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Single Idea 13680
[filed under theme 5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
]
Full Idea
There is a 'primitivist' account, according to which logical consequence is a primitive notion.
Gist of Idea
Maybe logical consequence is a primitive notion
Source
Theodore Sider (Logic for Philosophy [2010], 1.5)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.9
A Reaction
While sympathetic to substitutional views (Idea 13674), the suggestion here pushes me towards thinking that truth must be at the root of it. The trouble, though, is that a falsehood can be a good logical consequence of other falsehoods.
The
45 ideas
from 'Logic for Philosophy'
13678
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The most popular account of logical consequence is the semantic or model-theoretic one
[Sider]
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13679
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Maybe logical consequence is more a matter of provability than of truth-preservation
[Sider]
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13680
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Maybe logical consequence is a primitive notion
[Sider]
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13682
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Maybe logical consequence is impossibility of the premises being true and the consequent false
[Sider]
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13683
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A relation is a feature of multiple objects taken together
[Sider]
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13724
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In model theory, first define truth, then validity as truth in all models, and consequence as truth-preservation
[Sider]
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13684
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The semantical notion of a logical truth is validity, being true in all interpretations
[Sider]
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13685
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Natural deduction helpfully allows reasoning with assumptions
[Sider]
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13686
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We can build proofs just from conclusions, rather than from plain formulae
[Sider]
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13687
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No assumptions in axiomatic proofs, so no conditional proof or reductio
[Sider]
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13688
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Good axioms should be indisputable logical truths
[Sider]
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13689
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'Theorems' are formulas provable from no premises at all
[Sider]
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13691
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Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'
[Sider]
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13690
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Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
[Sider]
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13693
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A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification
[Sider]
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13692
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A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation
[Sider]
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13694
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We can 'sharpen' vague terms, and then define truth as true-on-all-sharpenings
[Sider]
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13695
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Supervaluational logic is classical, except when it adds the 'Definitely' operator
[Sider]
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13696
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When a variable is 'free' of the quantifier, the result seems incapable of truth or falsity
[Sider]
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13697
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Valuations in PC assign truth values to formulas relative to variable assignments
[Sider]
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13698
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In a complete logic you can avoid axiomatic proofs, by using models to show consequences
[Sider]
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13699
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Compactness surprisingly says that no contradictions can emerge when the set goes infinite
[Sider]
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13700
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A 'total' function must always produce an output for a given domain
[Sider]
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13702
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The identity of indiscernibles is necessarily true, if being a member of some set counts as a property
[Sider]
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13701
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A single second-order sentence validates all of arithmetic - but this can't be proved axiomatically
[Sider]
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13703
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λ can treat 'is cold and hungry' as a single predicate
[Sider]
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13704
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It is hard to say which are the logical truths in modal logic, especially for iterated modal operators
[Sider]
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13705
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Truth tables assume truth functionality, and are just pictures of truth functions
[Sider]
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13706
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Intuitively, deontic accessibility seems not to be reflexive, but to be serial
[Sider]
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13707
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Maybe metaphysical accessibility is intransitive, if a world in which I am a frog is impossible
[Sider]
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13708
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S5 is the strongest system, since it has the most valid formulas, because it is easy to be S5-valid
[Sider]
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13709
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Logical truths must be necessary if anything is
[Sider]
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13710
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In D we add that 'what is necessary is possible'; then tautologies are possible, and contradictions not necessary
[Sider]
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13711
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System B introduces iterated modalities
[Sider]
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13712
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Epistemic accessibility is reflexive, and allows positive and negative introspection (KK and K¬K)
[Sider]
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13714
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We can treat modal worlds as different times
[Sider]
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13715
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You can employ intuitionist logic without intuitionism about mathematics
[Sider]
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13716
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'If B hadn't shot L someone else would have' if false; 'If B didn't shoot L, someone else did' is true
[Sider]
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13717
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Transworld identity is not a problem in de dicto sentences, which needn't identify an individual
[Sider]
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13719
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Barcan Formula problem: there might have been a ghost, despite nothing existing which could be a ghost
[Sider]
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13720
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Converse Barcan Formula: □∀αφ→∀α□φ
[Sider]
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13718
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The Barcan Formula ∀x□Fx→□∀xFx may be a defect in modal logic
[Sider]
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13721
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'Strong' necessity in all possible worlds; 'weak' necessity in the worlds where the relevant objects exist
[Sider]
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13723
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System B is needed to prove the Barcan Formula
[Sider]
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13722
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A 'theorem' is an axiom, or the last line of a legitimate proof
[Sider]
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