MODUS PONENS, CONDITIONALCIRCULARITY AND MATERIAL IMPLICATION Draft of August 19, 2020 Matheus Silva The modus ponens can be interpreted as an answer to a circularity charge, but this strategy is only feasible if the additional conditional premise is interpreted as a claim to a material implication. Here's how it works. Suppose one argues that a premise A implies a conclusion B. This argument is arbitrary, so I press the arguer for further justification. In response, the arguer adds the premise 'A implies B' and makes some qualifications. The result of this newfound premise is the following argument: P1. A implies B; P2. A is true; therefore, C. B is true. Now, this won't solve the circularity worries because we are not obliged to accept C on the basis of P1 and P2 unless we first concede the conditional (S) if P1 and P2 are true, C must be true . So we must add S as an additional premise in order to accept the implication. 1 But this won't get us far because we are not obliged to accept C due to P1, P2 and S unless we concede that (T) if P1 and P2 and S are true, C must be true. But once (T) is added I can demand a further justification, and so and so forth. The modus ponens can be interpreted as a failed attempt to avoid a circularity accusation. It is the result of adding a conditional premise to justify an earlier inference, but this brings additional charges of circularity, which lead to more conditional premises. The conditional premises work like fraudulent promissory notes: their truth ensure the validity of the earlier implication statement, but in order to ascertain their truth you need to accept their truth in advance. It is a strategy that is bound to fail because each conditional premise begs the question. It seems that the conditional premise is just a different way of reinstating the initial implication allegation. It doesn't bring new information to the table, but repeats what was said before in different words. We need to look carefully at the first modus ponens, since it contains some quirks that need to be addressed before we can stop this regress. First, in order to avoid the circularity, accusation the arguer was compelled to assert the truth of the second premise. This is weird, because I can accept that A implies B even if A is false. The other weird aspect of the initial modus ponens is that this will only work if the first premise is a strict implication, but this means that the premise will be false if A and B are not conceptually connected, and most conditional sentences that are presented as a premise of a modus ponens argument are not like that. Finally, the revised argument remains circular since there are no reasons to accept that A implies B other than the premise that asserts that A strictly implies B. Perhaps if we interpret the premise as a material implication we might provide a solution for this circularity problem. The statement that A implies B should be interpreted as 'it is not the case that A is true and B is false in a given parameter world'. This reference to a parameter world is justified by the fact that when we evaluate arguments we consider all the possible worlds in which the premises are true. The set of these possible worlds might include Carroll, L. 1895. What the Tortoise Said to Achilles? Mind 4, 278–80. 1 !1 the actual world, but it doesn't need to be restricted by it. The premises and the conclusion need to be evaluated in the same parameter worlds, otherwise I could select a world where the conclusion is false and compare it with another world where the premises are true. Thus, we have: P1*. It is not the case that A is true and B is false in the parameter world. P2*. A is true in the parameter world. Ergo, C*. B is true in the parameter world. This reformulation not only seems to correct all the flaws of the initial reconstruction, as it puts an end to the justification regress. P1* is not a strict implication anymore and the arguer doesn't need to accept that P2* is true in order for the implication to hold, but only that when P1* and P2* are both true, C* is also true. The premises strictly imply the conclusion, as they should. More importantly, it is not circular or arbitrary to think that the premises necessitate the conclusion, since this is self-evident. If we have a certain states of things that ensure the truth of P1* and a consistency in truth values attribution, C* will be true in all the P1*&P2*worlds. The conditional that results from the conjunction of premises as an antecedent and the conclusion as a consequent is also self-evident: 'If it is not the case that A is true and B is false in the parameter world, and A is true in the parameter world, then B is true in the parameter world'. This puts a firm halt to the justification regress. The material implication is a claim to a relation of implication inside a parameter world. The strict implication is a claim to an unrestricted relation of implication that takes all the parameter worlds where the material implication occurs. The claim to a strict implication is not circular since it can be independently assessed and verified by the truth values of the material implication and its possible combinations with other premises and conclusion. !