Partial-valued logic

PART I (Partial-valued Languages): In Chapter I we consider modes of sentence composition and ask what 'truth-functionality' is functionality in two values, when there is also a third classification for meaningful sentences, An answer to this question might, we suggest, be seen as spelling out the very idea of the third classification as lack-of-a-value rather than a third value. We go on to ask how fully to exploit partial-valued semantics and motivate the need for modes of composition which themselves actually introduce non-trivial truth-value preconditions. There are two particularly interesting connectives which provide this expressive resourcefulness. The relation between partial- and total-valued languages is then considered. In Chapter II we consider subsentential modes of composition, and extensionality in general. Hence we come to see the shape of a partial-valued semantics which admits of undefinedness in a uniform way in all categories. But we restrict special attention to a simple kind of first-order language with terms and quantifiers and a discrete and determinate identity relation. There is, finally, a section devoted to definite descriptions.PART II (Logic): In Chapter III we set up laws for sentential composition, and in Chapter IV for the simple languages mentioned above. A theory is defined to be a (consequence) relation satisfying these laws, and we investigate the connection between theories and their models. Thus we are able to answer (some) questions about the logic: familiar questions and also ones peculiar to our partial-valued framework.PART III: In Chapter V we suggest that the features of our logic make it apt for deployment in a natural-language semantics which not only accommodates but actually gives a proper systematic treatment to 'presupposition'. The emphasis is on exhibiting presuppositions (as truth-value preconditions). We attempt, furthermore, to outline an account of linguistic practice which meshes with the semantic details in an illuminating way. 'Presupposing' is taken to be constitutive of assertion making, and what is presupposed constitutive of what is asserted.