1 Denying the antecedent probabilized: a dialectical view FRANK ZENKER Department of Philosophy and Cognitive Science Lund University, LUX, Box 192, 221 00 Lund Sweden frank.zenker@fil.lu.se ABSTRACT: This article provides an analysis and evaluation for probabilistic version of arguments that deny the antecedent (DA P ). Stressing the effects of premise retraction vs. premise subtraction in a dialectical setting, the cogency of DA P arguments is shown to depend on premises that normally remain implicit. The evaluation remains restricted to a Pascalian notion of probability, which is briefly compared to its Baconian variant. Moreover, DA P is presented as an exam-question plus evaluation that can be deployed as a learning assessment-instrument at graduate-level. KEYWORDS: affirming the consequent, delay tactic, denying the antecedent, dialectics, inductive logic, modus ponens, modus tollens, probabilistic independence, probabilistic relevance, retraction, subtraction 1. INTRODUCTION We treat the evaluation of DA P , a probabilistic version of what classical logic correctly treats as the formal fallacy of denying the antecedent (DA), i.e, the deductively invalid attempt at inferring the conclusion ~c from the premises a->c and ~a, where a stands for antecedent, c for consequent, and ~ for negation. Examples include: (1) Had my client been at the crime scene (a), then he would probably be guilty (c). But he wasn’t (~a), so he probably isn’t (~c). (2) If the lights are on (a), then probably someone’s at home (c). But the lights are out (~a), so probably no one is (~c). (3) If the product sells (a), then our marketing measures should probably be trusted (c). But it doesn’t (~a), so measures should be reviewed (~c). Here, (1) states a counterfactual conditional (“had”), (2) an indicative one (“are”), and that in (3) might even sustain a deontic reading (“should”). Disregarding such differences, we proceed to treat such DA P -arguments on the following schema, its formal version becoming clearer soon: (DA P ) If a then probably c. But not a, so probably not c. P f (c)=P i (c|a)>P i (c). But P i (a)=0, so P f (~c)>P i (c). As should be uncontroversial, if natural language instances of DA P instantiate a probabilistically valid inference, or argument, then only if the relevant probability values are right. A probabilistic version of modus ponens (MP P ) can be stated as the conditional probability of c given a, i.e., P(c|a), where P(c|a) directly depends on P(~c|a) whenever P(c|a)=1−P(~c|a) holds, which is the complement-relation of