Abstract
In this chapter we will spell out all the relevant rules of dialogues for immanent reasoning, that is, the dialogical framework incorporating features of Constructive Type Theory—a dialogical framework making the players’ reasons for asserting a proposition explicit. The rules can be divided, just as in the standard framework, into rules determining local meaning and rules determining global meaning. These include:
-
1.
Concerning local meaning (Sect. 7.1):
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(a)
formation rules (p. 105);
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(b)
rules for the synthesis of local reasons (p. 108); and
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(c)
rules for the analysis of local reasons (p. 109).
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(a)
-
2.
Concerning global meaning, we have the following (structural) rules (Sect. 7.2):
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(a)
rules for the resolution of instructions (p. 112);
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(b)
rules for the substitution of instructions (p. 113);
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(c)
equality rules determined by the application of the Socratic rules (p. 113); and
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(d)
rules for the transmission of equality (p. 115).
-
(a)
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Notes
- 1.
This rule is an expression at the level of plays of the rule for the substitution of variables in a hypothetical judgement . See (Martin-Löf, 1984, pp. 9–11).
- 2.
As a matter of fact, increasing her repetition rank would allow O to play the two alternatives for move 3 within a single play. But increasing the Opponent’s rank usually yields redundancies (Clerbout, 2014a, 2014b) making things harder to understand for readers not familiar with the dialogical approach; hence our choice to divide the example into different simple plays.
- 3.
- 4.
- 5.
Note that P is allowed to make an elementary statement only as a thesis (Socratic rule); he will be able to respond to the challenge on an elementary statement only if O has provided the required local reason in her initial concessions.
- 6.
See below, Sect. 7.3, for an illustration.
- 7.
See for instance \( {\mathcal{P}}_1 \)in Sect. 7.6.10 for an illustration.
- 8.
The symmetry used here is not the same notion as the symmetry of Sect. 4.3.
- 9.
This assumption is analogous to the assumption in Sect. 5.2 that O chooses a new constant when she can. The reason is the same: it is the best possible choice for O. Indeed, here also P is restricted by the Socratic rule , so he needs to rely on O’s choices in order to copy a move. In such a context , the only way to (try to) block the use of this kind of equality is to always choose, whenever possible, a new local reason.
- 10.
The reason is similar to the previous one: it is better for O to force P to makes his choice as soon as possible.
- 11.
- 12.
Chapter 9 will focus on proving that dialogues for immanent reasoning is an admissible logical framework; it will therefore be rather technical and will address problems that mainly concern logicians, though we will here take care to outline the main philosophical aspects involved.
- 13.
Among these variations can be counted cooperative games, non-monotony, the possibility of player errors or of limited knowledge or ressources, to cite but a few options the play level offers, making the dialogical framework very well adapted for history and philosophy of logic.
- 14.
The table which follows is in fact the dialogical analogue to the introduction rules in CTT: dialogically speaking, these rules display the duties required by P’s assertions —we will come back to this issue in Sect. 9.1.
- 15.
See (Martin-Löf, 1984, pp. 66–67).
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Rahman, S., McConaughey, Z., Klev, A., Clerbout, N. (2018). The Dialogical Roots of Equality: Dialogues for Immanent Reasoning. In: Immanent Reasoning or Equality in Action. Logic, Argumentation & Reasoning, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-91149-6_7
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