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Single Idea 10766
[filed under theme 5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
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Full Idea
One can distinguish at least two quite different senses of logic: as an instrument of demonstration, and perhaps as an instrument for the characterization of structures.
Gist of Idea
Logic is either for demonstration, or for characterizing structures
Source
Leslie H. Tharp (Which Logic is the Right Logic? [1975], §2)
Book Ref
'Philosophy of Logic: an anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.37
A Reaction
This is trying to capture the proof-theory and semantic aspects, but merely 'characterizing' something sounds like a rather feeble aspiration for the semantic side of things. Isn't it to do with truth, rather than just rule-following?
Related Idea
Idea 15407
Formalising arguments favours lots of connectives; proving things favours having very few [Burgess]
The
16 ideas
from 'Which Logic is the Right Logic?'
10762
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In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and'
[Tharp]
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10763
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Completeness and compactness together give axiomatizability
[Tharp]
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10770
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If completeness fails there is no algorithm to list the valid formulas
[Tharp]
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10771
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Compactness is important for major theories which have infinitely many axioms
[Tharp]
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10772
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Compactness blocks infinite expansion, and admits non-standard models
[Tharp]
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10764
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A complete logic has an effective enumeration of the valid formulas
[Tharp]
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10768
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Effective enumeration might be proved but not specified, so it won't guarantee knowledge
[Tharp]
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10765
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Soundness would seem to be an essential requirement of a proof procedure
[Tharp]
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10773
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The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals)
[Tharp]
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10766
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Logic is either for demonstration, or for characterizing structures
[Tharp]
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10767
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Elementary logic is complete, but cannot capture mathematics
[Tharp]
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10769
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Second-order logic isn't provable, but will express set-theory and classic problems
[Tharp]
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10775
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The axiom of choice now seems acceptable and obvious (if it is meaningful)
[Tharp]
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10774
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There are at least five unorthodox quantifiers that could be used
[Tharp]
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10776
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The main quantifiers extend 'and' and 'or' to infinite domains
[Tharp]
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10777
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Skolem mistakenly inferred that Cantor's conceptions were illusory
[Tharp]
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