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
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
The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ
The prefix σ names a possible world, and σ.n names a world accessible from that one
Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic
We let 'R' be the accessibility relation: xRy is read 'y is accessible from x'
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / b. Terminology of ML
Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x'
A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R >
A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- >
Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual)
A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate
4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / c. Derivation rules of ML
If a proposition is possibly true in a world, it is true in some world accessible from that world
Conj: a) if σ X∧Y then σ X and σ Y b) if σ (X∧Y) then σ X or σ Y
Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails]
Existential: a) if σ ◊X then σ.n X b) if σ □X then σ.n X [n is new]
Negation: if σ X then σ X
Implic: a) if σ (X→Y) then σ X and σ Y b) if σ X→Y then σ X or σ Y
Disj: a) if σ (X∨Y) then σ X and σ Y b) if σ X∨Y then σ X or σ Y
B symmetric: a) if σ.n □X then σ X b) if σ.n ◊X then σ X [n occurs]
4 transitive: a) if σ □X then σ.n □X b) if σ ◊X then σ.n ◊X [n occurs]
4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ◊X then σ ◊X [n occurs]
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
T reflexive: a) if σ □X then σ X b) if σ ◊X then σ X
Universal: a) if σ ◊X then σ.m X b) if σ □X then σ.m X [m exists]
D serial: a) if σ □X then σ ◊X b) if σ ◊X then σ □X
If a proposition is necessarily true in a world, it is true in all worlds accessible from that world
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / b. System K
The system K has no accessibility conditions
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
The system D has the 'serial' conditon imposed on its accessibility relation
□P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T
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
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
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
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
System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation
4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
Modality affects content, because P→◊P is valid, but ◊P→P isn't
4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
In epistemic logic knowers are logically omniscient, so they know that they know
Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P'
4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
F: will sometime, P: was sometime, G: will always, H: was always
4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability
The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity
5. Theory of Logic / F. Referring in Logic / 3. Property (λ-) Abstraction
'Predicate abstraction' abstracts predicates from formulae, giving scope for constants and functions
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
The Indiscernibility of Identicals has been a big problem for modal logic
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
□ must be sensitive as to whether it picks out an object by essential or by contingent properties
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
Counterpart relations are neither symmetric nor transitive, so there is no logic of equality for them