Without Termination Guarantee

As it is usually the case for dialogues, we distinguish two sets of game rules, namely the particle rules and the structural rules (see Chapter 2.3.2 for details). Our particle rules are actually the same of Figure 2.17 (p. 72), but without those for the quantifiers ∀ and ∃, because we consider only propositional logic here. Additionally, as we let the proponents state atoms and O attack these (just like Barth and Krabbe [10]), we need a particle rule for prime formulas as well. We also consider an extra rule for ⊥. Note that the ¬-rule is then actually redundant, because ¬A ≡ A ⊃ ⊥, but nevertheless we offer both options for historical reasons. The particle rules are displayed in Figure 3.1. We write P for an arbitrary prime formula and A and B for propositional formulas. Double exclamation marks (!!) represent the ipse dixisti remark. We do not have an absurdum dixisti but instead there is simply no possibility to defend against a ⊥-attack.

We consider two parties. On the one hand, we have the single opponent O, on the other hand the P-agents Propos =df {Pi | i ∈ N}.

The structural rules for MPID are as follows [139]:

I1 At the beginning of a dialogue, O states initial concessions and a single

I2 A round consists of a sequence of moves by O, followed by moves of all active P-agents. A dialogue run is a sequence of such rounds. The first round starts after the assertion of the hypothesis.

I3 If possible, all players are obliged to perform moves. A P-agent may postpone a move until succeeding rounds if he is forced to react to a critical attack (see rule I6), but commitments by O made with the crit- ical attack are only conceded as soon as the critically attacked agent reacts. Whenever a P-agent has several possibilities of how to react to an O-move, new P-agents are introduced to take out these remaining possibilities.

I4 A dialogue is won by the proponents iff the opponent cannot react to all of the proponents’ moves of the previous round. The opponent wins iff no P-agent can react to any of O’s statement of the same round (either with an attack or a defence).

I5 OnlyOis allowed to attack prime formulas. P-agents may defend against these attacks only if O has stated the prime formula herself towards a

P-agent who is not deactivated in the same round.

I6 Attacks on negations and implications are considered to be critical at- tacks. Other attacks are non-critical.

I7 Whenever aP-agent reacts to a critical attack, all other active proponent agents are immediately deactivated, i.e., they may not perform defences or counter-attacks.

I8 A P-agent may repeat critical attacks on the same assertion only after any P-agent reacted to a critical attack performed by O. Other repeti- tions are not allowed.

Rules I6 to I8 are significant for intuitionism. I6 and I7 (in combination with I3) put the necessary restrictions on the proponents which correspond to the non-invertible rules (⊃r and ¬r) of multi-conclusion sequent calculi like

O P0 P1 P2 1 (A∧ B) ⊃ (A ⊃ B) 2 [?, 1]0 A∧ B [!, 2] A⊃ B [?, 2] ?l [?, 2] ?r 3 [?, 2]0 A [!, 3] B [!, 2]1 A — — [!, 2]2 B — — 4 [?, 3]0 ?B [!, 4] !! — — — —

Figure 3.2: A simple MPID-example

G3im

. Rule I8 allows the P’s to repeat attacks on implications and negations which correspond to the duplication rules (⊃l and ¬l) that are however restricted here as well, as these repetitions may only be performed after a

P-agent reacted to a critical attack. Note that rule I6 refers to specific connec- tives, namely ⊃ and ¬. This is quite unusual, as in the literature, structural rules are normally independent of the particle rules and are therefore not to be related to specific logical operators. However, we give up this convention to ensure higher flexibility of our dialogical system. Later, we will change the set of critical moves and thereby generate other dialectics for other logics. The dialogical tableau of the first example is shown in Figure 3.2. In the first row the initial proponent P0 states the hypothesis (A∧ B) ⊃ (A ⊃ B). In row 2, O attacks this assertion (the 0 attached to the brackets is the pro- ponent number O refers to). With her attack she states the antecedent of the hypothesis. P0 can now either defend with the consequent or counter-attack

O’s commitment. He defends with the antecedent but calls in the new agents

P1 andP2 to perform the counter-attacks. There are two of these. One agent asks for the left conjunct of O’s concession, the other one asks for the right one. In the next round, O has to react to all of the proponents’ moves. She attacks P0’s implication and defends against the attacks ofP1 andP2 stating the conjuncts towards them. P0 reacts with a defence stating the consequent B. He is not allowed to counter-attack A, as only Omay attack primes. That

O P0 P1 1 A∨ ¬A 2 [?, 1]0 ?_∨ [!, 2] A [!, 2] ¬A 3 [?, 2]0 _? A — — [?, 2]1 _{A — —}

Figure 3.3: The Excluded Middle in MPID

is also the reason why P1 and P2 cannot move now and miss that turn.2 In the last round,OattacksP0’s B who answers thatOstated it herself (towards

P2) which is enough to win the game, as O cannot react to this.

The second example (Figure 3.3) concerns again the law of the excluded middle which is not valid in intuitionistic logic. The first attack by O is defended by P0 with the left and by P1 with the right disjunct. Then O

attacks both in row 3. The attack against ¬A is critical. This means that if

P1 reacts to this, P0 is deactivated. On the other hand, O’s concession of A towards P1 is not conceded until the agent reacts to this which means that

P0 cannot make use of it to perform an ipse dixisti defence. Also P1 cannot react, as it is not possible for him to attack primes and the particle rules also do not allow defences against ¬-attacks. No proponent is able to move and therefore the opponent wins.