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 | 13112 | In epistemic logic knowers are logically omniscient, so they know that they know |
1.11 | p.28 | 13111 | Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P' |
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 | 9735 | A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > |
1.6 | p.12 | 9736 | A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > |
1.6 | p.12 | 9738 | Each line of a truth table is a model |
1.6 | p.12 | 9737 | The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ |
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 | 9748 | System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions 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 | 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 |
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.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 | 13136 | The prefix σ names a possible world, and σ.n names a world accessible from that one |
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 | 13142 | Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new] |
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 |
2.3 | 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 | 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 | 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 | 13145 | D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X |
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 |