Game Rules

Semantically, we interpret the 2-modality in a similar way as Krabbe (see Chapter 2.4.1) but without different strictness levels. As there is no obvi- ous dialogical interpretation for the 3-modality and because of the duality between 2 and 3 in classical and normal modal logics, we propose two different sets of structural rules: one simplified that does not cope with the

3-operator, and one complete. The complete one is necessary, as later, when we establish a system for CS4, the duality of 2 and 3 is given up and there- fore the 3must be handled as an independent logical operator.

Particle Rules

As particle rule for the2we adapt that proposed by Lorenzen (Chapter 2.4.1) and adapt a 3-rule accordingly. The modal particle rules are shown in Fig- ure 4.1 and are simply added to those of MPID (Figure 3.1, p. 100). There is actually nothing special or surprising about these rules. Like Lorenzen [102] and almost all of his successors, we define the properties about the modalities in the structural rules.

Simplified Structural Rules

The simplified structural rules for MPMD/S4 without the 3-operator were already presented in [139]. They are based on those of MPID (p. 100), but

implication and negation are not critical operators anymore. Instead, critical attacks are only attacks on 2-statements. Like in Krabbe’s attempts [83, 86]

O may withdraw a commitment when a P-agent defends against a 2-attack. We distinguish weak commitments and strong ones: strong commitments are assertions by O prefixed by a2. By contrast, weak commitments are all other concessions by O.

The rules I1 to I4 and I8 of MPID (pp. 100, 101) are adapted without any changes. Rules I5 to I7 are replaced by the following [139]:

S4-5 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if O has stated the prime formula herself towards a P-agent who is not deactivated in the same round and if O has not withdrawn it.

S4-6 Attacks on 2 are considered to be critical attacks. Other attacks are non-critical.

S4-7 Whenever aP-agent reacts to a critical attack, all other active proponent agents are immediately deactivated, i.e., excluded from the rest of the dialogue run. O may immediately withdraw all weak commitments she made before in same the run.

As we see, these rules are very close to those of MPID. The set of critical attacks is changed and the concept of withdrawals is introduced and con- nected to these attacks. Deactivating is also a bit stricter, as the concerned agents are excluded completely .

Complete Structural Rules

The structural rules which involve the 3 are more complex and considered separately because there is currently no dialogical interpretation for the 3 that fits to the interpretation of the 2. We now call assertions starting with a 3 hypothetical assertions. Hypothetical assertions stated by the opponent

are accordingly called hypothetical concessions. We choose these terms as they give an idea of how the 3-operator can be interpreted semantically in a dia- logical setting.2 We do not claim that this is the best choice but rather a possible suggestion, as our focus lies on proof theory and not on philosoph- ical issues or argumentation theory. Attacks against hypothetical assertions are called co-critical attacks.

For the complete structural rules for MPMD/S4 we adapt again rules I1, I2 and I4 of MPID and rule S4-5 of the simplified rules. We have to adjust rules I3, S4-6, S4-7, and I8 as follows:

S4-3∗ 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 or to perform a co-critical attack. 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.

When a proponent agent states a hypothetical assertion, he is protected and stunned, i.e., he stays active but does not perform moves anymore. Instead, reactions to the (co-critical) attacks towards him are performed by new colleagues.3

S4-6∗ Attacks on 2 are considered to be critical attacks. Attacks on 3 are considered to be co-critical attacks. Other attacks are non-critical.

S4-7∗ Whenever a P-agent reacts to a critical attack, all other non-protected proponent agents are immediately deactivated, i.e., excluded from the rest of the dialogue run. The remaining proponent agents miss their turn in the round. O may immediately withdraw all weak commitments she made before in the same run.

2_{In the alethic interpretation of the modal operators,2 corresponds to necessarily and 3 to}

possibly which we interpret here as hypothetically.

3_{In this variant of MPMD/S4, a P-agent can be either stunned and protected, or neither}

stunned nor protected at the same time. For CS4 (Section 4.2.2) we need to distinguish these properties.

O P0 P1 P2 1 (3A⊃2B) ⊃2(A⊃ B) 2 [?, 1]0 ₍₍3A((⊃ 2(B( [!, 2] 2(A⊃ B) [?, 2] 3A 3 [?, 2]0 ?₂ [!, 3] A⊃ B stunned [?, 2]1 ?₃ protected 4 [?, 3]0 A [!, 4] B [?, 2]1 ?₃ [!, 4] A 5 [?, 4]0 ?B — — [?, 4]2 ?A [!, 5] !!

Figure 4.2: An example of an MPMD/S4-Dialogue

Whenever the opponent reacts to a co-critical attack, all non-protected

P-agents are immediately deactivated, i.e., excluded from the rest of the dialogue run. O may immediately withdraw all weak commitments she made before in the same run.

S4-8∗ A P-agent may repeat critical attacks on the same assertion only after someP-agent reacted to a critical attack performed byOor Oreacted to a co-critical attack.

The opponent may repeat an attack against the same protected agent only after someP-agent reacted to a critical attack performed byOor O

reacted to a co-critical attack. Other repetitions are not allowed.

With the new concepts of hypothetical assertions, co-critical attacks, protected and stunned agents, these rules are much more complex than the simplified ones for which the 3-operator is not regarded. Figure 4.2 shows a dialogue where the hypothesis is theIK5-axiom which is valid in classicalKand there- fore also in S4. Only one of O’s strategies is displayed (she counter-attacks

P1 in row 3).4 _{Actually, a new P-agent should defend here with A while P1}

is stunned and protected, but this agent is deactivated immediately as P0

defends against a critical attack. O may then attack P1 again (row 4). This is defended by P2 who eventually finished the game with ipse dixisti. Note that

O withdraws her implication of row 2 after P0 defends in row 3, although this is not necessary as the proponents may only repeat critical attacks after defending against a critical attack, and implications are not critically here.