Abstract
We present a new proof of soundness/completeness of tableaux with respect to dialogical games in Classical First-Order Logic. As far as we know it is the first thorough result for dialogical games where finiteness of plays is guaranteed by means of what we call repetition ranks.
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Notes
Actually Felscher [1, Section 4] noticed the difference and proposed a “deformation operation” to overcome it. But this operation is defined for dialogues where the E rule constraints the order of the Opponent’s moves. By contrast, there is no such constraint in the dialogical games as we define them.
It should be emphasized that this is of no harm on the Equivalence result proven in Felscher [1] because it includes finiteness of the play in the conditions for P’s victory.
By rule D13 or its liberalized version D13\(_{k}\) [1, p. 227].
See Section 3.6 in Felscher [2] and in particular p. 359.
Lorenz used this distinction for his claim that Intuitionistic Logic is characterized by the smallest repetition rank (for defences) because the so-called Last Duty First rule of intuitionistic dialogues bounds the possible number of defences to one.
It seems that in more recent works Lorenz’s account of repetition ranks came closer to the one we introduce. See Lorenz [8, p. 260].
Because it is not needed in the cases we consider, not even to formulate the distinction between Classical and Intuitionistic Logic. See Section 2.2.
This is inspired by and adapted from Felscher [1].
The reason why atomic sentences are not included is related to \(\text {SR}2\).
Intuitionistic dialogical games are defined simply by modifying \(\text {SR}1\) so that the repetition ranks bounds only the number of challenges, and players can defend only once against the last non-answered challenge. In this way there is no interference with other rules and thus no need for priority criteria over structural rules.
By bounding the total number of challenges and defences, and not only the identical ones.
See Gale–Stewart [4].
Letz [6, Definition 74 and footnote 5].
If there are several formulas of same type, priority is given according to complexity.
We also add the symbol ‘\(!\)’ for assertions.
With the notable exception of O-implications which are dealt with in Step C.
The quantifier depth of \(\mathcal {T}\) is the total number of applications of the γ rule in \(\mathcal {T}\). This is more than enough for P to play the needed number of challenges and defences against universally and existentially quantified sentences. But it is also obvious that P’s repetition rank should most of the time be at least \({\tt 2}\) so that he can efficiently challenge conjunctions or defend disjunctions.
This does not make t strongly contentious: it simply means that the successor of t labelled with \(\mathbf {O}\,!\chi\) is not in \(\theta\) but in another branch passing through t.
By this we mean “of maximal level”, that is, as low as possible in the tree.
Nodes labelled with atomic P-formulas may belong to \(D(t_{2})\) as the result of the previous Step of the procedure, and we cannot let them be considered as superfluous and removed.
The possibility to add alphabetic-variants without changing anything in terms of existence of winning P-strategies is the reason why some works add as a condition that always chooses new individual constants.
This makes it easier to deal with O-implications afterwards.
\(\mathcal {B}'\) simply represents a play in which O starts by defending and subsequently counter-attacks when she has no other possible move.
References
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Acknowledgments
I am grateful to Shahid Rahman, Tero Tulenheimo, Laurent Keiff and two anonymous referees for their comments and remarks which led me to clarify various points.
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Clerbout, N. First-Order Dialogical Games and Tableaux. J Philos Logic 43, 785–801 (2014). https://doi.org/10.1007/s10992-013-9289-z
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DOI: https://doi.org/10.1007/s10992-013-9289-z