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The traditional semantics for First Order Logic (sometimes called Tarskian semantics) is based on the notion of interpretations of constants. Herbrand semantics is an alternative semantics based directly on truth assignments for ground sentences rather than interpretations of constants. Herbrand semantics is simpler and more intuitive than Tarskian semantics; and, consequently, it is easier to teach and learn. Moreover, it is more expressive. For example, while it is not possible to finitely axiomatize integer arithmetic with Tarskian semantics, this can be done easily with Herbrand Semantics. The downside is a loss of some common logical properties, such as compactness and completeness. However, there is no loss of inferential power. Anything that can be proved according to Tarskian semantics can also be proved according to Herbrand semantics. In this presentation, we define Herbrand semantics; we look at the implications for research on logic and rules systems and automated reasoning; and and we assess the potential for popularizing logic.
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