Dynamic Probabilistic Entailment. Improving on Adams' Dynamic Entailment Relation

The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.

Axiomatization of Some Basic and Modal Boolean Connexive Logics

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses (i) a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and (ii) if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets.

Semantics Processing of Conditional Connectives: German wenn ‘if’ Versus nur wenn ‘only if’

In this paper, the meaning and processing of the German conditional connectives (CCs) such as wenn ‘if’ and nur wenn ‘only if’ are investigated. In Experiment 1, participants read short scenarios containing a conditional sentence (i.e., If P, Q.) with wenn/nur wenn ‘if/only if’ and a confirmed or negated antecedent (i.e., P/not-P), and subsequently completed the final sentence about Q (with or without negation). In Experiment 2, participants rated the truth or falsity of the consequent Q after reading a conditional sentence with wenn or nur wenn and a confirmed or negated antecedent (i.e., If P, Q. P/not-P. // Therefore, Q?). Both experiments showed that neither wenn nor nur wenn were interpreted as biconditional CCs. Modus Ponens (If P, Q. P. // Therefore, Q) was validated for wenn, whereas it was not validated in the case of nur wenn. While Denial of the Antecedent (If P, Q. not-P. // Therefore, not-Q.) was validated in the case of nur wenn, it was not validated for wenn. The same method was used to test wenn vs. unter der Bedingung, dass ‘on condition that’ in Experiment 3, and wenn vs. vorausgesetzt, dass ‘provided that’ in Experiment 4. Experiment 5, using Affirmation of the Consequent (If P, Q. Q. // Therefore, P.) to test wenn vs. nur wenn replicated the results of Experiment 2. Taken together, the results show that in German, unter der Bedingung, dass is the most likely candidate of biconditional CCs whereas all others are not biconditional. The findings, in particular of nur wenn not being semantically biconditional, are discussed based on available formal analyses of conditionals.

Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

On logical inference over brains, behaviour, and artificial neural networks

In the cognitive, computational, and neuro- sciences, we often reason about what models (viz., formal and/or computational) represent, learn, or "know", as well as what algorithm they instantiate. The putative goal of such reasoning is to generalize claims about the model in question to claims about the mind and brain. This reasoning process typically presents as inference about the representations, processes, or algorithms the human mind and brain instantiate. Such inference is often based on a model's performance on a task, and whether that performance approximates human behaviour or brain activity. The model in question is often an artificial neural network (ANN) model, though the problems we discuss are generalizable to all reasoning over models. Arguments typically take the form "the brain does what the ANN does because the ANN reproduced the pattern seen in brain activity" or "cognition works this way because the ANN learned to approximate task performance." Then, the argument concludes that models achieve this outcome by doing what people do or having the capacities people have. At first blush, this might appear as a form of modus ponens, a valid deductive logical inference rule. However, as we explain in this article, this is not the case, and thus, this form of argument eventually results in affirming the consequent – a logical or inferential fallacy. We discuss what this means broadly for research in cognitive science, neuroscience, and psychology; what it means for models when they lose the ability to mediate between theory and data in a meaningful way; and what this means for the logic, the metatheoretical calculus, our fields deploy in high-level scientific inference.

The Problem with Impure Infinitism

It is generally believed that pure versions of infinitism face two problems, namely: 1) they are unable to distinguish between potential and actual series of justified reasons because they are defined strictly in terms of relations between beliefs in the series so that every succeeding belief is justified by the belief before it and so on ad infinitum and, 2) they are unable to mark the difference between a set of justified reasons that are connected to truth and one that is not because they are defined strictly in terms of a relation between beliefs in the series of reasons. However, Aikin argues that impure infinitism could surmount these problems without undermining the infinite regress condition because impure infinitism can solve the Modus Ponens Reductio, MPR, argument that threatens pure versions of infinitism. I argue that Aikin does not succeed because his impure infinitism faces some fatal consequences and any attempt to salvage it will undermine the infinite regress of justification

On the weak intuitionistic fuzzy implication based on △ operation

In this paper, a new weak intuitionistic fuzzy implication is introduced. Fulfillment of some axioms and properties, together with the Modus Ponens and Modus Tollens inference rules are investigated. Negation induced by the newly proposed implication is presented.