Three levels of translation into many-sorted logic (original) (raw)

Abstract

We assume the opinion by which "tranlation into classical logic" is a reliable methodology of Universal Logic in the task of comparing different logics. What we add in this paper, following Manzano [9], is some evidence for adopting the slightly different paradigm of "tranlation into many-sorted classical logic." Our own methodology, splitted into three levels of translation, is discussed in some detail.

Key takeaways

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1. The text proposes a three-level methodology for translating logics into many-sorted logic (MSL).
2. MSL offers advantages over unsorted logic (USL), including shorter deductions and richer interpolation properties.
3. The methodology aims to integrate MSL within the Universal Logic framework, contrasting with first-order logic.
4. Key properties of MSL and USL include strong completeness and the Löwenheim-Skolem theorem, emphasizing their equivalence.
5. Translation processes for logics can yield substantial benefits in consistency proofs and theorem proving efficiency.

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References (10)

1. van Benthem, J.: Modal Logic and Classical Logic, Bibliopolis, Naples, 1983
2. Béziau, J.-Y. (ed.): Logica Universalis, Birkhäuser, Berlin, 2005
3. Feferman, S.: "Lectures on proof theory," Proceedings of the Summer School in Logic (Leeds 1967), LNM 70, Springer, Berlin, 1968, pp. 1-107
4. Gabbay, D.M. (ed.): What is a Logical System?, Oxford University Press, Oxford, 1994
5. Gabbay, D.M.: Labelled Deductive Systems, Volume 1, Oxford University Press, Oxford, 1996
6. Gödel, K.: "Eine Interpretation des intuitionistischen Aussagenkalküls," Ergebnisse eines mathematischen Kolloquiums, 4 (1933), pp. 34-40
7. Henkin, L.: "Banishing the rule of substitution for functional variables," The Journal of Symbolic Logic, 15, 1953, pp. 81-91
8. Hook, J.L.: "A note on interpretations of many-sorted theories," The Jour- nal of Symbolic Logic, 50, 1985, pp. 372-374
9. Manzano, M.: Extensions of First Order Logic, Cambridge University Press, Cambridge, 1996
10. Meseguer, J.: "General Logics," in Ebbinghaus et al. (eds.): Logic Collo- quium '87, North-Holland, Amsterdam, 1989, pp. 275-239

FAQs

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What are the key advantages of many-sorted logic over first-order logic?add

Many-sorted logic (MSL) allows more natural expressions of multiple sorts, leading to shorter deductions. This advantage can result in more efficient theorem provers when using MSL compared to first-order logic.

How does the translation method for MSL differ from traditional methods?add

The paper proposes a three-level methodology for translating any logic XL into many-sorted logic MSL*. This contrasts traditional methods that typically use first-order or higher-order logic, highlighting novelty in its structured approach.

What role does Lindström's theorem play in comparing MSL and first-order logic?add

Lindström's theorem asserts that first-order logic is the strongest complete logic with specific properties, implying that MSL does not extend first-order logic properly. This establishes a foundational comparison in terms of completeness and compactness.

What are the implications of Craig's interpolation theorem in MSL?add

MSL exhibits a stronger version of Craig's interpolation theorem, enhancing its expressive power over its one-sorted counterparts. This suggests that MSL may facilitate richer reasoning capabilities than traditional unsorted logics.

What challenges arise when translating from MSL to USL?add

Translating from many-sorted logic (MSL) to unsorted logic (USL) poses challenges due to the many-to-one relation of MSL operations to USL operations, which can lead to loss of specific structural interpretations. These complications highlight the nuances of logical expressiveness across the two systems.