normal modal logic (original) (raw)

The study of modal logic is based on the concept of a logic, which is a set Λ of wff’s satisfying the following:

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• •

The last condition means: if A and A→B are in Λ, so is B in Λ.

Normal modal logics are the most widely studied modal logics. The smallest normal modal logic is called K. Other normal modal logics are built from K by attaching wff’s as axiom schemas. Below is a list of schemas used to form some of the most common normal modal logics:

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  4: □⁢A→□⁢□⁢A
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  5: ◇⁢A→□⁢◇⁢A
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  D: □⁢A→◇⁢A
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  T: □⁢A→A
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  B: A→□⁢◇⁢A
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  C: □⁢(A∧□⁢B)→□⁢(A∧B)
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  M: □⁢(A∧B)→□⁢A∧□⁢B
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  G: ◇⁢□⁢A→□⁢◇⁢A
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  L: □(A∧□A→B)∨□(B∧□B→A)
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  W: □(□A→A)→□A

For example, the normal modal logic D is the smallest normal modal logic containing D as its axiom schema.

Notation. The smallest normal modal logic containing schemas Σ1,…,Σn is typically denoted

K𝚺𝟏⁢⋯⁢𝚺𝐧.

It is easy to see that K𝚺𝟏⁢⋯⁢𝚺𝐧 can be built from the “bottom up”: call a finite sequence of wff’s a deduction if each wff is either a tautology, an instance of Σi for some i, or as a result of an application of modus ponens or necessitation on earlier wff’s in the sequence. A wff is deducible from if it is the last member of some deduction. Let Λk be the set of all wff’s deducible from deductions of lengths at most k. Then

K𝚺𝟏⁢⋯⁢𝚺𝐧=⋃i=1∞Λi

Below are some of the most common normal modal logics: