Concepts, Sense, and Ontology
What Happened to the Sense of a Concept-Word?
Carlo Penco
Abstract
In this paper I shall outline a short history of the ideas concerning sense and reference of a concept-word from Frege to model theoretic semantics. I claim that, contrary to what is normally supposed, a procedural view of sense may be compatible with model theoretic semantics, especially in dealing with problems at the boundary between semantics and pragmatics. A first paragraph on the paradox of the concept horse will clarify the attitude concerning the history of ideas that I assume in this paper. In the second paragraph I will discuss some misunderstandings in the shift from the sense/reference distinction in Frege to the intension/extension distinction in model theoretic semantics. In the third I will show how a particular interpretation of the Fregean sense of a concept word (and of cognitive sense in general) may be of interest for model theoretic semantics.
Introduction
Discussion on concepts both in philosophy and psychology have produced so many new ideas on the topic, that it becomes difficult to make any comparison between contemporary debates and the Fregean worries. After recent criticism of concepts as natural kinds (Frixione 2007, Macherie 2009) cognitive scientists, philosophers and psychologists are proposing new ways of treating different aspects of cognition in humans and other animals; are concepts developed from a prelinguistic ability to classify? How do they develop in children? If we do not define concepts as natural kinds, shall we define them as functional kinds? shall we define them epistemically, semantically o by their origin? (see for instance Sainsbury-Tye 2011). Although some Fregean âproblemsâ are still confronted, the contemporary debate on concepts seems to go far away from the original terminology used by Frege, that attracts mainly exegetic confrontation (we have excellent examples in KĂŒnne 2010 and Textor 2011). A possibility to find new suggestions in Fregeâs analysis of concepts may take two trends: on the one hand we may work on how his complex distinction of âlevelsâ of concepts present psychologists and computer scientists with new problems (cf. Brandom 2009); on the other hand we may work on the history of ideas1 and look inside the development of semantics after Frege, trying to reconstruct some of Fregeâs ideas in a new setting. I will follow the second trend, pointing out a blind spot in contemporary semantics, due to a failure to engage with the Fregeâs conception of the sense of a predicateâor in his terminology, a âconcept-wordâ (Begriffswort).2
In this paper I will try to show the compatibility of a procedural interpretation of the Fregean sense of a predicate with contemporary model theoretic semantics. I donât claim that Frege cannot suggest alternative perspectives in semantics and theories of meaning; however, as Eva Picardi (2005, 35) remarks, it is difficult to accept that radically different interpretations of Fregeâsuch as representationalist vs. inferentialist theoriesââdid equal justice to Fregeâs central concernsâ. Picardi 2005 has shown some difficulties of strong inferentialism to keep some basic Fregean desiderata; on the other hand most people agree that model theoretic semantics, although it has been developed on the track of Frege through Carnap, apparently abandoned some Fregean requirements on cognitive aspects. Nevertheless I think that some of Fregeâs most debated views on concepts are either preserved in new settings, like lambda calculus, or could be developed inside model theoretical semantics. I will then present (1) an assessment of one of the most famous problem concerning the Fregean theory of concepts as exemplifying a way to see its compatibility with develoments of logics after him; (2) a short historical presentation of the evolution of semantics after the Fregean distinctions of sense and reference for predicates in front of the âanomalyâ of the original Fregean tripartite classification; (3) a use of the Fregean requirement on the sense of predicates that impinges upon the problem of the boundary between semantics and pragmatics.
1 Frege on Concepts as âObjects of a Special Kindâ
Fregeâs original theory of concept is grounded on his analogy between concepts and functions: âwhat is called in logic a concept is connected with what we call a function ⊠a concept is a function whose value is always a truth valueâ (FC 15) Presented in this way the theory is certainly original with respect to the past; historically, it is a generalization of the idea of function. Stripped of its prose it can be considered the origin of the âclassicalâ view, where connectives can be considered as functions from truth values to truth values and predicates as functions from individuals to truth values: Px represents a function that has the value true when completed with a singular term referring to an object falling under the concept P, or belonging to the class denoted by P.
A great deal of the philosophical discussion on Fregeâs theory of concept has been devoted to his theory of the non-definability of (the notion of ) a concept. Frege gives a semantic definition of objects and concepts as what is referred to, respectively, by singular terms (proper names) and predicates (concept words). Predicates or concept words are for Frege unsaturated expressions, i.e. patterns given by a sentence fragment that needs to be completed by a singular term, as with â⊠is a horseâ.3 However, in natural language, we are almost compelled to refer to concepts using the definite article: âthe concept horseâ How can we make the connection between the expressions â⊠is a horseâ and âthe concept horseâ? How can we say that the concept horse is a concept? Our grammar suggests that an expression composed with the definite article âtheâ (a definite description) is a singular term, whose reference is an object and not a concept; therefore we should paradoxically assert âthe concept horse is not a conceptâ.4 This has been called âthe paradoxâ of the concept horse. Frege (1892b: 201) concludes that concepts are âobjects of a special kindâ, and asks the reader to accept this incongruence of natural language. Coming back on the issue years later, Frege (1906: 210) insists that grammar may mislead us, given that using a definite description to refer to concepts is âa mistake language forces upon usâ. However informal elucidations should be enough to clarify the intention of the writer in order to understand the sharp distinction between concepts and objects (functions and arguments) on which the construction of his formal system is grounded5.
Frege required âa pinch of saltâ of us in order to understand the difference between objects and concepts, remarking that not everything in a formal system can be explained, and that the elucidations of the signs preceding the presentation of the formal system are informal introductions, that cannot be expressed in terms of the formal system. Among many discussions (starting with Dummett 1973 until Davidson 20056) we find two extreme positions: on the one hand Crispin Wright claims that the paradox is not solvable unless we reject the application of the notion of sense and reference to predicates; on the other hand New-Wittgensteinians claim that Fregean elucidations are plain and ârobustâ nonsense. Both criticisms seem overstated.
On the one hand Wright 1998 claims that Fregeâs use of a singular term to refer to concepts clashes with his requirement for which two expressions with the same reference should be inter-substitutable in all extensional sentences salva veritate, and in all sentences salva congruitate (reference principle); in fact singular terms (âthe concept horseâ) and concept words (â⊠is a horseâ) have different grammatical roles and cannot substituted salva congruitate.7 Therefore, in the end, Frege was mistaken: singular terms refer, but predicates donât8. Wright criticizes Dummettâs attempt to solve the âparadoxâ finding a way to express the second order expression âconcept horseâ, but the discussion may probably be stopped before the beginning. One problem with Wrightâs interpretation is that he wonders âhow exactly Frege is to communicate his semantic proposals about predicatesâ; he asks for a âdecent semantic theoryâ (Wright 1998, §III) while Frege explicitly considers his elucidations something where exactness cannot be attained, becauseâused to introducing his formal systemâthey are not part of it. Instead of conceding Frege to give an informal introduction to the basic concepts of his semantics, Wright looks for a formal analysis, and comes to the conclusion that Fregeâs basic mistake is the application of the sense/reference distinction to predicates (concept words). Wright requires a strict formalism exactly where Frege was supposing that no formal definition is required: we cannot give definitions for primitive elements of the system. (E.g. Frege 1906: 301; 1924: 290). Wright is correct in sayingâafter Fregeâthat singular terms and predicates behave differently, and we may refer to predicates indirectly, by giving their extension. In fact we may use extensions (classes) as the semantic value of predicates (as contemporary semantics does); but this does not abolish the possibility of speaking of concepts.9 We touch here a point in whichâas Textor (2011, 253) remarksââreference as what we want to speak about and reference as semantic role come apartâ. Speaking of the reference of a predicate is not only defining a semantic value in a formal system, but alsoâbasicallyâa reminder for the distinction between a function and its extension, distinction on which Frege was insisting in all his remarks on the idea of function. We might be content to claim that, in our informal elucidations, we need to refer to entities that are not objects, but concepts.
On the other hand, since the connection between Fregeâs âelucidationsâ and Wittgensteinâs remarks on the unsayable discussed by Geach 1976 and later by Diamodn 1988, many authors, mainly âNew Wittgensteiniansâ, began to theorize the âineffabilityâ or ânonsenseâ of philosophical elucidations (the elucidations of Tractatus itself, or the elucidations in the introduction to Fregeâs Begriffsschrift). Certainly Frege was well aware that the basic concepts of the theory are not part of it and called the words âconceptâ and âfunctionâ with the term âpseudo-predicatesâ, and used to speak of ânonsenseâ (Unsinn) about attempts to define primitive elements of his system until his latest writings10. Wittgenstein in the Tractatus called âobjectâ and âfunctionâ âformal conceptsââi.e. not genuine, empirical conceptsâthat âshowâ their function in the use of the formalism. Anticipating Quineâs motto, Wittgenstein used to say that the correct use of the word âobjectâ is expressed in the formalism by a variable.11 However, although both Frege and Wittgenstein used the term ânonsenseâ (âUnsinnâ), it is plain that Frege used it in special cases, where the grammar of language clashes with theoretical intuitions as in the case of âthe concept horseâ. Instead of accepting the attempt to recognize the limitations of the grammar of our natural language to express some basic ideas of the formal system, the New Wittgensteinans consider that what elucidations attempt to say always issues in plain nonsense.
Fregeâs aim (followed to the extreme in Wittgensteinâs Tractatus) was more modest, and asked for informal agreement on basic concepts of his formal theory: âsince definitions are not possible for primitive elements, something else must enter in. I call it elucidation. It is this, therefore, that serves the purpose of mutual understanding among investigators, as well as of the communication of the science to others.â12 I am not alone in thinking that the so called âparadoxâ of the concept horse is not really a paradox13, but what the âsecondâ Wittgenstein would have called a âmisunderstandingâ due to the grammar of our natural language. Even in speaking of ânonsenseâ we should need a pinch of salt.
In what follows I suggest an attitude where some basic Fregean ideas can be considered not only as such, in contrast with logical systems developed after him, but also for their value to illuminate and being illuminated by more recent developments.
A first example is what happened of the Fregean suggestion that concepts are âobjects of a special kindâ (he could have said âentitiesâ). The suggestion has been developed by Alonzo Church with the lambda notation, where we may refer to concepts by an expression with bound variables which is formally analogous to the iota operator for definite descriptions (that Frege introduces in Grundgesetze § 11). In fact, facing the problem of the paradox of âthe concept horseâ, somebody might attempt to use a second order description operator such as: iF: (x) (F(x) iff Horse (x)), that is âthe F such that for all x, x is an F iff x is an horseâ. But Frege introduced the description operator for singular terms would have not accepted it for predicates that need to be represented as insaturated expressions (see also Dummett 1973: 244). Church breaks this prohibition and invents a new kind of operator, with the expression âlx. horse (x)â as a way to expressing the concept horse. Contrary to Fregeâs requirement, we have here an expression that is not literally âunsaturatedâ, that is with a gap. Is Churchâs solution radically different from Fregeâs? Certainly it is, from the point of view of strict literal interpretation, but, nevertheless other aspects of Fregeâs main tenets seem to be represented, especially in lambda abstraction and lambda application, including the sharp differenc...