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Single Idea 9738
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
]
Full Idea
Each line of a truth table is, in effect, a model.
Gist of Idea
Each line of a truth table is a model
Source
M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 1.6)
Book Ref
Fitting,M/Mendelsohn,R: 'First-Order Modal Logic' [Synthese 1998], p.12
A Reaction
I find this comment illuminating. It is being connected with the more complex models of modal logic. Each line of a truth table is a picture of how the world might be.
The
44 ideas
from M Fitting/R Mendelsohn
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9725
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'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions
[Fitting/Mendelsohn]
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13113
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F: will sometime, P: was sometime, G: will always, H: was always
[Fitting/Mendelsohn]
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13111
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Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P'
[Fitting/Mendelsohn]
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13112
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In epistemic logic knowers are logically omniscient, so they know that they know
[Fitting/Mendelsohn]
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13114
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□P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed
[Fitting/Mendelsohn]
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9404
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Modality affects content, because P→◊P is valid, but ◊P→P isn't
[Fitting/Mendelsohn]
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9727
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Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic
[Fitting/Mendelsohn]
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9726
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We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'
[Fitting/Mendelsohn]
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9734
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Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x'
[Fitting/Mendelsohn]
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9736
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A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- >
[Fitting/Mendelsohn]
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9740
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If a proposition is possibly true in a world, it is true in some world accessible from that world
[Fitting/Mendelsohn]
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9739
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If a proposition is necessarily true in a world, it is true in all worlds accessible from that world
[Fitting/Mendelsohn]
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9735
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A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R >
[Fitting/Mendelsohn]
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9737
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The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ
[Fitting/Mendelsohn]
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9738
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Each line of a truth table is a model
[Fitting/Mendelsohn]
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9741
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Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual)
[Fitting/Mendelsohn]
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9744
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The system T has the 'reflexive' conditon imposed on its accessibility relation
[Fitting/Mendelsohn]
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9746
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The system K4 has the 'transitive' condition on its accessibility relation
[Fitting/Mendelsohn]
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9745
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The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation
[Fitting/Mendelsohn]
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9747
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The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation
[Fitting/Mendelsohn]
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9748
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System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation
[Fitting/Mendelsohn]
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9742
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The system K has no accessibility conditions
[Fitting/Mendelsohn]
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9743
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The system D has the 'serial' conditon imposed on its accessibility relation
[Fitting/Mendelsohn]
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13136
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The prefix σ names a possible world, and σ.n names a world accessible from that one
[Fitting/Mendelsohn]
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13139
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Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y
[Fitting/Mendelsohn]
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13137
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Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y
[Fitting/Mendelsohn]
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13140
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Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails]
[Fitting/Mendelsohn]
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13138
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Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y
[Fitting/Mendelsohn]
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13143
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Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists]
[Fitting/Mendelsohn]
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13142
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Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new]
[Fitting/Mendelsohn]
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13141
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Negation: if σ ¬¬X then σ X
[Fitting/Mendelsohn]
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13144
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T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X
[Fitting/Mendelsohn]
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13148
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4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs]
[Fitting/Mendelsohn]
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13147
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4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs]
[Fitting/Mendelsohn]
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13146
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B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs]
[Fitting/Mendelsohn]
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13145
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D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X
[Fitting/Mendelsohn]
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13149
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S5: a) if n ◊X then kX b) if n ¬□X then k ¬X c) if n □X then k X d) if n ¬◊X then k ¬X
[Fitting/Mendelsohn]
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13725
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□ must be sensitive as to whether it picks out an object by essential or by contingent properties
[Fitting/Mendelsohn]
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13726
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Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them
[Fitting/Mendelsohn]
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13727
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A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate
[Fitting/Mendelsohn]
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13728
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The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability
[Fitting/Mendelsohn]
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13729
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The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity
[Fitting/Mendelsohn]
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13730
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The Indiscernibility of Identicals has been a big problem for modal logic
[Fitting/Mendelsohn]
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13731
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Objects retain their possible properties across worlds, so a bundle theory of them seems best
[Fitting/Mendelsohn]
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