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Single Idea 13720
[filed under theme 4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
]
Full Idea
The Converse Barcan Formula reads □∀αφ→∀α□φ (or an equivalent using ◊).
Gist of Idea
Converse Barcan Formula: □∀αφ→∀α□φ
Source
Theodore Sider (Logic for Philosophy [2010], 9.5.2)
Book Ref
Sider,Theodore: 'Logic for Philosophy' [OUP 2010], p.240
A Reaction
I would read that as 'if all the αs happen to be φ, then αs have to be φ'. Put like that, I would have thought that it was obviously false. Sider points out that some new object could turn up which isn't φ.
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]
|
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]
|
13679
|
Maybe logical consequence is more a matter of provability than of truth-preservation
[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]
|
13691
|
Induction has a 'base case', then an 'inductive hypothesis', and then the 'inductive step'
[Sider]
|
13690
|
Proof by induction 'on the length of the formula' deconstructs a formula into its accepted atoms
[Sider]
|
13692
|
A 'precisification' of a trivalent interpretation reduces it to a bivalent interpretation
[Sider]
|
13693
|
A 'supervaluation' assigns further Ts and Fs, if they have been assigned in every precisification
[Sider]
|
13695
|
Supervaluational logic is classical, except when it adds the 'Definitely' operator
[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
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λ can treat 'is cold and hungry' as a single predicate
[Sider]
|
13705
|
Truth tables assume truth functionality, and are just pictures of truth functions
[Sider]
|
13704
|
It is hard to say which are the logical truths in modal logic, especially for iterated modal operators
[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]
|