## Abstract

Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic (FOL). However, there are good reasons to avoid using full second-order logic (SOL) for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by augmenting it with the transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory (which is much more relevant for the task of mechanizing mathematics). Two natural Gentzen-style proof systems for ancestral logic are presented: one for the reflexive transitive closure, and one for the non-reflexive one. We show that these systems are sound for ancestral logic and provide evidence that they indeed encompass all forms of reasoning for this logic that are used in practice. The two systems are shown to be equivalent by providing translation algorithms between them. We end with an investigation of two main proof-theoretical properties: cut elimination and constructive consistency proof.

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## Notes

\(v\left[ x:=a\right] \) denotes the

*x*-variant of*v*which assigns to*x*the element*a*from*D*.\(\varphi \left\{ \frac{t_{1}}{x_{1}},...,\frac{t_{n}}{x_{n}}\right\} \) denotes the formula obtained from \(\varphi \) by simultaneously substituting \(t_{i}\) for each free occurrence of \(x_{i}\) in \(\varphi \), assuming that \(t_{1},...,t_{n}\) are free for \(x_{1},...x_{n}\) in \(\varphi \).

For a logic \(\mathcal {L}\) we write \(\mathcal {L^{\sigma }}\) to denote the logic where the language is based on the signature \(\sigma \).

In this section we refer to \(TC_{G}\), though similar considerations apply to \(RTC_{G}\) as well.

Gentzen’s proof is

*constructive*in the sense that it provides an effective algorithm for transforming any proof of the empty sequent into a cut-free one. There is a debate to what extent the method used to justify that Gentzen’s procedure always terminates (transfinite induction up to \(\varepsilon _{0}\)) is acceptable from a pure constructive point of view. We shall not enter this discussion here, but only note that in Takeuti (2013) there is a detailed argument that it is not only constructive, but in fact justified even from a finitist standpoint.The end-piece of a proof Gentzen (1969) consists of all the sequents of the proof encountered if we ascend each path starting from the end-sequent and stop when we arrive to an operational inference rule. Thus the lower sequent of this inference rule belongs to the end-piece, but its upper sequents do not.

Note that the addition of the axioms for multiplication to \(TC_{A}\) is not really necessary, as they are derivable using the

*TC*-formula which defines multiplication given in Avron (2003).If \(\mathcal {L}\) is a language that expands the language of

*PA*, and*S*and*T*are two systems expanding \(TC_{A}\) and \(PA_{G}\) for the language \(\mathcal {L}\) by the same set of additional axioms, then, using practically the same method we can prove that*S*and*T*are equivalent.

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## Acknowledgments

This research was supported by the Ministry of Science, Technology and Space, Israel.

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Cohen, L., Avron, A. The middle ground-ancestral logic.
*Synthese* **196**, 2671–2693 (2019). https://doi.org/10.1007/s11229-015-0784-3

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DOI: https://doi.org/10.1007/s11229-015-0784-3