Ideas of M Fitting/R Mendelsohn, by Theme

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

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4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
9738 Each line of a truth table is a model
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / a. Symbols of ML
9727 Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic
9726 We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'
9737 The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ
13136 The prefix σ names a possible world, and σ.n names a world accessible from that one
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / b. Terminology of ML
13727 A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate
9734 Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x'
9736 A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- >
9735 A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R >
9741 Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual)
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / c. Derivation rules of ML
13140 Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails]
13137 Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y
13143 Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists]
13141 Negation: if σ ¬¬X then σ X
13139 Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y
13138 Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y
13142 Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new]
9739 If a proposition is necessarily true in a world, it is true in all worlds accessible from that world
9740 If a proposition is possibly true in a world, it is true in some world accessible from that world
13145 D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X
13148 4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs]
13147 4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs]
13146 B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs]
13144 T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X
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. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / b. System K
9742 The system K has no accessibility conditions
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
13114 □P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed
9743 The system D has the 'serial' conditon imposed on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T
9744 The system T has the 'reflexive' conditon imposed on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / e. System K4
9746 The system K4 has the 'transitive' condition on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
9745 The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4
9747 The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
9748 System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
9404 Modality affects content, because P→◊P is valid, but ◊P→P isn't
4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
13111 Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P'
13112 In epistemic logic knowers are logically omniscient, so they know that they know
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
13113 F: will sometime, P: was sometime, G: will always, H: was always
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
13729 The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity
13728 The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
9725 'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
13730 The Indiscernibility of Identicals has been a big problem for modal logic
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
13725 □ must be sensitive as to whether it picks out an object by essential or by contingent properties
13731 Objects retain their possible properties across worlds, so a bundle theory of them seems best
10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
13726 Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them