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Automating Free Logic in HOL, with an Experimental Application in Category Theory

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Abstract

A shallow semantical embedding of free logic in classical higher-order logic is presented, which enables the off-the-shelf application of higher-order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category theory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments.

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Notes

  1. Cf. Sect. 4.4 for further information.

  2. Calculi for free logic are presented in [30]; see also the references therein.

  3. The predication \( E x\) represents that x is a member of E.

  4. The \(\star \) symbol is not to be confused with any other symbol in Isabelle/HOL.

  5. It is well known that we could work with a much smaller set of logical connectives, see e.g. Sect. 1.4 of Andrews’s overview article [2]. The choice here closely reflects the set of primitive connectives as chosen in higher-order automated theorem provers such as LEO-II [13], Leo-III [12], and Satallax [18].

  6. Similar soundness and completeness proofs for shallow semantical embeddings have been presented in [4] and [5].

  7. In fact, it may be safely assumed that there are no other constant symbols given in a HOL signature, except for the symbols in \(\widehat{F}\) and \(\widehat{P}\), the symbols and \(\varvec{\star _{i}}\) and the logical connectives.

  8. The fixings introduced in \(\widehat{M}\) are not in conflict with any of the requirements regarding frames and interpretations. The existence of a valuation function V for an HOL interpretation crucially depends on how sparse the function spaces have been chosen in frame \(\{D_\alpha \}_{\alpha \in {T}}\). Andrews [1] discusses criteria that are sufficient to ensure the existence of a valuation function; in \(\widehat{M}\) these requirements are met.

  9. In the remainder of this article, and inline with our text so far, we present the formulas of \({\text {FFOL}} \) in non-boldface font. These formulas have been encoded in Isabelle/HOL using the abbreviations as introduced in Fig. 2. In the actual source encoding, however, the usage of boldface and non-boldface is (for technical reasons) reversed.

  10. Technical remark: We have selected CVC4 in our experiments as the default SMT solver, since we did run into errors when working with Z3. These errors can easily be reconstructed in the provided source files when switching back to Z3 as default.

  11. An expert reviewer of this article, to whom we are very grateful, provided alternative proofs which can be fully replayed in the kernel of Isabelle.

  12. A recipe for this translation is as follows: (i) replace all \(x \circ y\) by \(y \cdot x\), (ii) rename the variables to get them again in alphabetical order, (iii) replace \(\varphi \Box \) by \(\textit{cod }\varphi \) and \(\Box \varphi \) by \(\textit{dom }\varphi \), and finally (iv) replace \(\textit{cod }y \cong \textit{dom }x\) (resp. \(\textit{cod }y \simeq \textit{dom }x\)) by \(\textit{dom }x \cong \textit{cod }y\) (resp. \(\textit{dom }x \simeq \textit{cod }y\)).

  13. Def. 1.11 in Freyd Scedrov: “The ordinary equality sign \(=\) [i.e., our \(\cong \)] will be used in the symmetric sense, to wit: if either side is defined then so is the other and they are equal. ...”

  14. This could perhaps be an oversight, or it could indicate that Freyd and Scedrov actually mean the Axioms Set discussed in Sect. 5.2 below.

  15. For this we have to inactivate the axiom that postulates that \(\star \) is an undefined/non-existing object.

  16. The discussion in our releated conference paper [9] was before the discovery of the above constricted inconsistency issue, which tells us that the system (in our setting) can even be reduced to axioms A1, A2a, and A3a (when we assume undefined objects).

  17. This minimal set of axioms has also been mentioned by Freyd in a note [22] and attributed to Martin Knopman. However, the proof sketch presented there seems to fail when the adapted version of A1 (with \(\simeq \)) is employed.

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Acknowledgements

We thank Günter Rote, Lutz Schröder and and Emil Weydert for their comments to [10], which together with [9] forms the basis for this article. We also want to express our gratitude to the reviewers of this article. Their fruitful feedback definitely helped to improve the final version.

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  1. Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  2. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Christoph Benzmüller

  3. University of California, Berkeley, USA

    Dana S. Scott

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Benzmüller received funding from the German National Research Foundation DFG under Heisenberg grant Towards Computational Metaphysics (BE 2501/9-2) and from VolkswagenStiftung under grant Consistent Rational Argumentation in Politics (CRAP).

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Benzmüller, C., Scott, D.S. Automating Free Logic in HOL, with an Experimental Application in Category Theory. J Autom Reasoning 64, 53–72 (2020). https://doi.org/10.1007/s10817-018-09507-7

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