## 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.

## References

Aho, A. V., & Ullman, J. D. (1979). Universality of data retrieval languages. In

*Proceedings of the 6th ACM SIGACT-SIGPLAN symposium on principles of programming languages*, (pp. 110–119). ACM.Avron, A. (2003). Transitive closure and the mechanization of mathematics. In Fairouz D. Kamareddine (Ed.),

*Thirty five years of automating mathematics, applied logic series*(Vol. 28, pp. 149–171). Netherlands: Springer.Avron, A. (2004). Formalizing set theory as it is actually used. In

*Mathematical knowledge management*, (pp. 32–43). Springer.Avron, A. (2008). A framework for formalizing set theories based on the use of static set terms. In

*Pillars of computer science*, (pp. 87–106). Springer.Campbell, J. J. J. A., Dos Reis, J. C. G., Wenzel, P. S. M., & Sorge, V. (2008). Intelligent computer mathematics.

Cohen, Liron, & Avron, Arnon. (2014). Ancestral logic: A proof theoretical study. In U. Kohlenbach, et al. (Eds.),

*Logic, language, information, and computation, volume 8652 of lecture notes in computer science*(pp. 137–151). Berlin, Heidelberg: Springer.Constable, R. L., Allen, S. F., Bromley, M., Cleaveland, R., Cremer, J. F., Harper, R. W., et al. (1986).

*Implementing mathematics with the Nuprl proof development system*. Upper Saddle River: Prentice Hall.Da Silva, J. J., D’Ottaviano, I. M. L., & Sette, A. M. (1999). Translations between logics.

*Lecture Notes in Pure and Applied Mathematics*,*203*, 435–448.D’Ottaviano, I. M. L., & Feitosa, H. A. (2012).

*On gödel’s modal interpretation of the intuitionistic logic. Universal Logic: An anthology*. Basel: Birkhäuser.Ebbinghaus, H.-D., Flum, J., & Ebbinghaus, H.-D. (1995).

*Finite model theory*(Vol. 2). New York: Springer.Fagin, R. (1974). Generalized first-order spectra and polynomial-time recognizable sets. In R. Karp (Ed.),

*S*IAM-AMS Proceedings 7, (pp. 27–41). Immerman.Gentzen, G. (1969). Neue fassung des widerspruchsfreiheitsbeweises für die reine zahlentheorie, forschungen zur logik. 4:19–44. (M. E. Szabo, English Trans.). The collected work of Gerhard Gentzen, Amsterdam.

Gentzen, G. (1935). Untersuchungen über das logische schließen. i.

*Mathematische Zeitschrift*,*39*(1), 176–210.Gentzen, G. (1943). Beweisbarkeit und unbeweisbarkeit von anfangsfällen der transfiniten induktion in der reinen zahlentheorie.

*Mathematische Annalen*,*119*, 140–161.Henkin, L. (1961). Some remarks on infinitely long formulas. In L. Henkin (Ed.),

*Infinistic methods*. New York: Pergamon Press.Kamareddine, F. D. (2003).

*Thirty five years of automating mathematics*(Vol. 28). New York: Springer.Martin, R. M. (1943). A homogeneous system for formal logic.

*The Journal of Symbolic Logic*,*8*(1), 1–23.Martin, R. M. (1949). A note on nominalism and recursive functions.

*The Journal of Symbolic Logic*,*14*(1), 27–31.Myhill, J. (1952). A derivation of number theory from ancestral theory.

*The Journal of Symbolic Logic*,*17*(3), 192–197.Pohlers, W. (2009).

*Proof theory: The first step into impredicativity*. New York: Springer.Prawitz, D., & Malmnäs, P.-E. (1968). A survey of some connections between classical, intuitionistic and minimal logic. In H. Arnold Schmidt, K. Schütte, & H. J. Thiele (Eds.),

*Contributions to mathematical logic, proceedings of the logic colloquium, Hannover 1966*(pp. 215–229). North-Holland: North-Holland Publishing Company.Rudnicki, P. (1992). An overview of the mizar project. In

*Proceedings of the 1992 workshop on types for proofs and programs*, (pp. 311–330).Shapiro, S. (1991).

*Foundations without foundationalism: A case for second-order logic*. Oxford: Oxford University Press.Smith, P. (2008). Ancestral arithmetic and Isaacson’s thesis.

*Analysis*,*68*(297), 1–10.Takeuti, G. (2013).

*Proof theory*. Mineola: Courier Dover Publications.Troelstra, A. S., & Schwichtenberg, H. (2000).

*Basic proof theory number 43*. Cambridge: Cambridge University Press.Yasuhara, M. (1966). Syntactical and semantical properties of generalized quantifiers.

*The Journal of Symbolic Logic*,*31*(4), 617–632.

## 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