/- Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects. (shrink)

The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬G(x, y)→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[G(x, y)→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's (...) indefinability theorem in mathematical logic. Gödel numbering is the key for arithmetic modeling applied in this paper. (shrink)

The paper examines two possible analyses of fictional names within Pavel Tichý’s Transparent Intensional Logic. The first of them is the analysis actually proposed by Tichý in his (1988) book The Foundations of Frege’s Logic. He analysed fictional names in terms of free variables. I will introduce, explain, and assess this analysis. Subsequently, I will explain Tichý’s notion of individual role (office, thing-to-be). On the basis of this notion, I will outline and defend the second analysis of fictional (...) names. This analysis is close to the approach known in the literature as role realism (the most prominent advocates of this position are Nicholas Wolterstorff, Gregory Currie, and Peter Lamarque). (shrink)

‘Quantified pure existentials’ are sentences (e.g., ‘Some things do not exist’) which meet these conditions: (i) the verb EXIST is contained in, and is, apart from quantificational BE, the only full (as against auxiliary) verb in the sentence; (ii) no (other) logical predicate features in the sentence; (iii) no name or other sub-sentential referring expression features in the sentence; (iv) the sentence contains a quantifier that is not an occurrence of EXIST. Colin McGinn and Rod Girle have alleged that (...) standard first-order logic cannot adequately deal with some such existentials. The article defends the view that it can. (shrink)

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

This paper investigates the connection between two recent trends in philosophy: higher-orderism and conceptual engineering. Higher-orderists use higher-order quantifiers (in particular quantifiers binding variables that occupy the syntactic positions of predicates) to express certain key metaphysical doctrines, such as the claim that there are properties. I argue that, on a natural construal, the higher-orderist approach involves an engineering project concerning, among others, the concept of existence. I distinguish between a modest construal of this project, on which it aims at engineering (...) higher-order analogues of the familiar notion of first-order existence, and an ambitious construal, on which it additionally aims at engineering a broadened notion of existence that subsumes first-order and higher-order existence. After identifying a substantial problem for the ambitious project, I investigate a possible response which is based on adopting a cumulative type theory as the background higher-order logic. While effective against the problem at hand, this strategy turns out to undermine a major reason to embrace higher-orderism in the first place, namely the idea that higher-orderism dissolves a range of otherwise intractable debates in metaphysics. Higher-orderists are therefore best advised to pursue their engineering project on the modest variant and against the background of standard type theory. (shrink)

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and (...) semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions. (shrink)

Is a naturalistic account of reference possible? Here is a simple argument to the effect that it is not: Let R be the relation that allegedly naturalizes reference, and consider the predicate "being an object that does not stand in the relation R to this expression". Call this predicate "P". On the face of it, P is a counterexample to the alleged naturalization, since it appears to refer to all and only those objects that do not stand in (...) the relation R to it. Actually, an argument like this was advanced more than twenty years ago by the late Paolo Casalegno. Although for various reasons it has not been given due attention by the philosophical community, the argument is interesting in its structure and remarkable in its conclusion. In this paper I shall reconstruct it in detail and discuss it. Then, I shall show that the argument fails, since, contrary to what Casalegno claimed, under certain conditions P is paradoxical, hence not a referring expression. My rejection builds on the fundamental distinction between simple and compound expressions, namely on the compositional structure of natural languages. If I am right, whether any attempt at naturalizing reference that takes care of this distinction will succeed remains an open empirical question. (shrink)

In this article I present some characteristics of logics and semantics of an uncertain world. I confront two-valued and fuzzy logic. I use Kafka’s novel Process as an example, which is designed as an uncertain context with words which are rigid designators without rigid meaning. That produces an uncertain world of logical and semantical relations. In presentation of problems I introduce basic concepts of Frege’s, Wittgenstein’s, Tarsky’s, Searle’s, Quine’s and Davidson’s philosophy of language. I differ the logical and semantical (...) identification of identity. I differ reference and inference, or representation and identification as two components which are fundamental for the identification of identity. I found this difference on the role of logical unification and granulation of predicates in the structure of thought and semantical unification and granulation of attributes in the structure of statements and their relation to ontology of context. Through the confrontation of the logical and semantical unification and granulation I find that the limits of logic are not also the limits of language. The semantical unification goes over the highest genre and below the lowest species. That enables the extra-logical, non-scientifical, con-fessional, pro-phetical, artistic, and ordinary use of language. (shrink)

We begin with the hypothetical assumption that Tarski’s 1933 formula ∀ True(x) φ(x) has been defined such that ∀x Tarski:True(x) ↔ Boolean-True. On the basis of this logical premise we formalize the Truth Teller Paradox: "This sentence is true." showing syntactically how self-reference paradox is semantically ungrounded.

It is commonly accepted, after Frege, that identity statements like “Tully is Cicero” differ from statements like “Tully is Tully”. For the former, unlike the latter, are informative. One way to deal with the information problem is to postulate that the terms ‘Tully’ and ‘Cicero’ come equipped with different informative values. Another approach is to claim that statements like these are of the subject/predicate form. As such, they should be analyzed along the way we treat “Tully walks”. Since proper (...) names can appear in predicative position we could go as far as to dismiss the sign of identity altogether, some told us. I will try to discuss the advantages and/or disadvantages of this approach and investigate whether Frege’s view that the ‘is’ of identity must be distinguished from the ‘is’ of predication can be reconciled with the fact that names can appear in predicative position. (shrink)

This paper deals with the logical form of quantified sentences. Its purpose is to elucidate one plausible sense in which quantified sentences can adequately be represented in the language of first-order logic. Section 1 introduces some basic notions drawn from general quantification theory. Section 2 outlines a crucial assumption, namely, that logical form is a matter of truth-conditions. Section 3 shows how the truth-conditions of quantified sentences can be represented in the language of first-order logic consistently with some (...) established undefinability results. Section 4 sketches an account of vague quantifier expressions along the lines suggested. Finally, section 5 addresses the vexed issue of logicality. (shrink)

In the 17th century, Hobbes stated that we reason by addition and subtraction. Historians of logic note that Hobbes thought of reasoning as “a ‘species of computation’” but point out that “his writing contains in fact no attempt to work out such a project.” Though Leibniz mentions the plus/minus character of the positive and negative copulas, neither he nor Hobbes say anything about a plus/minus character of other common logical words that drive our deductive judgments, words like ‘some’, ‘all’, (...) ‘if’, and ‘and’, each of which actually turns out to have an oppositive, character that allows us, “in our silent reasoning,” to ignore its literal meaning and to reckon with it as one reckons with a plus or a minus operator in elementary algebra or arithmetic. These ‘logical constants’ of natural language figure crucially in our everyday reasoning. Because Hobbes and Leibniz did not identify them as the plus and minus words we reason with, their insight into what goes on in ‘ratiocination’ did not provide a guide for a research program that could develop a +/- logic that actually describes how we reason deductively. I will argue that such a +/- logic provides a way back from modern predicate logic—the logic of quantifiers and bound variables that is now ‘standard logic’—to an Aristotelian term logic of natural language that had been the millennial standard logic. (shrink)

We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...) iff Vy(y ≠ x → y ⊧ φ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒ $_{c}$ . Strong completeness of the normal ℒ $_{c}$ logics is proved with respect to models in which all worlds are named. Every ℒ $_{c}$ -logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from ℒ to ℒ $_{c}$ are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched. (shrink)

In this paper I sketch some arguments that underlie Hegel's chapter on judgment, and I attempt to place them within a broad tradition in the history of logic. Focusing on his analysis of simple predicative assertions or ‘positive judgments’, I first argue that Hegel supplies an instructive alternative to the classical technique of existential quantification. The main advantage of his theory lies in his treatment of the ontological implications of judgments, implications that are inadequately captured by quantification. The second (...) concern of this paper is the manner in which Hegel makes logic not only dependent on ontology generally, but also variant in regard to domains of objects. In other words, he offers a domain-specific logical theory, according to which the form of judgment or inference is specific to the subject of judgment. My third concern lies with the metaphilosophical consequences of this theory, and this includes some more familiar Hegelian themes. It is well known that Hegel frequently questioned the adequacy of the sentential form for expressing higher order truths. My reading of his theory of predication explains and contextualizes this tendency by demystifying notions like the so-called speculative proposition. (shrink)

It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, (...) say, Delta-3-1 comprehension axioms are not logical truths. What I am going to suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic". (shrink)

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor differ...

This paper is about compositionality, belief reports, and related issues. I begin by introducing Putnam’s proposal for understanding compositionality, namely that the sense of a sentence is a function of the sense of its parts and of its logical structure (section 1). Both Church and Sellars think that Putnam’s move is superfluous or unnecessary since there is no relevant puzzle to begin with (section 2). I will urge that Putnam is right in thinking that there is indeed a puzzle with (...) a discussion of translation and belief individuation (section 3). Later Salmon (2001/ 2007) reinforces Church’s position, but I will argue that it is still possible to make my case by clarifying the nature of my proposal, i.e., understanding explanations of action from the third-person point of view (section 4). Now, Fine (2007) agrees with Putnam that there is indeed a puzzle to be solved, but he argues that Putnam’s solution of it is problematic, and that his own semantic relationism is a better view. In response to this, I will recast the notion of compositionality based on a certain conception of belief individuation, namely that the semantic content of a sentence is a function of the semantic contents of its parts and of the structure of intensional discourses (sections 3 and 5). Finally the paper will end with a reconsideration of the recalcitrant Kripke’s puzzle about belief (1979/1988), since it might seem to put some pressure on my account. It turns out that my understanding of this puzzle is again different from Fine’s perspective (section 6). (shrink)

Frege's essay "Der Gedanke.Eine logische Untersuchung" was first published in the Beitrage zur Philosophie des Deutschen Idealismus for 1918-1919 and is one of three related logical studies published as a complete work by Gunther Patzig entitled Logische Untersuchungen in Gottingen, 1966 .

Gottlob Frege maintained that two name-containing identity sentences, represented schematically as a=a and a=b,can both be true in virtue of the same object’s self-identity but nonetheless, puzzlingly, differ in their epistemic profiles. Frege eventually resolved his puzzlement by locating the source of the purported epistemic difference between the identity sentences in a difference in the Sinne, or senses, expressed by the names that the sentences contain. -/- Thus, Frege portrayed himself as describing a puzzle that can be posed prior to (...) and independently of any particular theoretical position regarding names, and then resolving that puzzle with his theory of Sinn and Bedeutung. In this paper, I suggest that Frege’s presentation is problematic. If attempt is made to characterize the epistemic status of true identity sentences without appeal to Frege’s theoretical commitments, then what initially seemed puzzling largely dissolves. It turns out that, in order to generate puzzlement, Frege must invoke the theoretical account that he uses the puzzle to establish the purported necessity of. (shrink)

On Kripke’s intended definition, a term designates an object x rigidly if the term designates x with respect to every possible world in which x exists and does not designate anything else with respect to worlds in which x does not exist. Kripke evidently holds in Naming and Necessity, hereafter N&N (pp. 117–144, passim, and especially at 134, 139–140), that certain general terms – including natural-kind terms like ‘‘water’’ and ‘‘tiger’’, phenomenon terms like ‘‘heat’’ and ‘‘hot’’, and color terms like (...) ‘‘blue’’ – are rigid designators solely as a matter of philosophical semantics (independently of empirical, extra-linguistic facts). As a consequence, Kripke argues, identity statements involving these general terms are like identity statements involving proper names (e.g., ‘‘Clark Kent=Superman’’) in that, solely as a matter of philosophical semantics, they express necessary truths if they are true at all. But whereas it is reasonably clear what it is for a (first-order) singular term to designate, Kripke does not explicitly say what it is for a general term to designate. General terms are standardly treated in modern logic as predicates, usually monadic predicates. There are very forceful reasons – due independently to Church and Godel, and ultimately to Frege – for taking predicates to designate their semantic extensions. But insofar as the extension of the general term ‘‘tiger’’ is the class of actual tigers (or its characteristic function), it is clear that the term does not rigidly designate its extension, since the class of tigers in one possible world may differ from the class of tigers in another. What, then, is it for ‘‘tiger’’ to be rigid? (shrink)

A predicate logic typically has a heterogeneous semantic theory. Subjects and predicates have distinct semantic roles: subjects refer; predicates characterize. A sentence expresses a truth if the object to which the subject refers is correctly characterized by the predicate. Traditional term logic, by contrast, has a homogeneous theory: both subjects and predicates refer; and a sentence is true if the subject and predicate name one and the same thing. In this paper, I will examine evidence (...) for ascribing to Aristotle the view that subjects and predicates refer. If this is correct, then it seems that Aristotle, like the traditional term logician, problematically conflates predication and identity claims. I will argue that we can ascribe to Aristotle the view that both subjects and predicates refer, while holding that he would deny that a sentence is true just in case the subject and predicate name one and the same thing. In particular, I will argue that Aristotle's core semantic notion is not identity but the weaker relation of constitution. For example, the predication ‘All men are mortal’ expresses a true thought, in Aristotle's view, just in case the mereological sum of humans is a part of the mereological sum of mortals. (shrink)

Gödel’s slingshot-argument proceeds from a referential theory of definite descriptions and from the principle of compositionality for reference. It outlines a metasemantic proof of Frege’s thesis that all true sentences refer to the same object—as well as all false ones. Whereas Frege drew from this the conclusion that sentences refer to truth-values, Gödel rejected a referential theory of definite descriptions. By formalising Gödel’s argument, it is possible to reconstruct all premises that are needed for the derivation of Frege’s thesis. For (...) this purpose, a reference-theoretical semantics for a language of first-order predicate logic with identity and referentially treated definite descriptions will be defined. Some of the premises of Gödel’s argument will be proven by such a reference-theoretical semantics, whereas others can only be postulated. For example, the principle that logically equivalent sentences refer to the same object cannot be proven but must be assumed in order to derive Frege’s thesis. However, different true (or false) sentences can refer to different states of affairs if the latter principle is rejected and the other two premises are maintained. This is shown using an identity criterion for states of affairs according to which two states of affairs are identical if and only if they involve the same objects and have the same necessary and sufficient condition for obtaining. (shrink)

Frege supposedly believes that vague predicates have no referent (Bedeutung). But given other things he evidently believes, such a position would seem to commit him to a suspect nihilism according to which assertoric sentences containing vague predicates are neither true nor false. I argue that we have good reason to resist ascribing to Frege the view that vague predicates have no Bedeutung and thus good reason to resist seeing him as committed to the suspect nihilism. In the process, I call (...) attention to several under-appreciated texts in which Frege suggests that a vague predicate, though lacking a Bedeutung of its own, can come to acquire a Bedeutung in certain contexts. The upshot of this suggestion is that vague predicates can serve the purposes of ordinary communication quite well, even if they are useless for logical purposes. (shrink)

Down through the ages, logic has adopted many strange and awkward technical terms: assertoric, prove, proof, model, constant, variable, particular, major, minor, and so on. But truth-value is a not a typical example. Every proposition, even if false, no matter how worthless, has a truth-value:even “one plus two equals four” and “one is not one”. In fact, every two false propositions have the same truth-value—no matter how different they might be, even if one is self-contradictory and one is consistent. (...) It is not such a big surprise that every true proposition, no matter how worthless, has a truth-value. Every proposition, whether valuable or worthless, has a truth-value.But it is a little strange that every two true propositions, no matter how different they might be, have the same truth-value. The Pythagorean Theorem has the same truth-value as the proposition that one is one. It is a stretch to think of truth-values as values in any of the normal senses of the word ‘value’. (shrink)

● Sergio Cremaschi, The non-existing Island. I discuss the way in which the cleavage between the Continental and the Anglo-American philosophies originated, the (self-)images of both philosophical worlds, the converging rediscoveries from the Seventies, as well as recent ecumenic or anti-ecumenic strategies. I argue that pragmatism provides an important counter-instance to both the familiar self-images and to the fashionable ecumenic or anti-ecumenic strategies. My conclusions are: (i) the only place where Continental philosophy exists (as Euro-Communism one decade ago) is America; (...) (ii) less obviously, also analytic philosophy does not exist, or does no more exist as a current or a paradigm; what does exist is, on the one hand, philosophy of language and, on the other, philosophy of mind, that is, two disciplines; (iii) the dissolution of analytic philosophy as a school has been extremely fruitful, precisely in so far as it has left room for disciplines and research programmes; (iv) what is left, of the Anglo-American/Continental cleavage is primarily differences in styles, depending partly on intellectual traditions, partly owing to sociology, history, institutional frameworks; these differences should not be blurred by rash ecumenism; besides, theoretical differences are alive as ever, but within both camps; finally, there is indeed a lag (not a difference) in the appropriation of intellectual techniques by most schools of 'Continental' philosophy, and this should be overcome through appropriation of what the best 'analytic' philosophers have produced. ● Michael Strauss, Language and sense-perception: an aspect of analytic philosophy. To test an assertion about one fact by comparing it with perceived reality seems quite unproblematic. But the very possibility of such a procedure is incompatible with the intellectualistic basis of logical positivism and atomism (as it is for example to be found in Russell's Analysis of Mind). According to the intellectualistic approach pure sensation is meaningless. Sensation receives its meaning and order from the intellect through interpretation, which is performed with the help of linguistic tools, i.e. words and sentences. Before being interpreted, sensation is not a picture or a representation, it is neither true nor false, neither an illusion nor knowledge; it does not tell us anything; it is a lifeless and order-less matter. But how can a thought (or a proposition) be compared with such a lifeless matter? This difficulty confronts the intellectualist, if on the one hand he admits the necessity of comparing thought with sense-perception, and on the other hand presupposes that we possess only intellectual and no immediate perceptual understanding of what we see and hear. In this paper I give a critical exposition of three attempts, made by Russell, Neurath and Wittgenstein, to solve this problem. The first attempt adheres to strict conventionalism, the second tends to naturalism and the third leads to an amended, very moderate version of conventionalism. This amended conventionalism looks at sense impressions as being a peculiar language, which includes primary symbols, i.e. symbols not founded on convention and not being in need of interpretation. ● Ernst Tugendhat, Phenomenology and language analysis. The paper, first published in German in 1970, by which Tugendhat gave a start to the German rediscovery of analytic philosophy. The author stages a confrontation between phenomenology and language analysis. He argues that language analysis does not differ from phenomenology as far as the topics dealt with are concerned; instead, both currents are quite different in method. The author argues that language-analytic philosophy does not simply lay out of the mainstream of transcendental philosophy, but that instead it challenges this tradition on the very level of foundations. The author criticizes the linguistic-analytic approach centred on the subject as well as any object-centred approach, while proposing inter-subjective understanding through language as the new universal framework. This is, when construed in so general terms, the same program of hermeneutics, though in a more basic version. ● Jürgen Habermas, Language game, intention and meaning. On a few suggestions by Sellars and Wittgenstein. The paper, first published in German in 1975, in which Habermas announces his own linguistic turn through a discovery of speech acts. In this essay the author wants to work out a categorical framework for a communicative theory of society; he takes Wittgenstein's concept of language game as a Leitfade and, besides, he takes advantage also of Wilfried Sellars's quasi-transcendental account of the genesis of intentionality. His goal is to single out the problems connected with a theory of consciousness oriented in a logical-linguistic sense. ● Zvie Bar-On, Isomorphism of speech acts and intentional states. This essay presents the problem of the formal relationship between speech acts and intentional states as an essential part of the perennial philosophical question of the relation between language and thought. I attempt to show how this problem had been dealt with by two prominent philosophers of different camps in our century, Edmund Husserl and John Searle. Both of them wrote extensively about the theory of intentionality. I point out an interesting, as it were unintended, continuity of their work on that theory. Searle started where Husserl left off 80 years earlier. Their meeting point could be used as the first clue in our search. They both adopted in effect the same distinction between two basic aspects of the intentional experience: its content or matter, and its quality or mode. Husserl did not yet have the concept of a speech act as contradistinguished from an intentional state. The working hypothesis, however, which he suggested, could be used as a second clue for the further elaboration of the theory. The relationship of the two levels, the mental and the linguistic, which remained for Husserl in the background only, became the cornerstone of Searle' s inquiry. He employed the speech act as the model and analysed the intentional experience by means of the conceptual apparatus of his own theory of speech acts. This procedure enabled him to mark out a number of parallelisms and correlations between the two levels. This procedure explains the phenomenon of the partial isomorphism of speech acts and intentional states. ● Roberta de Monticelli, Ontology. A dialogue among the linguistic philosopher, the naturalist, and the phenomenological philosopher. This paper proposes a comparison between two main ways of conceiving the role and scope of that fundamental part of philosophy (or of "first" philosophy) which is traditionally called "ontology". One way, originated within the analytic tradition, consists of two main streams, namely philosophy of language and (contemporary) philosophy of mind, the former yielding "reduced ontology" and the latter "neo-Aristotelian ontology". The other way of conceiving ontology is exemplified by "phenomenological ontology" (more precisely, the Husserlian, not the Heideggerian version). Ontology as a theory of reference ("reduced" ontology, or ontology as depending on semantics) is presented and justified on the basis of some classical thesis of traditional philosophy of language (from Frege to Quine). "Reduced ontology" is shown to be identifiable with one level of a traditional, Aristotelian ontology, namely the one which corresponds to one of the four "senses of being" listed in Aristotle's Metaphysics: "being" as "being true". This identification is justified on the basis of Franz Brentano's "rules for translation" of the Aristotelian table of judgements in terms of (positive and negative) existential judgments such as are easily translatable into sentences of first order predicate logic. The second part of the paper is concerned with "neo-Aristotelian ontology", i.e. with naturalism and physicalism as the main ontological options underlying most of contemporary discussion in the philosophy of mind. The qualification of such options as "neo-Aristotelian" is justified; the relationships between "neo-Aristotelian ontology" and "reduced ontology" are discussed. In the third part the fundamental tenet of "phenomenological ontology" is identified by the thesis that a logical theory of existence and being does capture a sense of "existing" and "being" which, even though not the basic one, is grounded in the basic one. An attempt is done of further clarifying this "more basic" sense of "being". An argument making use of this supposedly "more basic" sense is advanced in favour of a "phenomenological ontology". ● Kuno Lorenz, Analytic Roots in Dialogic Constructivism. Both in the Vienna Circle ad in Russell's early philosophy the division of knowledge into two kinds (or two levels), perceptual and conceptual, plays a vital role. Constructivism in philosophy, in trying to provide a pragmatic foundation - a knowing-how - to perceptual as well as conceptual competences, discovered that this is dependent on semiotic tools. Therefore, the "principle of method" had to be amended by the "principle of dialogue". Analytic philosophy being an heir of classical empiricism, conceptually grasping the "given", and constructive philosophy being an heir of classical rationalism, perceptually providing the "constructed", merge into dialogical constructivism, a contemporary development of ideas derived especially from the works of Charles S. Peirce (his pragmatic maxim as a means of giving meaning to signs) and of Ludwig Wittgenstein (his language games as tools of comparison for understanding ways of life). 7. Albrecht Wellmer, "Autonomy of meaning" and "principle of charity" from the viewpoint of the pragmatics of language. In this essay I present an interpretation of the principle of the autonomy of meaning and of the principle of charity, the two main principles of Davidson's semantic view of truth, showing how both principles may fit in a perspective dictated by the pragmatics of language. I argue that (I) the principle of the autonomy of meaning may be thoroughly reformulated in terms of the pragmatics of language, (ii) the principle of charity needs a supplement in terms of pragmatics of language in order to become really enlightening as a principle of interpretation. Besides, I argue that: (i) on the one hand, the fundamental thesis of Habermas on the pragmatic theory of meaning ("we understand a speech act when we know what makes it admissible") is correlated with the seemingly intentionalist thesis according to which we understand a speech act when we know what a speaker means; (ii) on the other hand, to say that the meaning competence of a competent speaker is basically a competence about a potential of reasons (or also of possible justifications) which is inherently connected with the meaning of statements, or with their use in utterances. ● Rüdiger Bubner, The convergence of analytic and hermeneutic philosophy This paper argues that the analytic philosophy does not exist, at least as understood by its original programs. Differences in the analytic camp have always been bigger than they were believed to be. Now these differences are coming to the fore thanks to a process of dissolution of dogmatism. Philosophical analysis is led by its own inner logic towards questions that may be fairly qualified as hermeneutic. Recent developments in analytic philosophy, e.g. Davidson, seem to indicate a growing convergence of themes between philosophical analysis and hermeneutics; thus, the familiar opposition of Anglo-Saxon and Continental philosophy might soon belong to history. The fact of an ongoing appropriation of analytical techniques by present-day German philosophers may provide a basis for a powerful argument for the unity of philosophizing, beyond its strained images privileging one technique of thinking and rejecting the remainder. Actual philosophical practice should take the dialogue between the two camps more seriously; in fact, the processes described so far are no danger to philosophical work. They may be a danger for parochial approaches to philosophizing; indeed, contrary to what happens in the natural sciences, Thomas Kuhn's "normal science" developing within the framework of one fixed paradigm is not typical for philosophical thinking. And in philosophy innovating revolutions are symptoms more of vitality than of crisis. ● Karl-Otto Apel, The impact of analytic philosophy on my intellectual biography. In my paper I try to reconstruct the history of my Auseinandersetzung mit - as I called it - "language-analytical" philosophy (including even Peircean semiotics) since the late Fifties. The heuristics of my study was predetermined by two main motives of my beginnings: the hermeneutic turn of phenomenology and the transformation of "transcendental philosophy" in the light of the "language a priori". Thus, I took issue with the early and the later Wittgenstein, logical positivism, and post-Wittgensteinian and post-empiricist philosophy of science (i.e. G.H. von Wright and the renewal of the "explanation vs understanding controversy" as well as the debate between Th. Kuhn and Popper/Lakatos); besides, with speech act theory and the debate about "transcendental arguments" since Strawson. The "pragmatic turn", started already by C.L. Morris and the later Carnap, led me to study also the relationship between Wittgensteinian "use" theory of meaning and of truth. This resulted on my side in something like a program of "transcendental semiotics", i.e. "transcendental pragmatics" and "transcendental hermeneutics". ● Ben-Ami Scharfstein, A doubt on both their houses: the blindness to non-western philosophies. The burden of my criticism is that contemporary European philosophers of all kinds have continued to think as if there were no true philosophy but that of the West. For the most part, the existentialists have been oblivious of their Eastern congeners; the hermeneuticians have yet to stretch their horizons beyond the most familiar ones; and the analysts remain unaware of the analyses and linguistic sensitivities of the ancient non-European philosophers. Briefly, ignorance still blinds almost all contemporary Western philosophers to the rich, variegated philosophical traditions outside of their familiar orbit. Both Continental and Anglo-Americans have lost the breadth of view that once characterized such thinkers as Herder and the Humboldts. The blindness that has resulted is not simply that of individual Western philosophers but of our whole, still parochial philosophical culture. (shrink)

Tarski’s Convention T—presenting his notion of adequate definition of truth (sic)—contains two conditions: alpha and beta. Alpha requires that all instances of a certain T Schema be provable. Beta requires in effect the provability of ‘every truth is a sentence’. Beta formally recognizes the fact, repeatedly emphasized by Tarski, that sentences (devoid of free variable occurrences)—as opposed to pre-sentences (having free occurrences of variables)—exhaust the range of significance of is true. In Tarski’s preferred usage, it is part of the meaning (...) of true that attribution of being true to a given thing presupposes the thing is a sentence. Beta’s importance is further highlighted by the fact that alpha can be satisfied using the recursively definable concept of being satisfied by every infinite sequence, which Tarski explicitly rejects. Moreover, in Definition 23, the famous truth-definition, Tarski supplements “being satisfied by every infinite sequence” by adding the condition “being a sentence”. Even where truth is undefinable and treated by Tarski axiomatically, he adds as an explicit axiom a sentence to the effect that every truth is a sentence. Surprisingly, the sentence just before the presentation of Convention T seems to imply that alpha alone might be sufficient. Even more surprising is the sentence just after Convention T saying beta “is not essential”. Why include a condition if it is not essential? Tarski says nothing about this dissonance. Considering the broader context, the Polish original, the German translation from which the English was derived, and other sources, we attempt to determine what Tarski might have intended by the two troubling sentences which, as they stand, are contrary to the spirit, if not the letter, of several other passages in Tarski’s corpus. (shrink)

Truth is a matter of interest not only to philosophers, but to scientists and other researchers in various branches of knowledge. This paper examines Davidson’s views of the concept of truth. In the first section, I provide a brief account of the basic ideas of Davidson’s philosophy. An understanding of Davidson’s philosophy is essential for anyone who wishes to follow recent debates in the philosophy of language, the philosophy of action, the philosophy of logic, and the philosophy of the (...) mind. I give also a brief sketch of the leading traditional theories of truth, i.e., coherence, correspondence and pragmatic theories. In the second section, I give a comprehensive account of Davidson’s main contributions to the truth - the conditions theory of meaning. That the meaning of a sentence can be given by stating the conditions under which it is true, is not a novel idea. It is at least as old as the work of Frege. In addition to Frege, this idea has been held by, among others, Wittgenstein, Carnap, Quine and - as its leader apologist and most recent proponent - Davidson. Before I look at the details of Davidson’s contributions, I show how they relate to the work of Frege, Tarski, and Quine, all of whom greatly influenced Davidson’s theory. Putnam, Dummett, Rorty and many other philosophers rank Davidson among the deflationists. They all think that Tarski’s truth predicates completely capture the concept of truth. This is a mistake. The third section is concerned with this mistake. In the fourth section, I explain Davidson’s criticisms of what he regards as the main other anti-deflationary attempts to characterize truth: views which epistemologize truth on the one hand, and the correspondence theory of truth, on the other hand. Finally, in the fifth section, I make some remarks about Davidson’s treatment of the concept of truth. (shrink)

In this paper I introduce the idea of a higher-order modal logic—not a modal logic for higher-order predicate logic, but rather a logic of higher-order modalities. “What is a higher-order modality?”, you might be wondering. Well, if a first-order modality is a way that some entity could have been—whether it is a mereological atom, or a mereological complex, or the universe as a whole—a higher-order modality is a way that a first-order modality could have been. (...) First-order modality is modeled in terms of a space of possible worlds—a set of worlds structured by an accessibility relation, i.e., a relation of relative possibility—each world representing a way that the entire universe could have been. A second-order modality would be modeled in terms of a space of spaces of (first-order) possible worlds, each space representing a way that (first-order) possible worlds could have been. And just as there is a unique actual world which represents the way that things actually are, there is a unique actual space which represents the way that first-order modality actually is. -/- One might wonder what the accessibility relation itself is like. Presumably, if it is logical or metaphysical modality that is being dealt with, it is reflexive; but is it also symmetric, or transitive? Especially in the case of metaphysical modality, the answer is not clear. And whichever of these properties it may or may not have, could that itself have been different? Could at least some rival modal logics represent different ways that first-order modality could have been? -/- To be clear, the idea behind my proposal is not just that some things which are possible or necessary might not have been so at the first order, as determined by the actual accessibility relation, but also that the actual accessibility relation, and hence the nature or structure of actual modality, could have been different at some higher order of modality. Even if the accessibility relation is actually both symmetric and transitive, perhaps it could (second-order) have been otherwise: There is a (second-order) possible space of worlds in which it is different, where it fails to be symmetric, or transitive. We must, therefore, introduce the notion of a higher-order accessibility relation, one that in this case relates spaces of first-order worlds. The question then arises as to whether that relation is symmetric, or transitive. We can then consider third-order modalities, spaces of spaces of spaces of possible worlds, where the second-order accessibility relation differs from how it actually is. I can see no reason why there should be a limit to this hierarchy of higher-order modalities, any more than I can see a reason why there should be a limit to the hierarchy of higher-order properties. There will thus be an infinity of orders, one for each positive integer, and each order will have an accessibility relation of its own. To keep things as clear as possible, a space of first-order points (i.e., of possible worlds) shall be called a galaxy, a space of second-order points, a universe, and a space of any higher order, a cosmos. However, to keep things as simple as possible, in what follows I will deal with but a single cosmos, and hence will not deal with modalities higher than the third order. -/- The accessibility relation is not the only thing that might be thought to vary between spaces of worlds: Perhaps the contents of the spaces can vary as well. While I presume that the contents of the worlds themselves remain constant—it makes doubtful sense to suppose that in one space some entity e exists in a world w and in another space e doesn’t exist in that same world w—we may suppose that different spaces differ as to which worlds they contain, just as different worlds may differ as to which objects they contain. Thus we might have a higher-order analogue of a variable-domain modal logic. There seem, then, to be three ways in which spaces can differ: First, as to the properties of the accessibility relation; second, as to which worlds the relation relates; and third, as to which worlds or spaces are parts of their domains. -/- The paper will be structured as follows. In Section 2 I provide some reasons why one might want to pursue this kind of project in the first place. In Section 3 I outline the syntax and semantics of my proposed logic. Section 4 covers semantic tableaux for this system; and after giving the rules for their construction, I construct a few of them myself to establish some logical consequences of the system and give the reader a feel for how it works. In Sections 5, 6 and 7 I explore some of its potential philosophical implications for areas besides logic, namely the philosophy of language; metaphysics, including the metaphysics of modality and the philosophy of time, and finally the philosophy of religion, before concluding the paper in Section 8. (shrink)

W. V. Quine’s criterion of ontological commitment is commonly presented as the slogan: “to be is to be the value of a variable”. More specifically, to be is to be in the domain of values of a bound variable attached to a existential quantifier in the body of a theory. If a sentence of our best available theory quantifies over bachelors, so we conclude that, according to it, there are bachelors. Simplicity and the use of logical apparatus for determining this (...) criterion are responsible for many philosophers to adhere to this standpoint within analytical metaphysics. I will argue that despite this strong influence of the Quinean criterion of ontological commitment in many debates, we find at least three interlinked problems in its original formulation: (1) The criterion depends of a clear definition of theory for the determination of the quantifications counting as entailers of ontological commitments. However, Quine does not present such clear concept of what would be a theory and of how to formalize it. Quine develops this notion under just the metaphor of the knowledge fabric, in which we have a group of sentences, which does not suffice for the required account of ontological commitment. (2) There are distinct possibilities of quantifying over given sentences, which leaves room for different ontological commitments associated with the same statement. In that sense, someone could argue that the Quinean solution would be to endorse just the first order predicate logic. Nonetheless, the Quinean should explain why we ought to be restricted to this logic. (3) The previous problems lead to a possible explosion of the Quinean criterion of ontological commitment, making it trivial, because it permits the quantification over any supposed entities and, hence, that anything can exist. Thus, I will discuss Quine’s criterion of ontological commitment in the light of the abovementioned problems, showing that, unless the Quinean presents a more specific definition of theory beyond just a group of sentences, he leads to an explosion of commitments with different entities. I will conclude that, if you understand ‘theory’ as Alonzo Church’s concept of ‘formalized language’, this knot can be undone. However, Church’s concept is much more akin to Carnap’s linguistic framework than to a notion of knowledge fabric “without borders” as the one proposed by Quine. (shrink)

Subverting a once widely held Quinean paradigm, there is a growing consensus among philosophers of logic that higher-order quantifiers (which bind variables in the syntactic position of predicates and sentences) are a perfectly legitimate and useful instrument in the logico-philosophical toolbox, while neither being reducible to nor fully explicable in terms of first-order quantifiers (which bind variables in singular term position). This article discusses the impact of this quantificational paradigm shift on metaphysics, focussing on theories of properties, propositions, and (...) identity, as well as on the metaphysics of modality. (shrink)

The paper is Part III of the large research, dedicated to both the revision of the system of basic logical categories and the generalization of modern predicate logic to functional logic. We determinate and contrapose modern Fregean logistics and proposed by the author ultra-Fregean logistics, next we describe values and arguments of functions, arguments of relations, relations themselves, sets (classes), and subsets (subclasses) as derivative categories (concepts) of ultrafregean logistics. Logistics is a part of metalogic, independent of (...) semantics. Fregean logictics is a metalogical theory, based on the quadruple <particular (individual), predicate, equality, sequence>; it generates predicate logic. Ultrafregean logictics is based on the quadruple <particular, function, representation, sequence>, where the notion of a function is a generalization of the notion of a predicate and the notion of representation is a generalization of the notion of equality; this logictics generates functional logic. For the completely correct denotation of the functional values, we need the Churchian symbolics with parenthesis. Predicates are usually identified with relations. A relation is the derived and even definable category of ultra-Fregean logictics. Namely, relations are representations by functions (of one of their arguments). We show that Frege could really establish this definition and the notion (category) of representation but, unfortunately, rejected this course of thought. Next, we show that every n-ary relation can be resolved for some of its arguments via some (n-1)-ary function. A set, or class, is a derived and not definable category of ultra-Fregean logistics. The universal way to introduce the sets is Frege’s abstraction principle. We formulate this principle for functional logic and show that the notion of a set is quantified, so there is the dual existential notion of a nonempty subset, involved by the same abstraction principle. (shrink)

This paper provides a new approach to a family of outstanding logical and semantical puzzles, the most famous being Frege's puzzle. The three main reductionist theories of propositions (the possible-worlds theory, the propositional-function theory, the propositional-complex theory) are shown to be vulnerable to Benacerraf-style problems, difficulties involving modality, and other problems. The nonreductionist algebraic theory avoids these problems and allows us to identify the elusive nondescriptive, non-metalinguistic, necessary propositions responsible for the indicated family of puzzles. The algebraic approach is also (...) used to defend antiexistentialism against existentialist prejudices. The paper closes with a suggestion about how this theory of content might enable us to give purely semantic (as opposed to pragmatic) solutions to the puzzles based on a novel formulation of the principle of compositionality. (shrink)

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient (...) class='Hi'>logic texts. A renaissance in ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and Greg Restall’s (...) bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness (...) is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)

This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. (...) It is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

Despite the wide acceptance of standard modal logic, there has always been a temptation to think that ordinary modal discourse may be correctly analyzed and adequately represented in terms of predicates rather than in terms of operators. The aim of the formal model outlined in this paper is to capture what I take to be the only plausible sense in which ‘possible’ and ‘necessary’ can be treated as predicates. The model is built by enriching the language of standard modal (...) logic with a quantificational apparatus that is “substitutional” rather than “objectual”, and by obtaining from the language so enriched another language in which constants for such predicates apply to singular terms that stand for propositions. (shrink)

Imagine an equilateral triangle “pointing upward”—its horizontal base under its apex angle. A semiotic triangle has the following three “vertexes”: (apex) an expression, (lower-left) one of the expression’s conceptual meanings or senses, and (lower-right) the referent or denotation determined by the sense [1, pp. 88ff]. One example: the eight-letter string ‘coleslaw’ (apex), the concept “coleslaw” (lower-left), and the salad coleslaw (lower-right) [1, p. 84f]. Using Church’s terminology [2, pp. 6, 41]—modifying Frege’s—the word ‘coleslaw’ expresses the concept “coleslaw”, the word ‘coleslaw’ (...) denotes or names the salad coleslaw, and the concept “coleslaw” determines the salad coleslaw—recalling Frege’s principle that sense determines denotation. Church [2, p. 6] wrote: -/- We shall say that a name denotes or names its denotation and expresses its sense. […] Of the sense we say that it determines its denotation, or is a concept of the denotation. -/- Aristotle seems cognizant of distinctions going beyond those in semiotic triangles. The expression Aristotle’s semiotic pyramids seem warranted by Aristotle’s Categories, 1a1: -/- When [two] things have a name (onoma) in common and the concept (logos) of being (ousia) which corresponds to the name in each case is different, they are called same-named (homonuma). Thus, for example, both a man and a picture [of an animal] are called animals. These have only a name in common. In each case the name’s concept of being [an animal] is different; for if one says what being an animal is for each of them, one will give two distinct concepts. -/- Semiotic triangles and pyramids in Aristotle’s logic are compared to those in Church’s [2]. [1] JOHN CORCORAN, Sentence, proposition, judgment, statement, and fact, Many Sides of Logic, College Publications, 2009. [2] ALONZO CHURCH, Introduction to Mathematical Logic, Princeton, 1956. -/- The semiotic pyramid in Categories, 1a1 has a square base under the vertex ‘animal’. On the corners of the square are: the concept “animal”, the concept “animal picture”, the animals, and the animal pictures. The animals are homonymous with the animal pictures. People find Aristotle’s example far-fetched or inept even if the experience of pointing to a picture while saying “That is Tarski” is familiar. Imagine looking at a painting while thinking “That is an animal”. Without putting too fine a point on this, notice that in Aristotle’s sense it is individual things that are homonymous, not words. It would be natural to say also in his sense that two things are homonyms if one is homonymous with the other. In contrast, we use the words homonym and homonymous to relate words that are spelled the same and pronounced the same but have different meaning. Consider the noun ‘center’ and the verb ‘center’. Consider the noun ‘smell’ and the verb ‘smell’. The spelling of two homonyms is an ambiguity, or an ambiguous spelling. We need appropriate adjectives to distinguish the Categorical senses of ‘homonym’ and ‘homonymous’ from the current English sense just mentioned. I propose ‘ontological’ for the sense relating things and ‘linguistic’ for that relating words. Given that all words are things but not all things are words, we ask are words that are linguistically homonymous also ontologically homonymous? END OF POST ABSTRACT. (shrink)

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.