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Single Idea 13840

[filed under theme 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic ]

Full Idea

First-order logic is the strongest complete compact theory with a Löwenheim-Skolem theorem.

Gist of Idea

First-order logic is the strongest complete compact theory with Löwenheim-Skolem

Source

Ian Hacking (What is Logic? [1979], §13)

Book Ref

'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.245

The 19 ideas from Ian Hacking

7447 Probability was fully explained between 1654 and 1812 [Hacking]
7448 Probability is statistical (behaviour of chance devices) or epistemological (belief based on evidence) [Hacking]
7459 Follow maths for necessary truths, and jurisprudence for contingent truths [Hacking]
7449 Epistemological probability based either on logical implications or coherent judgments [Hacking]
7450 In the medieval view, only deduction counted as true evidence [Hacking]
7451 Formerly evidence came from people; the new idea was that things provided evidence [Hacking]
7452 An experiment is a test, or an adventure, or a diagnosis, or a dissection [Hacking, by PG]
7454 Gassendi is the first great empiricist philosopher [Hacking]
13833 'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking]
13834 Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking]
13835 Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking]
13837 With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically [Hacking]
13838 A decent modern definition should always imply a semantics [Hacking]
13839 Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers [Hacking]
13843 If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking]
13840 First-order logic is the strongest complete compact theory with Löwenheim-Skolem [Hacking]
13844 A limitation of first-order logic is that it cannot handle branching quantifiers [Hacking]
13842 Second-order completeness seems to need intensional entities and possible worlds [Hacking]
13845 The various logics are abstractions made from terms like 'if...then' in English [Hacking]