Page 290 - Encyclopedia of Philosophy of Language

Language and Logic
term 'deictic-nomologicaF for this theory of natural- kind words, to call attention to the context- dependence of their introduction, together with the lawful (nomological) necessity of identity statements like 'water is H2O.'
See also: Necessity; Possible Worlds. Bibliography
Brennan A 1988 Conditions of Identity. Clarendon Press, Oxford
Kripke S 1980 Naming and Necessity, rev. edn. Basil Blackwell, Oxford
Morris T V 1984 Understanding Identity Statements. Aber- deen University Press, Aberdeen
Putnam H 1973 Meaning and reference. Journal of Phil- osophy 70: 699-711
Quine W V O 1960 Word and Object. MIT Press, Cambridge, MA
Russell B 1905 On denoting. Mind 14: 479-93
Salmon N 1982 Reference and Essence. Basil Blackwell.
Oxford
Wiggins D 1967 Identity and Spatio-Temporal Continuity.
Basil Blackwell,Oxford
Wiggins D 1980 Sameness and Substance. Basil Blackwell,
Oxford
Intuitionism is a school in the philosophy of math- ematics founded by the Dutch mathematician L. E. J. Brouwer. The name derives from Brouwer's agree- ment with Kant that arithmetic deals with mental constructions derived from a priori intuitions con- cerning the structure of temporal succession. Brouwer's constructivism led him to reject large parts of standard mathematics as illegitimate. In particular, the intuitionists reject the idea of completed infinite sequences or totalities. Since arithmetic deals with finite mental constructions, infinity must always be conceived of as potential. An infinite sequence is to be interpreted as a rule for indefinitely extending finite initial segments without ever reaching a completed infinite totality. This has radical consequences which become obvious in analysis, many of whose standard results are rejected by the intuitionist as false of the continuum, correctly conceived; hence some intuition- ist theorems contradict (at least at face value) those of classical mathematics. These radical mathematical ideas led Brouwer to argue against standard classical logic, claiming that certain of its principles, whilevalid for finite surveyable domains, were not legitimately extendible to the infinite. Brouwer's impact on con- temporary philosophy of language has perhaps arisen mainly from the formalization of his logical ideas by Kolmogorov and, more influentially, Heyting, though many who espouse intuitionism in logic do so from a perspective far removed from the Kantianism which motivated Brouwer.
1. Intuitionist Logic: Bivalent* and Excluded Middk
Whereas the father of modern logic, Gottlob Frege, had sought to find a foundation for mathematics in logic, Brouwer's approach to logic is almost the
reverse. For him, one first establishes the correct prin- ciples of reasoning in mathematics and then distills logic out of reflection on those. Hence Brouwer was led to reject the law of excluded middle: the law that for any proposition p, either p holds or p does not
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hold. If one ascends to a metalinguistic formulation 4
of the excluded middle principle and equates s is false' with 's is not true' it becomes the principle of bivalence: every sentence is either true or false.
Now one can reject the principle of bivalence and still retain classical logic and all its theorems. One might, for instance, propose intermediate grades of truth in order to cope with such phenomena as vague- ness; there are ways of doing this (though not perhaps the most plausible), which retain classical logic. But the intuitionistic rejection of classical logic is moti- vated by entirely different considerations and holds even for propositions which are taken to have per- fectly definite meanings, such as the propositions of mathematics.
In particular, intuitionists agreewithclassical math- ematicians that there is no third possibility between truth and falsity (tertium non datur). Although they deny that every proposition is either true or false, it does not follow from this, in intuitionistic logic, that there is some proposition which is neither. In fact, the double negation of each instance of the principle of excluded middle, namely, n~l(p v ~ip) (here '-]' is the negation symbol, 'v' the disjunction symbol) is a theorem of intuitionist logic. For the intuitionist, as for the classicist, to reject (p v ~ip) is absurd, equi- valent to affirming the existence of true contradictions. But whilst accepting, for any p, that n~l(p v ~ip) holds, intuitionsts do not thereby infer that (p v -jp), for they reject the principle of double negation elim- ination, from ~|~IP conclude p.