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Ideas of M Fitting/R Mendelsohn, by Text
[American, fl. 1998, Two logicians working in New York.]
1998

FirstOrder 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

9737

The symbol  is the 'forcing' relation; 'Γ  P' means that P is true in world Γ

1.6

p.12

9736

A 'model' is a frame plus specification of propositions true at worlds, written < G,R, >

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

9738

Each line of a truth table is a model

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' (selfreferring), 'transitive' (carries over), or 'symmetric' (mutual)

1.8

p.19

9743

The system D has the 'serial' conditon imposed on its accessibility relation

1.8

p.19

9742

The system K has no accessibility conditions

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

9744

The system T has the 'reflexive' conditon imposed 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

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

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.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

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

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.52

13148

4r revtrans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs]

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 antimonotonicity, 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
