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Compiled by Anthony Harrison-Barbet
FREGE
LOGICISM
Born in Wismar, where his father was Headmaster of a girls' school, Gottlob Frege was educated at the Universities of Jena and Göttingen, studying mathematics, natural sciences, and philosophy. After gaining his doctorate at Göttingen in 1873 he returned to Jena as a Privatdozent in mathematics and was appointed 'ausserordentlicher' professor in 1879 and 'full' professor in 1896. He was largely neglected by the academic world (apart from Russell and Wittgenstein on both of whom he was a major influence); and the significance of his pioneering work was not fully appreciated until the nineteen fifties.
PHILOSOPHY OF MATHEMATICS
[1] What are the foundations of mathematics? What are numbers? These are the questions that initially interested Frege. [See The Foundations of Arithmetic.] He rejected 'psychologistic' and empiricist accounts of numbers as abstractions from our perceptions of groups of objects, and the so-called 'formalist' and 'conventionalist' views that they are just arbitrary signs or marks on a page, perhaps governed by rules of use, as in a game. And he disagreed with the claim that arithmetical truths are synthetic a pnon judgments (although he considered that geometrical truths were) [a]. Each of the above three theories fails in one of these respects. Arithmetic, Frege argued, must be in some sense objective and certain, and must of course be applicable to the world. This requires that the signs of arithmetic must have some 'reference' — just as we cannot talk about the application of, say, the pieces in a game of chess without having already assigned some 'representative' function to them and the moves that may be made with them. His solution was to regard numbers as applicable to concepts, that is, "objects of reason", which are subject to a criterion of identity [b]. This can be understood by means of an example. The statement 'Jupiter's moons are four' should be interpreted as 'The number of Jupiter's moons is four'; and in this sentence 'the number of Jupiter's moons' and 'four' refer to identical objects. How then is number defined? Suppose we have two concepts A and B, and that the objects covered by one concept correspond one-to-one to the things covered by the other. The number (as object) belonging to A is then 'equinumerous', that is, the same as the number belonging to B. And 'having the same number' is to say that the two concepts A and B have the same extension, that is, covers the same class of objects. The number 0, Frege adds, is the number which belongs to the concept 'not identical with itself'. In this way he was led to the view that the definitions and laws of arithmetic could be derived solely from the laws of logic. The truths of arithmetic are thus 'analytic' in so far as anayticity is defined by Frege as truths of logic, or as truths which can be reduced to such truths through the use of definitions in logical terms. Synthetic truths, by contrast, are not truths of logic [c].
PHILOSOPHY OF LOGIC AND LANGUAGE
[2] Frege was critical of what he saw as the limitations of traditional formal logic in that (1) not all judgements or propositions are of the subject-predicate form, and (2) that the syllogism is not a universal pattern of logical inference. His development of formal logic showed further that the grammar of ordinary language (on which traditional Aristotelian logic was based) is seriously misleading [a]. In his earliest work ['Concept Script'] he had set out a comprehensive system of formal logic as a formalized language of pure thought, which would exhibit what was essential to and underlying our discourse ('natural language') [b]. Frege is thus implicitly giving primacy to 'philosophy of language' as the means by which thought might be analysed. He is not concerned with philosophy as involving just ad hoc or piecemeal clarification and elimination of errors. His approach can be seen from his analysis of predication, proper names, and meaning in later articles.
(1) ['Function and Concept'] Consider a mathematical function, say ( )2+ 2( ). This, he says, is 'unsaturated' — it cannot stand alone; it needs an 'argument' x to make an algebraic sentence and complete the sense: x2 + 2x. A predicative expression is like this. Thus ' — is round' has to be completed by a proper name (the argument), for example, 'the Earth'. The proper name relates to a concept. Names cannot be used as predicates. As for the meaning of words, proper names (which for Frege include both ordinary names and definite descriptions) 'mean' in so far as they refer to objects: but predicate expressions have meaning by virtue of the role they play in the sentence [c].
(2) ['On Sense and Reference'] The form 'x is y' obscures some important distinctions. If I say, for example, 'Venus is the morning star', the 'is' here is neither the 'is' of predication ('Venus is bright') nor the 'is' of existence (as in 'Venus exists') — Frege shows clearly that 'existence' is not a predicate — but is the 'is' of identity, although the statement is contingent and known a posteriori [d]. I am not attributing a quality to Venus but am asserting that 'Venus' (an ordinary name) and 'morning star' (a definite description) have the same object. Now consider 'The morning star is the evening star' (or 'Phosphorus is Hesperus'). Again I am asserting an identity (though this was not known to ancient astronomers). Yet 'morning star' and 'evening star' surely have a different sense. There is a problem here given the view he held in earlier work [see Concept Script], namely, that identity is a relation between the signs or names themselves, whereas we generally think of identity statements as saying something about the world. To deal with this he therefore now distinguished between sense (Sinn) and reference (Bedeutung) ['On Sense and Reference']. The two names, though differing in sense, have the same reference (or denotative meaning); they pick out the same object. They are said to be 'extensionally equivalent'. And sense 'determines' reference. As for the sense of a name, Frege considers this to be given by a 'definite description' associated with that name and known by the user [e]. To meet the objection that different people might apply different — and subjective — descriptions to a name, he said that there are some descriptions which are 'public' and grounded in our language [f]. [See also 'The Thought'.] What of sentences as a whole? They too, Frege says, must have both a sense and a reference. Suppose we say 'The morning star is a body illuminated by the sun' and 'The evening star is a body illuminated by the sun'. These two sentences clearly have the same reference, but they express a different 'idea' or 'thought'. ('It is raining' and the German sentence 'Es regnet': are different sentences but for Frege have the same thought.) It is the Thought which constitutes the sense of the whole sentence. (And Frege considers the judgement as a functional unity, and not just a linking of logically prior and separable terms.) What then of the reference? A reference is required if the sentence as a whole is to be considered as being true or false. And Frege argues that the reference of a sentence is a truth-value — the True or the False. This thus belongs to the content or object of the sentence or proposition not to any mental act of judging [g]. The sense, or thought of the sentence can then be further identified with the conditions which make it true. To understand a sentence, to know what it means, is therefore to know what its truth-conditions are [h]. The thought (Gedanke), which can also be understood as 'proposition', is supposed by Frege to occupy a realm of Sense or Meaning (together with numbers and classes). Meaning is thus analysable in 'internalist' or 'mentalist' terms. This realm is real but not in the same way that the physical realm of objects or the mental realm of subjective ideas and images are [see 'The Thought]. (To the extent that propositions are expressed by sentences he also seems to consider them as composite names) [i]. In the case of sentences containing names or definite descriptions which do not refer to any entity the sentence as a whole is said to have no truth-value. For a name or definite description to be taken as having a reference, a denotatum must be presupposed as existing [j]. Thus, 'The present King of France' cannot be said to be either true or false unless the existence now of a King of France is presupposed. We can say it is meaningful in so far as when we communicate we assert it to be true. While assertion seems to be the main function of utterances, Frege recognised ['Concept Script'] that sentences can also be used to formulate definitions, ask questions, give commands, or tell stories. Sentences thus variously used are then said to have a different force. He also noted a third qualification — the 'colouring' (Förbung) of a sentence [k]. This refers to that part of a sentence's meaning which is not relevant to the determination of its truth-value. [See 'On Sense and Reference'.]
CRITICAL SUMMARY
The impact of Frege on twentieth century philosophy was as great — if less immediate — as Descartes' philosophy was on the seventeenth. Having rejected the psychologism of Mill and (probably) early Husserl he sought to ground mathematics in logic (albeit unsuccessfully as it later turned out). But more importantly he revolutionized modern logic, offering a new account of predication and quantification. He has also been a major influence on the philosophy of language. Central in his writings are his distinction between sense and reference and his treatment of problems arising out of identity and predication. These and other issues have, however, engendered a great deal of controversial discussion in recent years. Do we need sense as well as reference? Is meaning to be determined by truth conditions? Does this commit us to some form of realism? Some philosophers have been critical of Frege's account of predication. And some have tended to dismiss his emphasis on logical structures supposedly underlying informal language and have argued in favour of the adequacy of the latter, linking it with assertion (as use of sentences) rather than on Frege's key concept of truth. These matters continue to be much debated.
READING
Frege: Begriffschrift (1879) (Concept Script or Notation) (trans. as Conceptual Notation by T. W. Bynum); Die Grundlagen der Arithmetik (1884) (The Foundations of Arithmetic, trans. J. L. Austin); Funktion und Begriff (1891) ('Function and Concept'); 'Über Sinn und Bedeutung' (1892) ('On Sense and Reference'); 'Über Begriff und Gegenstand' (1892) ('On Concept and Object') — these and other articles are in P. T. Geach and M. Black (eds) Philosophical Writings of Gottlob Frege or B. McGuinness (ed.), Collected papers on Mathematics, Logic and Philosophy; Der Gedanke: eine logische Untersuchungen (1918) ('The Thought: a Logical Investigation' — in P. F. Strawson (ed.), Philosophical Logic). A useful collection is M. Beaney (ed.), The Frege Reader.
Studies:
Introductory
G. Currie, Frege: An Introduction to His Philosophy.
Advanced
M. Dummett, Frege: Philosophy of Language.
Collections of essays
E. D. Klemke (ed.), Essays on Frege (1968)
T. Ricketts, Cambridge Companion to Frege, Cambridge: Cambridge University Press.
C. Wright (ed.), Frege: Tradition and Influence.
CONNECTIONS
[1a; also 1c] Rejection of psychologism, empiricism, formalism, conventionalism; geometry but not arithmetic synthetic a priori Kant→
Mill→
→Husserl→
[1d]
[1b]
[1a d f]
[1a]
[1a]
[1a]
[1a f]
[2f]
[2b]
[1b] Arithmetic objective and certain, applies to world; numbers apply to concepts (objects of reason), subject to criterion of identity Kant→
Mill→
[1d]
[1b]
[1d]
[1b]
[1c] Arithmetic claimed to be derivable from logic and is thus analytic; analyticity in terms of logical truth Kant→
Mill→
[1a 1b]
[1f]
[1b]
[1a]
[2b]
[1b f]
[2a] Limitations of traditional logic: rejection of primacy of subject-predicate form, not all inference syllogistic; traditional logic grounded in ordinary language which can mislead Mill→
[1b]
[1c e]
[1b 2a]
[1d]
[3a]
[2b] Logic ('philosophy of language'), formalized language to exhibit thought underlying discourse ('natural' language) [1e]
[1b 3b]
[1d]
[3a]
[1c]
[1i]
[1g]
[1a]
[2c] Proper names relate to concepts and 'mean' by reference to objects; meaning of predicate expressions through function Mill→
[1b]
[1b]
[1c]
[1a c]
[3c]
[2b]
[1c]
[2d] Ambiguity of 'is' — existence (not a predicate), predication, identity (contingent) Kant→
[CSa]
[1b]
[5d]
[1g]
[1e]
[1e]
[2e] Sense and reference: names may differ in sense but have same reference (extensional equivalence); sense determines reference; proper names have sense in terms of definite descriptions Mill→
[1b]
[1d]
[1c]
[1c]
[3c]
[1a e]
[1b]
[1d]
[1b]
[1d 2c]
[1a]
[2f] Private and public (guided) descriptions [2c]
[1d]
[1d]
[1b]
[2g] Sentences as whole have sense (= the Thought) and reference (= truth-value) (judgements/ propositions as functional unity not 'prior' linking terms); truth-values belong to 'object' of sentence not to any mental act of judging Mill→
[2a]
[1c]
[1c 2b]
[1b]
[1d]
[1c]
[1b]
[1c]
[2h] Sense to be determined by truth-conditions [1i]
[1c]
[1a c]
[1f]
[1a]
[2i] Three realms — physical, mental, and meaning (Sinn); ('internalist' analysis); propositions (thoughts) belong to the last; propositions qua sentences are also names [3a e 7f]
[1i]
[1a c 2a]
[1a]
[1c]
[1a]
[2j] Denotatum's existence presupposed if sentence to have truth-value [1c]
[1c]
[2k] Sentences have many functions not just assertion, therefore different 'force'; notion of 'colouring [3a]
[2b]
[1e]
[1c]
[1a]