8 - Further topics

Summary

In our final chapter we briefly discuss two topics we feel deserve to be mentioned in spite of space constraints that prevent us from giving them fuller treatment. In the first half of the chapter, we address the debate concerning the possibility of giving a compositional semantics for IF logic. In the second half, we investigate the effect of introducing imperfect information to modal logic.

Compositionality

The original game-theoretic semantics for IF logic assigned meanings only to IF sentences [30]. Thus IF logic was immune to a common complaint lodged against Tarski's semantics for first-order logic, namely that truth is defined in terms of satisfaction, rather than truth alone. However, it also meant that one was not able to analyze IF sentences by looking at the meanings of their subformulas. Furthermore, Hintikka famously claimed that there could be no compositional semantics for IF logic:

… there is no realistic hope of formulating compositional truth-conditions for [IF sentences], even though I have not given a strict impossibility proof to that effect.

[28, pp. 110ff]

Hintikka's assertion inspired Hodges to develop his trump semantics, which gives meanings to all IF formulas [32, 33]. In Chapter 4, we defined two other semantics for IF formulas: game-theoretic semantics and Skolem semantics. In order to emphasize the similarities between IF logic and first-order logic, we introduced both semantics in terms of single assignments.