Page 281 - Encyclopedia of Philosophy of Language

2.6 Relevance Logic (also called 'Relevant Logic')
According to the classical conception an argument is valid just in case it is impossible for its premises to be true and its conclusion false; relevance logicians (Parry, Anderson, Belnap, Routley, etc.) hold that this is too weak, that it is also required that the premises be relevant to the conclusion. There are rival systems of relevance logic, the best known being Anderson and Belnap's system, R, the system of 'relevant impli- cation,' and E, the combination of R with the modal system S4 to represent entailment. Relevance logics extend but also restrict classical logic; in virtue of their stronger conception of validity Anderson and Belnap disallow the classical principle of inference, modus ponens ('MPP'; from 'if A then B' and 'A' to infer 'B') for material implication.
2.7 ParaconsistentLogic
As a limit case, the classical conception admits as valid arguments the conclusions of which are necessarily true, and arguments the premises of which are necess- arily false. In classical logic, therefore, anythingwhat- ever follows from a contradiction. In R and some other relevance logics, however, the principle ex con- tradictione quodlibet no longer holds. A system which can tolerate inconsistency without trivialization is a 'paraconsistent logic.' Paraconsistent systems,antici- pated by V asiliev and by Jaskowski, have been developed both within (e.g., by Routley) and without (e.g., by da Costa) the concern for relevance as a necessary condition for validity.
'Dialethic' logics are paraconsistent in a stronger sense; they not only deny that everything whatever follows from a contradiction, but also allow that con- tradictions are sometimes true. It is argued by Priest, for example, that dialethism is the appropriate response to the semantic paradoxes: the liar sentence is both true and false.
2.8 FreeLogic
'Free logic,' so-called because its motivation is the idea that logic should be free of ontological commitments, usually deviates from classical logic only at the level of predicate calculus. Classical predicate calculus is not valid for the empty domain, since the theorem '(3x) (Fx v -Fx)'—'there is something which either is or is not F'—implies that something exists; nor, in virtue of the rule of existential generalization (from 'Fa' to infer '(3x) Fx') is it valid for non-referring terms. The first system of predicate logic valid in the empty domain was developed by Jaskowski; the first system also to restrict the rule of existential gen- eralization (to: from 'Fa, a exists' to infer '(3x) Fx') by Leonard. V an Fraassen's 'presuppositional languages' are intended to supply a prepositional basis for a free predicate calculus which is as it were quasi-classical; the 'supervaluational' semantics allow
truth-value gaps, but, since they assign 'true' to any compound sentence to which all classical valuations would assign 'true,' exactly the classical theorems and inferences are sustained.
2.9 Meinongian Logic
Free logics restrict classical predicate calculus in the interests of ontological neutrality; a development of the 1980s is 'Meinongian logic,' which extends classi- cal predicate calculus in the interests of ontological tolerance. The idea is that (as in the 'theory of Objects' proposed by Alexius Meinong) the domain is to include not only actual, existent objects, but also non- existent and even impossible objects (the golden mountain, the round square). Meinongianism may (Routley) but need not (Parsons) be combined with paraconsistency.
3. Challenges to Classical Logic from Considerations of Language
Deviant logics have been developed sometimes out of purely formal interest; sometimes in anticipation of practical application in, especially, computing; some- times prompted by considerations from metaphysics, philosophy of science or philosophy of mathematics; sometimes by considerations internal to the phil- osophy of logic; but frequently classical logic has been thought to stand in need of revision in order to accom- modate such specifically linguistic phenomena as vagueness, reference failure, or semantic paradox.
3.1 Regimentalism versus Naturalism inLogic
Classical logic has the elegance of simplicity: a sim- plicity achieved in part by a high degree of abstraction and schematization relative to the complexities of natural languages. Regimentalists, as one might call them, such as Frege, Tarski, or Quine, hold that, from the point of view of the rigorous representation of what is valid in the way of argument, natural lan- guages are not only unnecessarily complex but also in various ways defective, prone to ambiguity, vague- ness, etc. Formal languages should be austere and simple, and must avoid such defects. Regimentalists tend to be hostile to proposals to revise logic so as to accommodate it better to natural language, preferring, when they cannot rule the unwelcome aspects of natu- ral language outside the scope of logic, to impose a sometimes Procrustean regimentation to bring them within the classical apparatus. By contrast, Natu- ralists, as it seems appropriate to call the other party, think that formal logic should aspire to mirror the subtleties of natural languages, and tend to be more sympathetic to such proposals.
3.2 Vagueness
Frege and Russell deplored the vagueness of natural languages. When Peirce protested that '[l]ogicians
Deviant Logics
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