From proofs in any classical first-order theory that proves the existence of at least two elements, one can eliminate definitions in polynomial time. From proofs in any classical first-order theory strong enough to code finite functions, including sequential theories, one can also eliminate Skolem functions in polynomial time.

In the history of philosophy, especially its recent history, a number of definitions of necessity have been ventured. Most people, however, find these definitions either circular or subject to counterexamples. I will show that, given a broadly Fregean conception of properties, necessity does indeed have a noncircular counterexample-free definition.

According to the standard view of definition, all defined terms are mere stipulations, based on a small set of primitive terms. After a brief review of the Hilbert-Frege debate, this paper goes on to challenge the standard view in a number of ways. Examples from graph theory, for example, suggest that some key definitions stem from the way graphs are presented diagramatically and do not fit the standard view. Lakatos's account is also discussed, since he provides further examples that suggest (...) many definitions are much more than mere convenient abbreviations. (shrink)

I aim to show how and why some definitions can be benignly circular. According to Lloyd Humberstone, a definition that is analytically circular need not be inferentially circular and so might serve to illuminate the application-conditions for a concept. I begin by tidying up some problems with Humberstone's account. I then show that circular definitions of a kind commonly thought to be benign have inferentially circular truth-conditions and so are malign by Humberstone's test. But his test is too demanding. The (...) inferences we actually use to establish the applicability of, e.g., colour concepts are designed to establish warranted assertability and not truth. Understood thus, dispositional analyses are not inferentially circular. (shrink)

In the present paper I shall first summarize Popper's criticism of the traditional method of definition, and then go on to comment critically on his own views on the form and function of so-called nominalist definitions.

Socrates' greatest philosophical contribution was to have initiated the search for definitions. In Definition in Greek Philosophy his views on definition are examined, together with those of his successors, including Plato, Aristotle, the Stoics, Galen, the Sceptics and Plotinus. Although definition was a major pre-occupation for many Greek philosophers, it has rarely been treated as a separate topic in its own right in recent years. This volume, which contains fourteen new essays by leading scholars, aims to reawaken interest in a (...) number of central and relatively unexplored issues concerning definition. These issues are briefly set out in the Introduction, which also seeks to point out scholarly and philosophical questions which merit further study. (shrink)

Those who have a brief against the analytic-synthetic distinction raise problems for what seem to supporters of the distinction to be some of the clearest cases. That bachelors are unmarried seems to many to be analytically true. But to hold this seems to imply that there is a definition of "bachelor" that includes being unmarried. But critics of the analytic-synthetic distinction, such as Jerry Fodor, deny that there are true definitions (reportive, not stipulative). So there can be no definition of (...) "bachelor". And many have noted that defining "bachelor" is not as easy as appears at first blush. (shrink)

Jerry Fodor ( Concepts: Where cognitive science went wrong . New York: Oxford University Press, 1998 ) famously argued that lexical concepts are unstructured. After examining the advantages and disadvantages of both the classical approach to concepts and Fodor’s conceptual atomism, I argue that some lexical concepts are, in fact, structured. Roughly stated, I argue that structured lexical concepts bear a necessary biconditional entailment relation to their structural constituents. I develop this account of the structure of lexical concepts within the (...) framework of Pavel Tichý’s ( The foundations of Frege’s logic . Berlin, New York: De Gruyter, 1988 ) theory of constructions. I argue that concepts are constructions which can be combined by way of Tichý’s construction-forming operations of composition and closure and an additional operation, simplification , which I propose in section 6. The last of these construction-forming operations plays a central role in my account of lexical concept structure. Stated generally, structured lexical concepts are a result of simplifying their structural constituents. (shrink)

Paul Boghossian advocates a version of the analytic theory of a priori knowledge. His defense of an "epistemic" notion of analyticity is based on an implicit definition account ofthe meaning of the logical constants. Boghossian underestimates the power of the classical Quinean criticisms, however; the challenge to substantiate the distinction between empirical and non-empirical sentences, as forcefully presented in Two Dogmas, still stands, and the regress from Truth by Convention still needs to be avoided. Here, Quine also showed that there (...) are no implicit definers for the logical constants. Moreover, even if they existed, their epistemic analyticity would, on Boghossian's own account, be doubtful. (shrink)

The book begins by focusing on the psychological constraints governing Frege's notion of sense, or meaning, and argues that, given these constraints, even the ...

Operational definitions were once considered the backbone of semantics of natural science. Still in 1955 A. W. Burks published an explication of the general scheme of these definitions. In the fifties of the last century however they became outmoded, while high school teachers for presumably good reasons were still in favour of them. I consider the banishment of this kind of definitions premature, and try to improve the explication of Burks in a way which qualifies them for a rehabilitation. In (...) the improved explication the concept of intrinsic property becomes important, for which I give a new definition on the basis of a structuralist account of physical processes. German Die operationalen Definitionen galten einst als der Kern der Semantik der Naturwissenschaft. Eine letzte Explikation des allgemeinen Schemas der operationalen Definitionen hat A. W. Burks noch 1955 vorgelegt. In den fünf ziger Jahren kamen sie indessen aus der Mode, während sie noch weiterhin von Gymnasiallehrern - wohl aus berechtigten Gründen - geschätzt wurden. Ich halte die Verbannung dieser Art von Definitionen für voreilig und versuche die Explikation von Burks in einer Weise zu verbessern, dass man sie wieder in der Wissenschaftstheorie rehabilitieren kann. In der verbesserten Explikation wird der Begriff der inneren Eigenschaft eines Dinges oder Vorgangs wichtig, den ich auf der Basis einer strukturalistischen Darstellung physikalischer Vorgänge neu definieren möchte. (shrink)

The neo-Fregeans have argued that definition by abstraction allows us to introduce abstract concepts such as direction and number in terms of equivalence relations such as parallelism between lines and one-one correspondence between concepts. This paper argues that definition by abstraction suffers from the fact that an equivalence relation may not be sufficient to determine a unique concept. Frege’s original verdict against definition by abstraction is thus reinstated.

Kripke maintains that one who stipulatively introduces the term ‘one meter’ as a rigid designator for the length of a certain stick s at time t is in a position to know a priori that if s exists at t then the length of s at t is one meter. Some (e.g., Soames 2003) have objected to this alleged instance of the contingent a priori on the grounds that the stipulator's knowledge would have to be based in part on substantive (...) metalinguistic knowledge. I examine Soames's argument for the a posteriority of the relevant metalinguistic knowledge, and I argue that its main premise is false. (shrink)

Kripke maintains that one who stipulatively introduces the term ‘one meter’ as a rigid designator for the length of a certain stick s at time t is in a position to know a priori that if s exists at t then the length of s at t is one meter. Some (e.g., Soames 2003) have objected to this alleged instance of the contingent a priori on the grounds that the stipulator's knowledge would have to be based in part on substantive (...) metalinguistic knowledge. I examine Soames's argument for the a posteriority of the relevant metalinguistic knowledge, and I argue that its main premise is false. (shrink)

The first part deals with the problem of the external form of ostensive definition. It is concluded that the definition statement is not complete. The proper form of this statement is not a sentence, but a sentential function, namely a sentential function of the type: ``Π x [N(x)=x is in the respect R and in the degree D such as A, B... and not such as K, L...]" where "N" stands for the term being defined. Thus the ostensive definition informs (...) about the criteria of applicability of the defined term in a partial way only, and the rest must be supplied by the addressee for whom the given definition was destined. In the second part the conditions are analysed on which depends the possibility of solving that problem, and consequently the conditions on which depend the informational value and the efficacy of ostensive definition. The concluding remarks deal with the properties of the terms introduced by the ostensive method. (shrink)

Much of The Reason’s Proper Study is devoted to defending the claim that simply by stipulating an abstraction principle for the “number-of” functor, we can simultaneously fix a meaning for this functor and acquire epistemic entitlement to the stipulated principle. In this paper, I argue that the semantic and epistemological principles Hale and Wright offer in defense of this claim may be too strong for their purposes. For if these principles are correct, it is hard to see why they do (...) not justify platonist strategies that are not in any way “neo-Fregean,” e.g. strategies that treat “the number of Fs” as a Russellian definite description rather than a singular term, or employ axioms that do not have the form of abstraction principles. (shrink)

Chapter I: The Problem Stated Section: The Paradox of Definition i) Here is the problem which is the main concern of this book. ...

Merely conceptual knowledge, not based on specific sensitivity to the referential domain, is not seriously a priori. It is argued here that it is either weakly and superficially a priori, or downright a posteriori. This is done starting from the fact that many of our definitions (or concepts) are recognizably empirically established, and pointing out that recognizably empirical grounding yields superficial apriority. Further, some (first-order) concept analyzing propositions are empirically false about their referents and thus empirically refutable. Therefore, our empirical (...) definitions (or concepts) are fallible and empirically revisable: they can turn out to be incorrect about the intended satisfiers of the concept defined, and their concept analyzing propositions to be false. Now, empirical revisability is incompatible with strong apriority (and entails at best a weak apriority or aposteriority). The result is quite shocking: analyticity does not entail apriority. (shrink)

Gupta"s and Belnap"s Revision Theory of Truth defends the legitimacy of circular definitions. Circularity, however, forces us to reconsider our conception of meaning. A readjustment of some standard theses about meaning is here proposed, by relying on a novel version of the sense–reference distinction.

The traditional definition per genus et differentiam is argued to be cognitively grounded in perception and in order to avoid needless argument, definitions are stipulated to assert boundaries. An analysis of the notion of perspective shows that a boundary is a composite of two distinctions: similarity that includes and difference that excludes. The concept is applied to the type-token distinction and percepts are shown to be the result of a comparison between a token as representing some phenomenon and a type (...) as representing the kind of object the phenomenon might be. The Principle of Connection is proposed as the mental architecture that gives rise to percepts. The value of a percept is shown to be one of Identical, Similar, Different or Contrary and its perceptual aspects represent the cognitive schema of Essence, Quality, Quantity and Relation. The two notions of similarity and difference are claimed to constitute the necessary and sufficient conditions for definitions. Accordingly, the qualitative elements of a definition signify the genus as representing the similarity of the defined entity in terms of others of its kind and the quantitative elements signify the differentia as representing the difference it has from everything else. Paying proper attention to both qualitative and quantitative aspects will enable formulation of definitions that are ‘good’. (shrink)

The arguments of Fodor, Garret, Walker and Parkes [(1980) Against definitions, Cognition, 8, 263-367] are the source of widespread skepticism in cognitive science about lexical semantic structure. Whereas the thesis that lexical items, and the concepts they express, have decompositional structure (i.e. have significant constituents) was at one time "one of those ideas that hardly anybody [in the cognitive sciences] ever considers giving up" (p. 264), most researchers now believe that "[a]ll the evidence suggests that the classical [(decompositional)] view is (...) wrong as a general theory of concepts" [Smith, Medin & Rips (1984) A psychological approach to concepts: comments on Rey, Cognition, 17, 272], and cite Fodor et al. (1980) as "sounding the death knell for decompositional theories" [MacNamara & Miller (1989) Attributes of theories of meaning, Psychological Bulletin, 106, 360]. I argue that the prevailing skepticism is unmotivated by the arguments in Fodor et al. Fodor et al. misrepresent the form, function and scope of the decompositional hypothesis, and the procedures they employ to test for the psychological reality of definitions are flawed. I argue, further, that decompositional explanations of the phenomena they consider are preferable to their primitivist alternatives, and, hence, that there is prima facie reason to accept them as evidence for the existence of decompositional structure. Cognitive scientists would, therefore, do well to revert to their former commitment to the decompositional hypothesis. (shrink)

According to Gupta and Belnap, the “extensional behavior” of ‘true’ matches that of a circularly defined predicate. Besides promising to explain semantic paradoxicality, their general theory of circular predicates significantly liberalizes the framework of truth-conditional semantics. The authors’ discussions of the rationale behind that liberalization invoke two distinct senses in which a circular predicate’s semantic behavior is explained by a “revision rule” carrying hypothetical information about its extension. Neither attempted explanation succeeds. Their theory may however be modified to employ a (...) relativized notion of extension. The resulting contextualist semantics for ‘true’ construes circularity as a pragmatic phenomenon. (shrink)

This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to (...) Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. (shrink)

7.3.1 Ostension 7.3.2 Extensional Definition by Naming 7.3.3 Extensional Definition by Unique Description 7.4 Two Case Studies in the Application of the Intension/Extension Distinction 7.4.1 "God exists, by definition" 7.4.2 The 'Width' of an Intensional Definition..

These are some of the rules of classification and definition. But although nothing is more important in science than classifying and defining well, we need say no more about it here, because it depends much more on our knowledge of the subject matter being discussed than on the rules of logic. (Arnauld and Nicole (1683/1996) p.128).

On a rather popular conception, the paradox of analysis suggests that the intersubstitutivity of analysans and analysandum should be restricted to non-psychological contexts. This is typically taken to be compatible with the idea that two sentences differing only in that one has the analysandum where the other has the analysans express exactly the same proposition. In this note we argue that this should be pondered upon in light of the view that many important ordinary concepts are circular. In particular, we (...) submit that if there are correct analyses grounding circular definitions, then we are bound to further restrict the substitutivity principle, for we must admit that it might fail even in non- psychological contexts. (shrink)

This original and enticing book provides a fresh, unifying perspective on many old and new logico-philosophical conundrums. Its basic thesis is that many concepts central in ordinary and philosophical discourse are inherently circular and thus cannot be fully understood as long as one remains within the confines of a standard theory of definitions. As an alternative, the authors develop a revision theory of definitions, which allows definitions to be circular without this giving rise to contradiction (but, at worst, to “vacuous” (...) uses of definienda). The theory is applied with varying levels of detail to a circular analysis of concepts as diverse as truth, predication, necessity, physical object, etc. The focus is on truth, and hope is expressed that a deeper understanding of the Liar and related paradoxes has been provided: “We have tried to show that once the circularity of truth is recognized, a great deal of its behavior begins to make sense. In particular, from this viewpoint, the existence of the paradoxes seems as natural as the existence of the eclipses” (p. 142). We think that this hope is fully justified, although some problems remain that future research in this field should take into account. (shrink)

The problem of the interpretation of theoretical terms is outlined, and some difficulties connected with the distinction between partial definitions and empirical postulates are discussed. A reconstruction is sketched which is intended to explicate the 'definitional' character of partial definitions. Finally, some implications for the methodology of theory construction are indicated.

"Socrates' Pursuit of Definitions" examines the manner in which Socrates pursues definitions in Plato's early definitional dialogues and advances the following claims. Socrates evaluates definitions (proposed by his interlocutors or himself) by considering their consistency with conditions of the identity of F (F-conditions) to which he is committed. In evaluating proposed definitions, Socrates seeks to determine their truth-value. Socrates evaluates the truth-value of a proposed definition by considering the consistency of the proposed definition with F-conditions that F he believes to (...) be true. (For instance, a proposed definition's inconsistency with one of these gives Socrates reason to believe that the definition is false.) Socrates' belief in the truth of a given F-condition to which he is committed may be based on self-evidence, its endoxic status, experience, or deduction from premises to which he is committed on the basis of any of the previous three. However, Socrates does not consider the epistemological grounds of his commitments to his F-conditions. This is part of a general avoidance of metaethical and ethical epistemological issues. Due to his avoidance of these, Socrates' pursuit of true definitions is theoretically naïve. However, Socrates recognizes a certain limitation to his manner of pursuing definitions. These results are applied to advancing the following further points. (1) Although Socrates has a distinctive manner or style of pursuing definitions, it is inappropriate to ascribe to him a method of doing so in the following sense. The concept of method implies a certain theoretical conception of procedure that Socrates lacks. Moreover, according to Socrates' own conceptual framework, only one who possessed the relevant τέχνη would have a method. (2) Furthermore, Socrates' manner of pursuing definitions is not elenctic just insofar as the word "elenchus" is interpreted to have adversarial connotations; that is inconsistent with Socrates' motives and interests. (3) Socrates' manner of pursuing definitions is consistent among the early definitional dialogues. More specifically, there is no "demise of the elenchus" in a set of transitional dialogues, as Vlastos describes it. First, Socrates' manner of pursuing definitions is not "elenctic" (in the sense described). And, second, the fact that Socrates himself proposes definitions in allegedly post-elenctic dialogues (that is, Lysis and Hippias Major) is consistent with his manner of pursuing definitions. (4) In the early definitional dialogues, Socrates does not have a theory of definition. In particular, he lacks a general theoretical ontology. Moreover, while his comments and implicit commitments entail beliefs about some conditions for a satisfactory definition (for example, that the definiens must be a uniquely identifying true verbal description), such conditions do not constitute a theory. (5) Although in other early dialogues and in other parts of the definitional dialogues Socrates may express concern over the psychological states and well-being of his interlocutors, in the process of pursuing definitions, Socrates' principal concern is the evaluation of definitions, not the psychologies or lives of his interlocutors. (6) Finally, Socrates is committed to the epistemological priority of definitional knowledge for pertinent non-definitional knowledge. This does present a methodological problem of the kind to which Geach first drew attention. Specifically, according to the manner in which Socrates pursues definitions, it is unclear how he can get from belief that p to knowledge that p. Although this problem is genuine, Socrates himself is not unaware of such limits of his approach. (shrink)

Functionalists about truth employ Ramsification to produce an implicit definition of the theoretical term _true_, but doing so requires determining that the theory introducing that term is itself true. A variety of putative dissolutions to this problem of epistemic circularity are shown to be unsatisfactory. One solution is offered on functionalists' behalf, though it has the upshot that they must tread on their anti-pluralist commitments.