I am learning a course from Stanford University, and it introduces the notion of Herbrand Logic. However in Wikipedia I cannot find a definition specifically for "Herbrand Logic", only for Herbrand Base or Herbrand Theorem, concepts that also exist in relational locic / predicate logic ?
What is the difference between Herbrand Logic and Relational Logic or Predicate Logic?
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logic
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predicate-logic
first-order-logic
3 Answers
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"Herbrand Logic = First-order syntax + Herbrand semantics".
Essentially, in Herbrand semantics a model M is a set of ground terms, which are exactly those ground terms that it satisfies. In other words, a model is always some subset of the Herbrand base.
This is very different from the Tarski semantics, where a model may contain a whole bunch of elements that cannot be written down as ground terms, but they can take part in the interpretation. In particular, the quantifiers range over such elements as well.
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See here for Relational Logic.
Is first-order predicate logic without function symbols, i.e. tehre are only $n$-ary relation constants.
The semantics is based on Herbrand semantics.
Due to the lack of function symbols, the interpretation has only a finite number of objcets; thus, we can use truth assignments, as in propositional logic, and use them to evaluate formulas.
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0I do not see how this answers the question. This is just an explanation of first-order logic. – 2017-07-08
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Consider a first order language L.
In first order logic: A formula F follows from a set of formulas T if every first order structure for L that satisfies T also satisfies F.
In Herbrand logic: A formula F follows from a set of formulas T if every first order Herbrand structure for L that satisfies T also satisfies F.