Modal logics for incidence geometries (Q2785842)

Let \(S=(P,L,\text{in})\) be an incidence plane by the well-known axioms: \(P,L\neq\emptyset\); \(\text{in}\subseteq P\times L\); \(P\cap L=\emptyset\); two points are together incident with one and only one line; each line contains at least two different points; each point belongs at least to two different lines. Obviously, projective planes and affine planes are specifically incidence planes.NEWLINENEWLINENEWLINEThe notion incidence plane as a two-sorted relational structure with an inter-sort binary relation is not suitable as a semantical basis of some modal logic. Therefore the authors introduce the ``incidence frame'' over \(S\) as the relational structure \(W(S)= (W,\equiv_1,\equiv_2)\) by the following conditions: \(W= \{(X,y): X\in P_S,\;y\in L_S,\;X\;\text{in}_S\;y\}\); \((X_1,y_1)\equiv_1(X_2,y_2)\) iff \(X_1=X_2\); \((X_1,y_1)\equiv_2(X_2,y_2)\) iff \(y_1=y_2\). On the other hand an abstract definition for the incidence frame \(\underline W=(W,\equiv_1,\equiv_2)\) is given by axioms; also the incidence plane \(S(\underline W)\) over \(\underline W\) is introduced.NEWLINENEWLINENEWLINEThe authors show the following representation theorem for incidence frames in incidence planes: Let \(\underline W\) be an incidence frame, \(S(\underline W)\) be the incidence plane over \(\underline W\) and \(W'=W(S(\underline W))\) be the incidence frame over \(S(\underline W)\). Then \(\underline W\) is isomorphic with \(W'\). In an analogous way the representation theorem for incidence planes in incidence frames is valid. The categories \(\Sigma_i\) of incidence planes and \(\Phi_i\) of incidence frames are equivalent.NEWLINENEWLINENEWLINENow the authors introduce a modal logic for incidence geometry (MIG) with standard Kripke semantics in the class of incidence frames (by language, semantics, axiomatics). The completeness theorem for MIG is proved. For any formula \(A\) of MIG the following two conditions are equivalent: (i) \(A\) is a theorem of MIG, (ii) \(A\) is true in all incidence frames. In particular, the completeness theorems for the modal logic of projective geometry and for the modal logic of affine geometry, respectively, are valid.NEWLINENEWLINENEWLINEThis paper gives a sufficient exposition of the modal logic of incidence geometry.