A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.

Paul Pietroski presents an original philosophical theory of actions and their mental causes. We often act for reasons: we deliberate and choose among options, based on our beliefs and desires. However, bodily motions always have biochemical causes, so it can seem that thinking and acting are biochemical processes. Pietroski argues that thoughts and deeds are in fact distinct from, though dependent on, underlying biochemical processes within persons.

Seventeen specially written essays by eminent philosophers and linguists appear for the first time in this anthology, all with the central theme of logical form -- a fundamental issue in analytic philosophy and linguistic theory. Logical Form and Language brings together exciting new contributions from diverse points of view, which illuminate the lively current debate about this topic.

Noam Chomsky's book on syntactic structures is a serious attempts on the part of a linguist to construct within the tradition of scientific theory-construction ...

A study of how syntax relates to meaning by a leader of the new generation of philosopher-linguists.

Provides comprehensive coverage of logical structure as well as the techniques of formal reasoning.

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...) the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?". (shrink)

This paper presents a slightly modified version of the compositional semantics proposed in Events and Semantic Architecture (OUP 2005). Some readers may find this shorter version, which ignores issues about vagueness and causal constructions, easier to digest. The emphasis is on the treatments of plurality and quantification, and I assume at least some familiarity with more standard approaches.

This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it.

In these essays the minimalist approach to linguistic theory is formulated and progressively developed.

Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued (...) that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (shrink)

The event analysis of action sentences seems to be at odds with plausible (Davidsonian) views about how to count actions. If Booth pulled a certain trigger, and thereby shot Lincoln, there is good reason for identifying Booths' action of pulling the trigger with his action of shooting Lincoln; but given truth conditions of certain sentences involving adjuncts, the event analysis requires that the pulling and the shooting be distinct events. So I propose that event sortals like 'shooting' and 'pulling' are (...) true of complex events that have actions (and various effects of actions) as parts. Combining this view with some facts about so-called causative verbs, I then argue that paradigmatic actions are best viewed as tryings, where tryings are taken to be intentionally characterized events that typically cause peripheral bodily motions. The proposal turns on a certain conception of what it is to be the Agent of an event; and I conclude by elaborating this conception in the context of some recent discussions about the relation of thematic roles to grammatical categories. (shrink)

Ernie Lepore and Jerry Fodor have published a series of original and controversial essays on issues relating to compositionality in language and mind; they have...

Frege's work was largely devoted to an attempt to argue that the'basic laws of arithmetic' are truths of logic. That attempt had both philosophical and formal aspects. The present note offers an introduction to both of these, so that readers will be able to appreciate contemporary discussions of the philosophical significance of 'Frege's Theorem'.

A certain metaphysical thesis about meaning that we'll call Informational Role Semantics (IRS) is accepted practically universally in linguistics, philosophy and the cognitive sciences: the meaning (or content, or `sense') of a linguistic expression1 is constituted, at least in part, by at least some of its inferential relations. This idea is hard to state precisely, both because notions like metaphysical constitution are moot and, more importantly, because different versions of IRS take different views on whether there are constituents of meaning (...) other than inferential role, and on which of the inferences an expression occurs in are meaning constitutive. Some of these issues will presently concern us; but for now it will do just to gesture towards such familiar claims as that: it's part and parcel of dog meaning dog2 that the inference from x is a dog to x is an animal is valid; it's part and parcel of boil meaning boil that the inference from x boiled y to y boiled is valid; it's part and parcel of kill meaning kill that the inference from x killed y to y died is valid; and so on. (See Cruse, Ch. 1 and passim.) IRS brings in its train a constellation of ancillary doctrines. Presumably, for example, if an inference is constitutive of the meaning of a word, then learning the word involves learning that the inference holds. If dog means dog because dog ---> animal is valid, then knowing that dog ---> animal is valid is part and parcel of knowing what the word dog means; and, similarly, learning that x boiled y ---> y boiled is part and parcel of learning what boil means, and so forth. IRS constrains grammatical theories. The semantic lexicon of a language is supposed to make explicit whatever one has to know to understand the lexical expressions of the language, so IRS implies that meaning constitutive inferences are part of the semantic lexical entries for items that have them. Lexical entries are thus typically complex objects (`bundles of inferences') according to standard interpretations of IRS.. (shrink)

The representation of quantification over relations in monadic third-order logic is discussed; it is shown to be possible in numerous special cases of foundational interest, but not in general unless something akin to the Axiom of Choice is assumed.

What constitutes our number concept? What makes it possible for us to employ numbers the way we do; which mental faculties contribute to our grasp of numbers? What do we share with other species, and what is specific to humans? How does our language faculty come into the picture? This 2003 book addresses these questions and discusses the relationship between numerical thinking and the human language faculty, providing psychological, linguistic and philosophical perspectives on number, its evolution and its development in (...) children. Heike Wiese argues that language as a human faculty plays a crucial role in the emergence of systematic numerical thinking. She characterises number sequences as powerful and highly flexible mental tools that are unique to humans and shows that it is language that enables us to go beyond the perception of numerosity and to develop such mental tools. (shrink)

Die Grundlagen gehören zu den klassischen Texten der Sprachphilosophie, Logik und Mathematik. Frege stützt sein Programm einer Begründung von Arithmetik und Analysis auf reine Logik, indem er die natürlichen Zahlen als bestimmte Begriffsumfänge definiert. Die philosophische Fundierung des Fregeschen Ansatzes bilden erkenntnistheoretische und sprachphilosophische Analysen und Begriffserklärungen. Studienausgabe aufgrund der textkritisch herausgegebenen Jubiläumsausgabe (Centenarausgabe). Mit Einleitung, Anmerkungen, Literaturverzeichnis und Namenregister.

Extension of the logical language to deliver plural reference and the logical relations that constitute knowledge of the singular and plural acquires empirical bite just in case it conforms with increasing precision to the syntax of the natural language and affords explanation of what speakers know about the distribution and meaning of plural expressions in their language. As for the syntax of natural language, this discussion, being none too precise, is guided throughout by just two considerations and their immediate consequences, (...) which is discussed at greater length in this article. The first, morpheme univocality, is that a morpheme despite its various syntactic and morphological contexts has a single meaning that supports all its occurrences. (shrink)

The book is an attempt at explaining to the nation the ideas of Frege's Grundlagen. It is wordy and trite, a paradigm case of a redundant piece of writing. The reader is advised to steer clear of it.

Precursors. 2.1 Introduction Thus far I have presented an approach to the semantics of plurals in the form of two rather similar grammars for a fragment of English. And I have given a few examples of the kinds of things one can say within this ...