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Ideas of M Fitting/R Mendelsohn, by Text

[American, fl. 1998, Two logicians working in New York.]

1998 First-Order Modal Logic
Pref p.-1 'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions
1.10 p.25 F: will sometime, P: was sometime, G: will always, H: was always
1.11 p.28 In epistemic logic knowers are logically omniscient, so they know that they know
1.11 p.28 Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P'
1.12.2 Ex p.34 □P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed
1.2 p.5 Modality affects content, because P→◊P is valid, but ◊P→P isn't
1.3 p.5 Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic
1.5 p.9 We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'
1.5 p.9 Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x'
1.6 p.12 The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ
1.6 p.12 Each line of a truth table is a model
1.6 p.12 A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R >
1.6 p.12 A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- >
1.6 p.13 If a proposition is possibly true in a world, it is true in some world accessible from that world
1.6 p.13 If a proposition is necessarily true in a world, it is true in all worlds accessible from that world
1.7 p.17 Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual)
1.8 p.19 The system T has the 'reflexive' conditon imposed on its accessibility relation
1.8 p.19 The system K has no accessibility conditions
1.8 p.19 The system K4 has the 'transitive' condition on its accessibility relation
1.8 p.19 The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation
1.8 p.19 The system D has the 'serial' conditon imposed on its accessibility relation
1.8 p.19 The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation
1.8 p.19 System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation
2.2 p.48 The prefix σ names a possible world, and σ.n names a world accessible from that one
2.2 p.48 Conj: a) if σ X∧Y then σ X and σ Y b) if σ (X∧Y) then σ X or σ Y
2.2 p.48 Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails]
2.2 p.49 Existential: a) if σ ◊X then σ.n X b) if σ □X then σ.n X [n is new]
2.2 p.49 Negation: if σ X then σ X
2.2 p.49 Disj: a) if σ (X∨Y) then σ X and σ Y b) if σ X∨Y then σ X or σ Y
2.2 p.49 Implic: a) if σ (X→Y) then σ X and σ Y b) if σ X→Y then σ X or σ Y
2.2 p.49 Universal: a) if σ ◊X then σ.m X b) if σ □X then σ.m X [m exists]
2.3 p.52 T reflexive: a) if σ □X then σ X b) if σ ◊X then σ X
2.3 p.52 4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ◊X then σ ◊X [n occurs]
2.3 p.52 4 transitive: a) if σ □X then σ.n □X b) if σ ◊X then σ.n ◊X [n occurs]
2.3 p.52 D serial: a) if σ □X then σ ◊X b) if σ ◊X then σ □X
2.3 p.52 B symmetric: a) if σ.n □X then σ X b) if σ.n ◊X then σ X [n occurs]
2.3 p.54 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
4.3 p.87 □ must be sensitive as to whether it picks out an object by essential or by contingent properties
4.5 p.93 Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them
4.5 p.93 A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate
4.9 p.113 The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability
6.3 p.136 The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity
7.1 p.141 The Indiscernibility of Identicals has been a big problem for modal logic
7.3 p.148 Objects retain their possible properties across worlds, so a bundle theory of them seems best