Editing Temporal logic (section)

=== A minimal axiomatic logic ===

Burgess outlines a logic that makes no assumptions on the relation <, but allows for meaningful deductions, based on the following axiom schema:<ref>{{Cite book|title=Philosophical logic|last=Burgess|first=John P.|publisher=Princeton University Press|year=2009|isbn=9781400830497|location=Princeton, New Jersey|page=21|oclc=777375659|author-link=John P. Burgess}}</ref>
# {{var|A}} where {{var|A}} is a [[Tautology (logic)|tautology]] of [[first-order logic]]
# G({{var|A}}→{{var|B}})→(G{{var|A}}→G{{var|B}})
# H({{var|A}}→{{var|B}})→(H{{var|A}}→H{{var|B}})
# {{var|A}}→GP{{var|A}}
# {{var|A}}→HF{{var|A}}
with the following rules of deduction:
# given {{var|A}}→{{var|B}} and {{var|A}}, deduce {{var|B}} ([[modus ponens]])
# given ''a tautology'' {{var|A}}, infer G{{var|A}}
# given ''a tautology'' {{var|A}}, infer H{{var|A}}

One can derive the following rules:
# '''Becker's rule''': given {{var|A}}→{{var|B}}, deduce T{{var|A}}→T{{var|B}} where T is a '''tense''', any sequence made of G, H, F, and P.
# '''Mirroring''': given a theorem {{var|A}}, deduce its '''mirror statement''' {{var|A}}<sup>§</sup>, which is obtained by replacing G by H (and so F by P) and vice versa.
# '''Duality''': given a theorem {{var|A}}, deduce its '''dual statement''' {{var|A}}*, which is obtained by interchanging ∧ with ∨, G with F, and H with P.