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Cut-Elimination for Quantified Conditional Logic

Abstract

A semantic embedding of quantified conditional logic in classical higher-order logic is utilized for reducing cut-elimination in the former logic to existing results for the latter logic. The presented embedding approach is adaptable to a wide range of other logics, for many of which cut-elimination is still open. However, special attention has to be payed to cut-simulation, which may render cut-elimination as a pointless criterion.

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Notes

  1. 1.

    It is a recommended exercise to verify that η-equality, and hence β η-equality, is implied by the rules of \(\mathcal {G}_{\mathfrak {fb}}\).

  2. 2.

    Obviously, any universally quantified predicate variable (occurring negatively in the above approach) is a possible source for cut-simulation. The challenge thus is to avoid those predicate variables as far as possible. An axiomatic approach based on cut-strong axioms, as proposed by several authors including e.g. [45, 46], is therefore hardly a suitable option for the automation of HOL.

  3. 3.

    Note the predicate argument A of f in the term for \(\Rightarrow _{\tau \rightarrow \tau \rightarrow \tau }\) and the second-order quantifier ∀P τ in the term for π(ττ)→τ . FOL encodings of both constructs, if feasible, will be less natural.

  4. 4.

    Sets are identified with their characteristic functions.

  5. 5.

    In fact, it may be safely assumed that there are no other typed constant symbols given, except for the symbols f iττ , e i w iuo , k u n τ , and the logical connectives.

  6. 6.

    In H M we have merely fixed D i = S, D u = D and D τ = Q, and the interpretations of k u n τ , f iττ and e i w iuo . These choices 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 α } αT . Andrews [3] discusses criteria that are sufficient to ensure the existence of a valuation function; they require that certain λ-abstractions have denotations in frame {D α } αT . Since it is explicitly required that every term denotes and since Q has been appropriately constrained in QCL models (and hence D io in H M) these requirements are met.

  7. 7.

    Experiments with these and other reasoners for THF are supported online via Sutcliffe’s SystemOnTPTP infrastructure [62]; cf. www.tptp.org/cgi-bin/SystemOnTPTP.

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Acknowledgments

I am grateful to several colleagues for fruitful discussions and contributions to earlier related work; this list of persons includes Chad Brown, Dov Gabbay, Valerio Genovese, Larry Paulson, Dale Miller, Nik Sultana, and Bruno Woltzenlogel Paleo. Moreover, I am grateful to Alexander Steen for proof reading this article.

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Correspondence to Christoph Benzmüller.

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This work has been supported by the German National Research Foundation (DFG) under grant BE 2501/9-1,2.

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Benzmüller, C. Cut-Elimination for Quantified Conditional Logic. J Philos Logic 46, 333–353 (2017). https://doi.org/10.1007/s10992-016-9403-0

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Keywords

  • Cut-elimination
  • Quantified conditional logics
  • Classical higher-order logic
  • Semantic embedding
  • Cut-simulation