The pure logic of ground. (English) Zbl 1250.03010
‘Ground’ is the relation under which one truth holds in virtue of others. As such, it provides an explanation or account of that truth. This paper presents a semantics and proof theory for a pure theory of this relation, ‘pure’ in that it does not address the internal structure of the propositions over which the ground relation holds. The proof theory is then much like Gentzen’s structural rules for logical consequence, but different insofar as it concerns a different relation.
In fact, two pairs of ground relations are specified: full or partial, and weak or strict for each of those. Thus, PLG, the pure logic of ground, comprises four types of sequents. To interpret the language of PLG, a fact-based semantics is introduced, whereby \(C\) is a consequence of \(A_1,A_2,\dots\) iff, whenever fact \(f_1\) verifies \(A_1\), \(f_2\) verifies \(A_2,\dots\), then \(f_1.f_2.\dots\) verifies \(C\), where \(f_1.f_2.\dots\) is the composite fact, or fusion, of \(f_1,f_2,\dots\) etc. PLG is proved to be sound and complete with respect to this semantics. Various fragments of PLG are also investigated.
In fact, two pairs of ground relations are specified: full or partial, and weak or strict for each of those. Thus, PLG, the pure logic of ground, comprises four types of sequents. To interpret the language of PLG, a fact-based semantics is introduced, whereby \(C\) is a consequence of \(A_1,A_2,\dots\) iff, whenever fact \(f_1\) verifies \(A_1\), \(f_2\) verifies \(A_2,\dots\), then \(f_1.f_2.\dots\) verifies \(C\), where \(f_1.f_2.\dots\) is the composite fact, or fusion, of \(f_1,f_2,\dots\) etc. PLG is proved to be sound and complete with respect to this semantics. Various fragments of PLG are also investigated.
Reviewer: Louis F. Goble (Salem)
MSC:
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 03B47 | Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) |
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