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Single Idea 13698

[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness ]

Full Idea

You can establish facts of the form Γ|-φ while avoiding the agonies of axiomatic proofs by reasoning directly about models to conclusions about semantic consequence, and then citing completeness.

Gist of Idea

In a complete logic you can avoid axiomatic proofs, by using models to show consequences

Source

Theodore Sider (Logic for Philosophy [2010], 4.5)

Book Ref

Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.105

A Reaction

You cite completeness by saying that anything which you have shown to be a semantic consequence must therefore be provable (in some way).

The 45 ideas from 'Logic for Philosophy'

13678 The most popular account of logical consequence is the semantic or model-theoretic one [Sider]
13679 Maybe logical consequence is more a matter of provability than of truth-preservation [Sider]
13682 Maybe logical consequence is impossibility of the premises being true and the consequent false [Sider]
13680 Maybe logical consequence is a primitive notion [Sider]
13683 A relation is a feature of multiple objects taken together [Sider]
13724 In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation [Sider]
13684 The semantical notion of a logical truth is validity, being true in all interpretations [Sider]
13685 Natural deduction helpfully allows reasoning with assumptions [Sider]
13686 We can build proofs just from conclusions, rather than from plain formulae [Sider]
13688 Good axioms should be indisputable logical truths [Sider]
13687 No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider]
13689 'Theorems' are formulas provable from no premises at all [Sider]
13690 Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms [Sider]
13691 Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step' [Sider]
13692 A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation [Sider]
13695 Supervaluational logic is classical, except when it adds the 'Definitely' operator [Sider]
13693 A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification [Sider]
13694 We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings [Sider]
13697 Valuations in PC assign truth values to formulas relative to variable assignments [Sider]
13696 When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity [Sider]
13698 In a complete logic you can avoid axiomatic proofs, by using models to show consequences [Sider]
13699 Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
13700 A 'total' function must always produce an output for a given domain [Sider]
13701 A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically [Sider]
13702 The identity of indiscernibles is necessarily true, if being a member of some set counts as a property [Sider]
13703 λ can treat 'is cold and hungry' as a single predicate [Sider]
13704 It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider]
13705 Truth tables assume truth functionality, and are just pictures of truth functions [Sider]
13706 Intuitively, deontic accessibility seems not to be reflexive, but to be serial [Sider]
13707 Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible [Sider]
13708 S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid [Sider]
13709 Logical truths must be necessary if anything is [Sider]
13710 In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary [Sider]
13711 System B introduces iterated modalities [Sider]
13712 Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K) [Sider]
13714 We can treat modal worlds as different times [Sider]
13715 You can employ intuitionist logic without intuitionism about mathematics [Sider]
13716 'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true [Sider]
13717 Transworld identity is not a problem in de dicto sentences, which needn't identify an individual [Sider]
13719 Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost [Sider]
13720 Converse Barcan Formula: □∀αφ→∀α□φ [Sider]
13718 The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic [Sider]
13721 'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist [Sider]
13723 System B is needed to prove the Barcan Formula [Sider]
13722 A 'theorem' is an axiom, or the last line of a legitimate proof [Sider]