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Single Idea 11023

[filed under theme 5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives ]

Full Idea

The introduction rules represent, as it were, the 'definitions' of the symbols concerned, and the elimination rules are no more, in the final analysis, than the consequences of these definitions.

Gist of Idea

The logical connectives are 'defined' by their introduction rules

Source

Gerhard Gentzen (works [1938]), quoted by Stephen Read - Thinking About Logic Ch.8

Book Ref

Read,Stephen: 'Thinking About Logic' [OUP 1995], p.229

A Reaction

If an introduction-rule (or a truth table) were taken as fixed and beyond dispute, then it would have the status of a definition, since there would be nothing else to appeal to. So is there anything else to appeal to here?

The 31 ideas with the same theme [general role and status of logical connectives]:

21586 The logical connectives are not objects, but are formal, and need a context [Russell]
23476 Logical constants seem to be entities in propositions, but are actually pure form [Russell]
23477 We use logical notions, so they must be objects - but I don't know what they really are [Russell]
21597 Logical connectives have the highest precision, yet are infected by the vagueness of true and false [Russell, by Williamson]
6563 'And' and 'not' are non-referring terms, which do not represent anything [Wittgenstein, by Fogelin]
10905 My fundamental idea is that the 'logical constants' do not represent [Wittgenstein]
11065 The inferential role of a logical constant constitutes its meaning [Gentzen, by Hanna]
11023 The logical connectives are 'defined' by their introduction rules [Gentzen]
11213 Each logical symbol has an 'introduction' rule to define it, and hence an 'elimination' rule [Gentzen]
13829 If logical truths essentially depend on logical constants, we had better define the latter [Hacking on Quine]
11021 Prior rejected accounts of logical connectives by inference pattern, with 'tonk' his absurd example [Prior,AN, by Read]
17896 We need to know the meaning of 'and', prior to its role in reasoning [Prior,AN, by Belnap]
17898 Prior's 'tonk' is inconsistent, since it allows the non-conservative inference A |- B [Belnap on Prior,AN]
13836 Maybe introducing or defining logical connectives by rules of inference leads to absurdity [Prior,AN, by Hacking]
14352 '¬', '&', and 'v' are truth functions: the truth of the compound is fixed by the truth of the components [Jackson]
13837 With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking]
13357 Truth-functors are usually held to be defined by their truth-tables [Bostock]
13825 Natural deduction introduction rules may represent 'definitions' of logical connectives [Prawitz]
11175 Logical concepts rest on certain inferences, not on facts about implications [Fine,K]
10212 Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro]
15407 Formalising arguments favours lots of connectives; proving things favours having very few [Burgess]
16974 The nature of each logical concept is given by a collection of inference rules [Correia]
14186 Logical connectives contain no information, but just record combination relations between facts [Read]
18782 The connectives are studied either through model theory or through proof theory [Mares]
15019 Define logical constants by role in proofs, or as fixed in meaning, or as topic-neutral [Sider]
4704 Wittgenstein reduced Russell's five primitive logical symbols to a mere one [O'Grady]
18489 Connectives link sentences without linking their meanings [MacBride]
18751 Natural language includes connectives like 'because' which are not truth-functional [McGee]
11210 Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt]
11212 The sense of a connective comes from primitively obvious rules of inference [Rumfitt]
18802 In specifying a logical constant, use of that constant is quite unavoidable [Rumfitt]