Possibility Frames and Forcing for Modal Logic

Abstract

This paper develops the model theory of normal modal logics based on partial "possibilities" instead of total "worlds," following Humberstone (1981) instead of Kripke (1963). Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames.

Figures (24)

• Figure 1.1: main categorical relationships.
• Figure 1.2: classes of BAOs and semantically equivalent frames---any BAO in the class before the / can be turned into a frame in the class after the / that validates the same formulas, and vice versa. A $\ast$ indicates there is also a dual equivalence between associated categories of BAOs and frames as described in the main text or references. See Thomason Thomason1975 on $\mathcal{CAV}$/Kripke, Došen Dosen1989 on $\mathcal{CA}$/neighborhood, ten Cate & Litak tenCate2007 on $\mathcal{AV}$/dicrete, and Goldblatt Goldblatt1974 on BAO/descriptive world frames. The $\dagger$ indicates that we will prove a dual equivalence involving a reflective subcategory of the category of frames (see § \ref{['DualEquiv']}).
• Figure 2.1: the up-$\boldsymbol{R}$ condition from Example \ref{['IntMod']}. Given $x'R_iy'$, we may go up in the first coordinate to any $x$ above $x'$ to obtain $xR_iy'$. A solid arrow from $s$ to $t$ indicates that $t$ is a refinement of $s$ ($t\sqsubseteq s$), while a dashed arrow indicates that $t$ is accessible from $s$ ($sR_i t$).
• Figure 2.2: the $\boldsymbol{R}$-down condition from Example \ref{['IntMod']}. Given $xR_iy$, we may go down in the second coordinate to any $y'$ below $y$ to obtain $xR_iy'$.
• Figure 2.3: the $\boldsymbol{R}$-com condition from Example \ref{['IntMod']}.

Theorems & Definitions (293)

• Definition 1.1: Modal Language
• Definition 1.2: Classical Normal Modal Logic
• Definition 1.3: Intuitionistic Modal Language
• Definition 2.1: Partial-State Frames and Models
• Remark 2.2: Flipped Notation
• Definition 2.3: Partial-State Semantics
• proof
• Example 2.6: World Frames
• Example 2.7: Intuitionistic Modal Frames
• Theorem 2.8: Bozic1984