Blending margins: The modal logic K has nullary unification type

0453140 - MÚ 2016 RIV GB eng J - Journal Article
Jeřábek, Emil
Blending margins: The modal logic K has nullary unification type.
Journal of Logic and Computation. Roč. 25, č. 5 (2015), s. 1231-1240. ISSN 0955-792X. E-ISSN 1465-363X
R&D Projects: GA AV ČR IAA100190902; GA MŠMT(CZ) 1M0545
Institutional support: RVO:67985840
Keywords : modal logic * unification type * rule of margins
Subject RIV: BA - General Mathematics
Impact factor: 0.585, year: 2015
http://logcom.oxfordjournals.org/content/25/5/1231

We investigate properties of the formula p->[]p in the basic modal logic K. We show that K satisfies an infinitary weaker variant of the rule of margins A->[]A / A,~A, and as a consequence, we obtain various negative results about admissibility and unification in K. We describe a complete set of unifiers (i.e., substitutions making the formula provable) of p->[]p, and use it to establish that K has the worst possible unification type: nullary. In well-behaved transitive modal logics, admissibility and unification can be analyzed in terms of projective formulas, introduced by Ghilardi; in particular, projective formulas coincide for these logics with formulas that are admissibly saturated (i.e., derive all their multiple-conclusion admissible consequences) or exact (i.e., axiomatize a theory of a substitution). In contrast, we show that in K, the formula p->[]p is admissibly saturated, but neither projective nor exact. All our results for K also apply to the basic description logic ALC.
Permanent Link: http://hdl.handle.net/11104/0254021