scholarly journals Dynamic Probabilistic Entailment. Improving on Adams' Dynamic Entailment Relation

2021 ◽  
pp. 1-26
Author(s):  
Robert Van Rooij ◽  
Patricia Mirabile
Keyword(s):  
Original System ◽  
Modus Ponens ◽  
Behavioral Experiment ◽  
Truth Conditions ◽  
Indicative Conditionals ◽  
Indicative Conditional ◽  
Entailment Relation ◽  
New Information ◽  
General Relevance ◽  
Hypothetical Syllogism

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.

Tempered Pragmatism

2017 ◽  
Author(s):  
Ian Rumfitt
Keyword(s):  
Modus Ponens ◽  
Modus Tollens ◽  
Truth Conditions ◽  
Indicative Conditionals ◽  
Logical Basis ◽  
Theory Of Meaning

This chapter assesses the prospects of a pragmatist theory of content. It begins by criticizing the theory presented in D. H. Mellor’s essay ‘Successful Semantics’, then identifies problems and lacunae in the pragmatist theory of meaning sketched in chapter 13 of Dummett’s The Logical Basis of Metaphysics. It contends that the prospects are brighter for a tempered pragmatism, in which the theory of content is permitted to draw upon irreducible notions of truth and falsity. It sketches the shape of such a theory and illustrates the role of its pragmatist elements by showing how they point towards a promising account of the truth conditions of indicative conditionals. A feature of the account is that it validates Modus Ponens whilst invalidating Modus Tollens.

scholarly journals On Basic Probability Logic Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1409
Author(s):  
Marija Boričić Joksimović
Keyword(s):  
Probability Theory ◽  
Inference Rule ◽  
Modus Ponens ◽  
Probability Logic ◽  
Modus Tollens ◽  
Short Overview ◽  
Basic Probability ◽  
Probabilistic Version ◽  
Hypothetical Syllogism

We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, and B→C have probabilities a, b, c, r, and s, respectively, then for probability p of A→C, we have f(a,b,c,r,s)≤p≤g(a,b,c,r,s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner.

Conditionals and Abduction

2020 ◽  
pp. 161-166
Author(s):  
Timothy Williamson
Keyword(s):  
Past Tense ◽  
Indicative Conditionals ◽  
Cognitive Differences ◽  
Indicative Conditional ◽  
Counterfactual Conditional ◽  
Counterfactual Conditionals ◽  
The Past ◽  
Testing Of Hypotheses ◽  
New Evidence

The chapter gives a preliminary sketch of some cognitive differences between indicative conditionals and counterfactual conditionals relevant to the testing of hypotheses by experiment. They especially concern cases where the indicative conditional can be decided without new evidence while the counterfactual conditional cannot. They also show that the antecedent of a ‘counterfactual’ conditional need not be presupposed to be false. Differences connected with the past tense morphology of ‘would’ are explored. Cases are given where the morphology should be understood as expressing a ‘fake past’, modal rather than temporal.

Conditional Acceptance

2019 ◽  
pp. 99-121
Author(s):  
Ofra Magidor
Keyword(s):  
Indicative Conditionals ◽  
Indicative Conditional ◽  
Conditional Acceptance ◽  
Conditional Claim ◽  
Suppositional View

What is the correct semantics for indicative conditionals, and under what circumstances should agents accept a conditional claim? This paper presents a new case which has important implications for attempts to address these questions. The case involves an utterance of a certain indicative conditional in a particular context. It is shown that at least three prominent theories of conditionals (the material conditional view, the suppositional view, and Stalnaker’s view) predict that you ought to assign a high credence to the conditional in this case, but, it is argued, this prediction is incorrect. Finally, the paper discusses what conclusions we can draw from this case, both on the semantics of conditionals and on the epistemology of inference on the basis of suppositions more generally.

Learning Conditional Evidence

2019 ◽  
pp. 107-130
Author(s):  
Jan Sprenger ◽  
Stephan Hartmann
Keyword(s):  
Probability Distribution ◽  
Bayesian Network ◽  
Conditional Independence ◽  
Degrees Of Belief ◽  
Indicative Conditionals ◽  
Indicative Conditional ◽  
Posterior Probability Distribution ◽  
Judy Benjamin Problem ◽  
Bayesian Conditionalization ◽  
Conditional Evidence

Learning indicative conditionals and learning relative frequencies have one thing in common: they are examples of conditional evidence, that is, evidence that includes a suppositional element. Standard Bayesian theory does not describe how such evidence affects rational degrees of belief, and natural solutions run into major problems. We propose that conditional evidence is best modeled by a combination of two strategies: First, by generalizing Bayesian Conditionalization to minimizing an appropriate divergence between prior and posterior probability distribution. Second, by representing the relevant causal relations and the implied conditional independence relations in a Bayesian network that constrains both prior and posterior. We show that this approach solves several well-known puzzles about learning conditional evidence (e.g., the notorious Judy Benjamin problem) and that learning an indicative conditional can often be described adequately by conditionalizing on the associated material conditional.

scholarly journals De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

2021 ◽  
Author(s):  
Paul Égré ◽  
Lorenzo Rossi ◽  
Jan Sprenger
Keyword(s):  
Proof Theory ◽  
Future Research ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Truth Conditions ◽  
Indicative Conditionals ◽  
Tarski Algebra ◽  
Soundness And Completeness ◽  
Tableau Calculi ◽  
The One

AbstractIn Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1935, 1944) on the one hand, and by Cooper (Inquiry, 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic, 49, 245–260, 2008) on the other. Here we provide the proof theory for the resulting logics and , using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: allows for algebraic completeness, but not for the construction of a canonical model, while fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.

scholarly journals The semantics of indicative conditionals: A counterexample to the horseshoe analysis

2012 ◽  
Vol 55 (2) ◽  
pp. 53-62
Author(s):  
Dusko Prelevic
Keyword(s):  
Thought Experiment ◽  
Two Dimensional ◽  
Truth Conditions ◽  
Indicative Conditionals ◽  
Material Implication ◽  
Inverted Spectrum

The paper examines a counterexample of the horseshoe analysis of indicative conditionals (according to which indicative conditionals ??? have the same truth-conditions as the material implication ???). The example is a modified and improved version of Jason Decker?s ?playground conditionals? case. The paper aims to show why Decker?s original example is wrong, and how it can be improved by using the inverted spectrum thought experiment. It is also shown in this paper that playground conditionals do not pose any problems to the epistemic version of the two-dimensional semantics (E2-D), which leads to the conclusion that we should prefer E2-D to the horseshoe analysis of indicative conditionals. [Projekat Ministarstva nauke Republike Srbije, br. 179067: Logicko-epistemoloski osnovi nauke i metafizike].

The Cost of Structure Learning

2017 ◽  
Vol 29 (10) ◽  
pp. 1646-1655 ◽  
Author(s):  
Anne G. E. Collins
Keyword(s):  
Computational Models ◽  
Structure Learning ◽  
Behavioral Experiment ◽  
Learning Problem ◽  
Short Term ◽  
New Information ◽  
Transfer Rules ◽  
Learning Flexibility ◽  
The Cost

Human learning is highly efficient and flexible. A key contributor to this learning flexibility is our ability to generalize new information across contexts that we know require the same behavior and to transfer rules to new contexts we encounter. To do this, we structure the information we learn and represent it hierarchically as abstract, context-dependent rules that constrain lower-level stimulus–action–outcome contingencies. Previous research showed that humans create such structure even when it is not needed, presumably because it usually affords long-term generalization benefits. However, computational models predict that creating structure is costly, with slower learning and slower RTs. We tested this prediction in a new behavioral experiment. Participants learned to select correct actions for four visual patterns, in a setting that either afforded (but did not promote) structure learning or enforced nonhierarchical learning, while controlling for the difficulty of the learning problem. Results replicated our previous finding that healthy young adults create structure even when unneeded and that this structure affords later generalization. Furthermore, they supported our prediction that structure learning incurred a major learning cost and that this cost was specifically tied to the effort in selecting abstract rules, leading to more errors when applying those rules. These findings confirm our theory that humans pay a high short-term cost in learning structure to enable longer-term benefits in learning flexibility.

Counterfactuals and Probability

2019 ◽  
pp. 182-202
Author(s):  
Robert C. Stalnaker
Keyword(s):  
Conditional Probability ◽  
Truth Conditions ◽  
Indicative Conditional ◽  
Truth Values ◽  
Counterfactual Conditionals ◽  
Retrospective Assessment ◽  
Objective Chance ◽  
The Relationship

This chapter continues the attempt, begun in chapter 10, to reconcile the thesis that conditionals have truth conditions with accounts such as Dorothy Edgington’s that aim to explain conditionals as expressing a distinctive kind of attitude represented by conditional probability. This time the focus is on subjunctive or counterfactual conditionals. It is argued that the propositional analysis helps to explain the cases, emphasized by Edgington, where counterfactual statements seem to be retrospective assessment of what was earlier said with an indicative conditional. It is also argued that the propositional analysis can allow for cases where counterfactuals have probability values but not truth-values, and more generally that it can help to explain the relationship between counterfactuals and objective chance.

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