Two Values, Three Values, Many Values, No Values
Charles G. Morgan · Studies in fuzziness and soft computing · 2003
Classical formal semantics is based on bivalence and truth-functionality. Historically, problems with these two notions have motivated the move from two values to three values and from three values to many values. We reflect on ordinary reasoning and systems of rational belief to motivate numerically based probabilistic semantics, which abandons truth-functionality. But numerically based probability theory is overly specific compared to real belief systems. So we develop a formal semantics based on comparative probability structures. Thus we develop a formal semantics which is not truth-functional and which does not use any values at all. The semantics is shown to be universal for any extension, with or without quantifiers, of classical sentence logic. We briefly discuss some areas for additional research.