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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 9725 'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions
1.10 p.25 13113 F: will sometime, P: was sometime, G: will always, H: was always
1.11 p.28 13111 Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P'
1.11 p.28 13112 In epistemic logic knowers are logically omniscient, so they know that they know
1.12.2 Ex p.34 13114 □P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed
1.2 p.5 9404 Modality affects content, because P→◊P is valid, but ◊P→P isn't
1.3 p.5 9727 Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic
1.5 p.9 9726 We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'
1.5 p.9 9734 Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x'
1.6 p.12 9736 A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- >
1.6 p.12 9737 The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ
1.6 p.12 9738 Each line of a truth table is a model
1.6 p.12 9735 A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R >
1.6 p.13 9739 If a proposition is necessarily true in a world, it is true in all worlds accessible from that world
1.6 p.13 9740 If a proposition is possibly true in a world, it is true in some world accessible from that world
1.7 p.17 9741 Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual)
1.8 p.19 9742 The system K has no accessibility conditions
1.8 p.19 9743 The system D has the 'serial' conditon imposed on its accessibility relation
1.8 p.19 9744 The system T has the 'reflexive' conditon imposed on its accessibility relation
1.8 p.19 9746 The system K4 has the 'transitive' condition on its accessibility relation
1.8 p.19 9745 The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation
1.8 p.19 9747 The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation
1.8 p.19 9748 System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation
2.2 p.48 13136 The prefix σ names a possible world, and σ.n names a world accessible from that one
2.2 p.48 13140 Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails]
2.2 p.48 13137 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.49 13143 Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists]
2.2 p.49 13141 Negation: if σ ¬¬X then σ X
2.2 p.49 13139 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 13138 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 13142 Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new]
2.3 p.52 13145 D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X
2.3 p.52 13148 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 13147 4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs]
2.3 p.52 13146 B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs]
2.3 p.52 13144 T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X
2.3 p.54 13149 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 13725 □ must be sensitive as to whether it picks out an object by essential or by contingent properties
4.5 p.93 13726 Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them
4.5 p.93 13727 A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate
4.9 p.113 13728 The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability
6.3 p.136 13729 The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity
7.1 p.141 13730 The Indiscernibility of Identicals has been a big problem for modal logic
7.3 p.148 13731 Objects retain their possible properties across worlds, so a bundle theory of them seems best