Combining Philosophers
Ideas for Brad W. Hooker, Ralph Waldo Emerson and M Fitting/R Mendelsohn
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39 ideas
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
9738
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Each line of a truth table is a model [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / a. Symbols of ML
9727
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Modal logic adds □ (necessarily) and ◊ (possibly) to classical logic [Fitting/Mendelsohn]
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9726
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We let 'R' be the accessibility relation: xRy is read 'y is accessible from x' [Fitting/Mendelsohn]
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9737
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The symbol ||- is the 'forcing' relation; 'Γ ||- P' means that P is true in world Γ [Fitting/Mendelsohn]
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13136
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The prefix σ names a possible world, and σ.n names a world accessible from that one [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / b. Terminology of ML
13727
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A 'constant' domain is the same for all worlds; 'varying' domains can be entirely separate [Fitting/Mendelsohn]
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9734
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Modern modal logic introduces 'accessibility', saying xRy means 'y is accessible from x' [Fitting/Mendelsohn]
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9736
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A 'model' is a frame plus specification of propositions true at worlds, written < G,R,||- > [Fitting/Mendelsohn]
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9735
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A 'frame' is a set G of possible worlds, with an accessibility relation R, written < G,R > [Fitting/Mendelsohn]
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9741
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Accessibility relations can be 'reflexive' (self-referring), 'transitive' (carries over), or 'symmetric' (mutual) [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 2. Tools of Modal Logic / c. Derivation rules of ML
13149
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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 [Fitting/Mendelsohn]
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13141
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Negation: if σ ¬¬X then σ X [Fitting/Mendelsohn]
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13138
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Disj: a) if σ ¬(X∨Y) then σ ¬X and σ ¬Y b) if σ X∨Y then σ X or σ Y [Fitting/Mendelsohn]
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13142
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Existential: a) if σ ◊X then σ.n X b) if σ ¬□X then σ.n ¬X [n is new] [Fitting/Mendelsohn]
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13144
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T reflexive: a) if σ □X then σ X b) if σ ¬◊X then σ ¬X [Fitting/Mendelsohn]
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13145
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D serial: a) if σ □X then σ ◊X b) if σ ¬◊X then σ ¬□X [Fitting/Mendelsohn]
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13146
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B symmetric: a) if σ.n □X then σ X b) if σ.n ¬◊X then σ ¬X [n occurs] [Fitting/Mendelsohn]
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13147
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4 transitive: a) if σ □X then σ.n □X b) if σ ¬◊X then σ.n ¬◊X [n occurs] [Fitting/Mendelsohn]
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13148
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4r rev-trans: a) if σ.n □X then σ □X b) if σ.n ¬◊X then σ ¬◊X [n occurs] [Fitting/Mendelsohn]
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9740
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If a proposition is possibly true in a world, it is true in some world accessible from that world [Fitting/Mendelsohn]
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9739
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If a proposition is necessarily true in a world, it is true in all worlds accessible from that world [Fitting/Mendelsohn]
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13137
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Conj: a) if σ X∧Y then σ X and σ Y b) if σ ¬(X∧Y) then σ ¬X or σ ¬Y [Fitting/Mendelsohn]
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13140
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Bicon: a)if σ(X↔Y) then σ(X→Y) and σ(Y→X) b) [not biconditional, one or other fails] [Fitting/Mendelsohn]
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13139
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Implic: a) if σ ¬(X→Y) then σ X and σ ¬Y b) if σ X→Y then σ ¬X or σ Y [Fitting/Mendelsohn]
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13143
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Universal: a) if σ ¬◊X then σ.m ¬X b) if σ □X then σ.m X [m exists] [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / b. System K
9742
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The system K has no accessibility conditions [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / c. System D
13114
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□P → P is not valid in D (Deontic Logic), since an obligatory action may be not performed [Fitting/Mendelsohn]
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9743
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The system D has the 'serial' conditon imposed on its accessibility relation [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T
9744
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The system T has the 'reflexive' conditon imposed on its accessibility relation [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / e. System K4
9746
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The system K4 has the 'transitive' condition on its accessibility relation [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / f. System B
9745
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The system B has the 'reflexive' and 'symmetric' conditions on its accessibility relation [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / g. System S4
9747
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The system S4 has the 'reflexive' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
9748
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System S5 has the 'reflexive', 'symmetric' and 'transitive' conditions on its accessibility relation [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
9404
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Modality affects content, because P→◊P is valid, but ◊P→P isn't [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 5. Epistemic Logic
13112
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In epistemic logic knowers are logically omniscient, so they know that they know [Fitting/Mendelsohn]
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13111
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Read epistemic box as 'a knows/believes P' and diamond as 'for all a knows/believes, P' [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 6. Temporal Logic
13113
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F: will sometime, P: was sometime, G: will always, H: was always [Fitting/Mendelsohn]
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4. Formal Logic / D. Modal Logic ML / 7. Barcan Formula
13728
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The Barcan says nothing comes into existence; the Converse says nothing ceases; the pair imply stability [Fitting/Mendelsohn]
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13729
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The Barcan corresponds to anti-monotonicity, and the Converse to monotonicity [Fitting/Mendelsohn]
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