Page 300 - Encyclopedia of Philosophy of Language

Language and Logic
functional expressions. That is, it aims to be a notation for expressing precisely complex concepts as con- structed from basic elements. Moreover, these analy- ses use the very same formal resources as the ones used in proof itself—resources such as quantification, implication and negation. Frege's formal notation thereby achieves a unification of two domains which traditional logics always treated separately, the construction of concepts and the derivation of propositions.
The Begriffsschrift contains the first full exposition of propositional calculus, presented in an axiomatic and fully formal way. But few among Frege's con- temporaries were able to appreciate the importance of his contribution, not least because of his choice of a cumbersome and daunting notation system. (The notation relied heavily on spatial arrangements and did not resemble the linear notations used in math- ematics and, for that matter, natural languages.) This notation did not survive Frege, and the current logical symbolism is largely derived from Peano's Formulaire Mathematique, which was taken up by Russell's Prin- cipia Mathematica.
7.2 Formal Notation and Natural Language
In a series of papers many of which remained unpub- lished in his lifetime, Frege used the resources of the conceptual notation to explore the logical structures of language and judgments that can be made within it. He considered his ideography as related to ordinary language in the way that a microscope is related to the naked eye: it enables one to perceive distinctions which remain blurred or even totally blacked out in the ordinary language. For example, in ordinary lan- guage the word 'is' expresses the following possible relations:
(a) the relationship between an object and the con- cept under which it can be subsumed. That is, 'Julius is a man' has the logical form: Fa.
(b) The relationship of subordination between concepts, as in 'man is a rational animal,' whose logical form contains the complex predi- cate: Fx^Gx.
(c) The relation between a first order and a second order concept, named 'inherence,' as in 'there is a mortal human' which is written as: (3.x) (Fx & GJC). Here the quantifier serves as a second order concept linking the two first order concepts.
(d) The relationship of identity between two objects, as in 'Hesperus is Phosphorus,' whose logical form is expressed by the mathematical symbol for equality, used here as an identity between two proper names: a = b.
Failure to distinguish such differences in the logical structures of sentences containing 'is' not infrequently led to the drawing of false inferences. Famous proofs of the existence of God, based upon a confusion of
(c) with (a) above, treated existence as a first order property of an individual. Not only did the analysis 'subject-copula-predicate'ofthestandard Aristotelian logic encourage, in Frege's view, this type of con- fusion; it was also responsible for a failure to reveal the ways in which sentences having a different subject and a different predicate may nevertheless have the same meaning, as in 'Titus killed Caius' and 'Caius was killed by Titus.'
In a paper published in 1892, 'On Sense and Refer- ence,' Frege examined what is required of the relation of identity. Contrary to what he himself had claimed in his 1879 Begriffsschrift, he revised his account of identity, and no longer thought it could be treated simply as a relationship between signs. In doing this, he realized that the very conception of a sign had to be revised. The new view was that every sign expresses both a sense and a reference. In the case of proper names, the sense of the name-sign is a way in which the reference is given, and the reference is the object named. However, in the case of predicates, the predi- cate sign refers to a concept, and the extension of the predicate (the items in the world to which it applies) plays a different role. Using a famous example in the arena of proper names, this distinction allowed Frege to claim that sentences such as (5 and 6):
Hesperus is Hesperus (5)
Hesperus is Phosphorus (6)
are identical in terms of reference though they differ in 'cognitive value,' that is, in the way that they enter into our understanding of beliefs and inferential pro- cesses. Thus, though the ancient astronomers would have assented to (5) on straightforward logical grounds, they would have dissented from (6) because they believed that the evening star (Hesperus) was a different heavenly body from the morning star (Phos- phorus). However, as it happens both 'stars' are in fact the planet Venus, so Frege's distinction between sense and reference allowed him to treat both (5) and (6) as trivially claiming the self-identity of Venus, whilst at the same time maintaining that (5) and (6) could be understood differently by the ancient astron- omers. This was because it was sense that mattered at the level of understanding, and the two proper names differ in sense, though not in reference. Frege extended this sense-reference distinction to what could be called propositions or sentences asserted. He claimed that the sense of a sentence was the thought it expressed, and its reference was either the value True or False.
Aside from providing a detailed and more con- vincing analysis of natural language structures than was possible in the traditional logic, Frege also made suggestive remarks about what might be called the 'nonstructural' aspects of ordinary language. In his view, a logical analysis of natural language should carefully distinguish the two levels or functions of an
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