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Single Idea 13730
[filed under theme 9. Objects / F. Identity among Objects / 7. Indiscernible Objects
]
Full Idea
Equality has caused much grief for modal logic. Many of the problems, which have struck at the heart of the coherence of modal logic, stem from the apparent violations of the Indiscernibility of Identicals.
Gist of Idea
The Indiscernibility of Identicals has been a big problem for modal logic
Source
M Fitting/R Mendelsohn (First-Order Modal Logic [1998], 7.1)
Book Ref
Fitting,M/Mendelsohn,R: 'First-Order Modal Logic' [Synthese 1998], p.141
A Reaction
Thus when I say 'I might have been three inches taller', presumably I am referring to someone who is 'identical' to me, but who lacks one of my properties. A simple solution is to say that the person is 'essentially' identical.
Related Idea
Idea 13726
Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them [Fitting/Mendelsohn]
The
44 ideas
from 'First-Order Modal Logic'
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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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13112
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In epistemic logic knowers are logically omniscient, so they know that they know
[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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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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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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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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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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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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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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13141
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Negation: if σ ¬¬X then σ X
[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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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