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Single Idea 7788
[filed under theme 4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
]
Full Idea
The four important logical equivalences in modal logic (the Modal Negation equivalences) are: ¬◊p↔□¬p, ◊¬p↔¬□p, □p↔¬◊¬p, and ◊p↔¬□¬p.
Gist of Idea
Modal logic has four basic modal negation equivalences
Source
Rod Girle (Modal Logics and Philosophy [2000], 1.2)
Book Ref
Girle,Rod: 'Modal Logics and Philosophy' [Acumen 2000], p.3
A Reaction
[Possibly is written as a diamond, necessarily a square] These are parallel to a set of equivalences between quantifiers in predicate logic. They are called the four 'modal negation (MN) equivalences'.
The
15 ideas
from 'Modal Logics and Philosophy'
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7786
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Propositional logic handles negation, disjunction, conjunction; predicate logic adds quantifiers, predicates, relations
[Girle]
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7787
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Possible worlds logics use true-in-a-world rather than true
[Girle]
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7788
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Modal logic has four basic modal negation equivalences
[Girle]
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7789
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Necessary implication is called 'strict implication'; if successful, it is called 'entailment'
[Girle]
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7790
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If an argument is invalid, a truth tree will indicate a counter-example
[Girle]
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7792
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A world has 'access' to a world it generates, which is important in possible worlds semantics
[Girle]
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7793
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◊p → □◊p is the hallmark of S5
[Girle]
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7795
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S5 has just six modalities, and all strings can be reduced to those
[Girle]
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7794
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There are seven modalities in S4, each with its negation
[Girle]
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7796
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Modal logics were studied in terms of axioms, but now possible worlds semantics is added
[Girle]
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7798
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There are three axiom schemas for propositional logic
[Girle]
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7799
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Proposition logic has definitions for its three operators: or, and, and identical
[Girle]
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7797
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Axiom systems of logic contain axioms, inference rules, and definitions of proof and theorems
[Girle]
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7800
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Analytic truths are divided into logically and conceptually necessary
[Girle]
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7801
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Possibilities can be logical, theoretical, physical, economic or human
[Girle]
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