Page 317 - Encyclopedia of Philosophy of Language

the notational ideas which are now universally used to articulate the logical form of sentences and the logical structure of reasoning. The essentials of this notation are variables, predicates, and quantifiers. Variables, like x, y, z, function in logical notation very much as variables function in mathematical expressions, i.e., they mark a 'gap' in an expression which can be filled by a name or which can be quan- tified over to yield a true or false sentence. 'Predicate letters,' like F, G, and H, stand for something different from ordinary grammatical predicates and something more like mathematical functions. For example, in All crocodiles are amphibious creatures the grammatical subject is 'all crocodiles' and the grammatical predi- cate is 'are amphibious creatures.' In logic, 'all' is a 'quantifier,' a word of quantity, and the logical predi- cates are ' . . . is a crocodile' and ' . . . is an amphibious creature.' Logical predicates are commonly thought of as 'what is left' when names are removed from simple sentences; for example, John is a crocodile, John is scalier than Mary, and John is between Mary and Peter yield the logical predicates 'x is a crocodile,' 'x is scalier than y,' and 'x is between y and z,' written Fx, Gxy, and Hxyz respectively, where the variables mark gaps as explained above. The quantifiers are the words 'all' and 'some' (and their synonyms),and are commonly written Vxand Ex respectively. To return to the simple example, All crocodiles are amphibious is construed as saying, 'For all x, if x is a crocodile then x is an amphibious creature' which is written in logical notation as Vx(Fx^Gx).
Frege's system provided a considerably more powerful and flexible instrument for representing pat- terns of reasoning involving quantifiers than did Ari- stotle's syllogistic, and the Fregean tradition, with important contributions from Bertrand Russell and others, dominated thinking about reasoning until the
1970s. It produced many remarkable results, especially about mathematical reasoning: some of the most notable of these were Church's theorem that elementary predicate logic is undecidable, Tarski's theorem that 'truth' cannot be defined within elemen- tary arithmetic, and G6del's theorems that one cannot prove the consistency of elementary number theory without assuming it and that one cannot completely axiomatize elementary number theory. This remark- able tradition is the theoretical basis of the whole of the modern computing and information technology industry and much research is continuing along these lines.
3. Recent Developments
However, since the 1970s there has been an upsurge of interest in the study of reasoning which derives from several quite different perspectives, including those of informal logic, argumentation theory, and cognitive psychology.
3.1 Informal Logic
The informal logic tradition has emerged mainly in North America, among logicians and philosophers who used to teach modern formal logic partly in the hope of improving students' reasoning skills. Partly because this hope was not realized, and partly because of the difficulty of applying modern logic to much 'real reasoning'—reasoning of the kind people actually use in order to try to convince others—modern informal logic focuses on the study of such real reasoning. It pays particular attention to the language and structure of reasoning, and to fallacies. Though there were earlier works in this tradition, the publication of Michael Scriven's book, Reasoning (1976), is widely regarded as the moment when the subject came of age. Good examples of works in this tradition are Govier (1985); Johnson and Blair (1977); and Free- man (1988) For a scholarly account of the theoretical problems in this tradition, and some possible solutions see Freeman (1991).
3.2 Argumentation
The 'argumentation' tradition has arisen mainly in Holland, and derives its inspiration particularly from the speech-act theory of J. L. Austin as developed by J. R. Searle. However, it also owes a great deal to modern logic, to the theory of dialectic, and to Perel- man's work on rhetoric. Argumentation theory also focuses on 'real argumentation'; it describes its pro- gram as belonging to 'normative pragmatics,' and it assesses argumentation by reconstructing it in terms of an ideal dialectical model (for this approach to the study of argument see especially Eemeren, et al. 1984; 1987).
3.3 Cognitive Psychology
In recent years cognitive psychologists have given increasing attention to the study of reasoning. Inter- esting developments here, with far-reaching impli- cations for the whole field, are particularly associated with the work of Philip Johnson-Laird, especially with his contention that people reason, not by any kind of reference to 'logical rules' (as Piaget and many others have thought), but by means of 'mental models' (for a good exposition of these ideas see Johnson-Laird 1983, 1991).
4. Summary
In summary, the study of reasoning began with the ancient Greeks, remained in the Aristotelian tradition for nearly two thousand years, then took a math- ematical turn, and has in the latter part of the twen- tieth century returned to a broad approach to the subject, and is focusing, in the 1990s, on real reason- ing—reasoning of the kind people actually use with a
Reasoning
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