MAKING SENSE OF DEDUCTION Draft of August 6, 2020 Matheus Silva An argument is deductive when the arguer believes that the truth of the premises necessitates the truth of the conclusion. A deductive argument is valid when the arguer's claim is true, i.e., when there are no possible worlds whether the premises are true and the conclusion is false. But in order to evaluate this claim in an accurate manner we need to consider three qualifications that have been repeatedly ignored in the literature, namely, consistency in the attribution of truth values, world consistency and the inclusion of the reasons offered by the arguer in support of the premises. These apparently innocuous restrictions have significant and far-reaching implications. First, let's take into account the requirement that there should be a consistency in truth values attribution. This self-evident prescription is integral to our practices of argument evaluation. For consider how we decide whether all the worlds will ensure that the truth of the premises are preserved. Surely, there is infinitely many worlds where the premises are true (if they can be true), so it is not as if we could count them per se or analyse each world individually. So we need to resort to a short-cut in the form of consistency in truth values attribution. The inference used is that there are no possible combinations where the premises are true and the conclusion is false due to the truth conditions of the premises and conclusion and, more importantly, the consistency in truth values attribution. By that I mean the requirement that each propositional variable should assume the same truth value throughout the argument. This seems intuitively obvious, but it is routinely ignored due to some erroneous modal intuitions. Let's take for instance a simple argumentative form such as the first paradox of material implication, ¬A ⊨ A → B. According to the possible world account, this argumentative form is invalid since B can be false in the closest A-world. This means that it does not matter that the conditional was inferred from the falsity of the antecedent, since we ascertain its truth-value in the closest world where the antecedent is true. But this is a violation of the consistency requirement. If ¬A is taken as true, the A in the antecedent of the conditional is false. Considering a counterfactual world where A is true is irrelevant to the truth values of A → B. Notice that this restriction should not be confused with the claim that the attribution of truth values should be restricted to the actual world, but only that there should be consistency in world evaluation. This lead us to the second restriction. There should a consistency in world evaluation during the analysis of a deductive argument. In the previous example this requirement was violated by the possible world account. In the analysis of the first paradox we should consider whether A → B is false in any possible world where ¬A is true. Why? Because if we could make random attributions of truth values based on different worlds, then any argumentative form would be rendered invalid. You could simply select a world where the conclusion of a valid argumentative form is false and use this information in another world where the premises of this argumentative form are true. That would be a crude form of modal cheating. But if the requirements of consistent truth value attribution and world evaluation are observed, all the counter-examples to classical argumentative forms involving the material implication are disarmed. !1 The fact that possible world enthusiasts openly ignore such concerns suggests that there is something amiss here. So either they are not really interested in mere logic issues or they have a different view of what logic should aim for. It is more likely the former. The discussions involving possible worlds are like science fiction. They are more imaginative and free-float than regular logical issues, which seem almost mundane in comparison. In most cases these imaginative discussions are oriented towards epistemological and metaphysical subjects that have nothing to do with the nature of logical consequence. If we already have determined consistency requirements for the truth conditions of conditional and unconditional sentences, it seems that there is nothing else to be said on the subject. But notice that in this case we decided to ignore the reasons that lead the arguer to assert the conditional and unconditional sentences. This is an erroneous proposition because a deductive argument is nothing more than an arguer's claim to validity. So it is obvious that her reasons to accept both the premises and the conclusion should be involved in the evaluation. Then the question of whether the premises of a given argument necessitate the conclusion or not will also depend on whether the reasons that support the premises together with the premises necessitate the conclusion or not. Consequently, a more accurate representation of argumentative forms will be more complex and require considerably more analysis of the arguers' reasons. !