ABSTRACTThe probabilistic truth table task involves assessing the probability of "If A then C" conditional sentences. Previous studies have shown that a majority of participants assess this probability as the conditional probability P while a substantial minority responds with the probability of the conjunction A and C. In an experiment involving 96 participants, we investigated the impact on the rate of conjunctive responses of the context in which the task is framed. We show that a (...) context intended to lead participants to consider all the possible cases elicited more conjunctive responses than a context assumed not to have this effect. These results suggest that the step of inferring the probability can distort our assessment of participants' interpretation of conditional sentences. This might compromise the validity of the probabilistic task in studying conditional reasoning. (shrink)

We study abductive, causal, and non-causal conditionals in indicative and counterfactual formulations using probabilistic truth table tasks under incomplete probabilistic knowledge (N = 80). We frame the task as a probability-logical inference problem. The most frequently observed response type across all conditions was a class of conditional event interpretations of conditionals; it was followed by conjunction interpretations. An interesting minority of participants neglected some of the relevant imprecision involved in the premises when inferring lower or (...) upper probability bounds on the target conditional/counterfactual ("halfway responses"). We discuss the results in the light of coherence-based probability logic and the new paradigm psychology of reasoning. (shrink)

Two types of truth table tasks are used investigating mental representations of conditionals: a possibilities-based and a truth-based one. In possibilities tasks, participants indicate whether a situation is possible or impossible according to the conditional rule. In truth tasks participants evaluate whether a situation makes the rule true or false, or is irrelevant with respect to the truth of the rule. Comparing the two-option version of the possibilities task with the truth task (...) in Experiment 1, the possibilities task yields logical answer patterns whereas the truth task yields defective patterns. Adding the irrelevant option to the possibilities task in Experiment 2 leads to a considerable amount of defective patterns in the possibilities task, but still to more logical patterns in the possibilities task than in the truth task. Experiment 3 shows that directionality matters since rule-to-situation tasks yield more logical answer patterns than do situation-to-rule tasks. We conclude that both task types are not comparable as such since wording, number of options and directionality influence the results. (shrink)

Two types of truth table task are used to examine people's mental representation of conditionals. In two within-participants experiments, participants either receive the same task-type twice (Experiment 1) or are presented successively with both a possibilities task and a truth task (Experiment 2). Experiment 3 examines how people interpret the three-option possibilities task and whether they have a clear understanding of it. The present study aims to examine, for both task-types, how (...) participants' cognitive ability relates to the classification of the truth table cases as irrelevant, and their consistency in doing so. Looking at the answer patterns, participants' cognitive ability influences their classification of the truth table cases: A positive correlation exists between cognitive ability and the number of false-antecedent cases classified as ?irrelevant?, both in the possibilities task and the truth task. This favours a suppositional representation of conditionals. (shrink)

Probabilistic models have started to replace classical logic as the standard reference paradigm in human deductive reasoning. Mental probability logic emphasizes general principles where human reasoning deviates from classical logic, but agrees with a probabilistic approach (like nonmonotonicity or the conditional event interpretation of conditionals). -/- This contribution consists of two parts. In the first part we discuss general features of reasoning systems including consequence relations, how uncertainty may enter argument forms, probability intervals, and probabilistic informativeness. These (...) concepts are of central importance for the psychological task analysis. In the second part we report new experimental data on the paradoxes of the material conditional, the probabilistic modus ponens, the complement task, and data on the probabilistic truth table task. The results of the experiments provide evidence for the hypothesis that people represent indicative conditionals by conditional probability assertions. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a (...) class='Hi'>truth-table task, and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

Iterated conditionals of the form If p, then if q, r are an important topic in philosophical logic. In recent years, psychologists have gained much knowledge about how people understand simple conditionals, but there are virtually no published psychological studies of iterated conditionals. This paper presents experimental evidence from a study comparing the iterated form, If p, then if q, r with the “imported,” noniterated form, If p and q, then r, using a probability evaluation task and a (...) class='Hi'>truth‐table task, and taking into account qualitative individual differences. This allows us to critically contrast philosophical and psychological approaches that make diverging predictions regarding the interpretation of these forms. The results strongly support the probabilistic Adams conditional and the “new paradigm” that takes this conditional as a starting point. (shrink)

We investigated how people interpret conditionals and how stable their interpretation is over a long series of trials. Participants were shown the colored patterns on each side of a six-sided die, and were asked how sure they were that a conditional holds of the side landing upwards when the die is randomly thrown. Participants were presented with 71 trials consisting of all combinations of binary dimensions of shape (e.g., circles and squares) and color (e.g., blue and red) painted onto the (...) sides of each die. In two experiments (N1 = 66, N2 = 65), the conditional event was the dominant interpretation, followed by conjunction, and material conditional responses were negligible. In both experiments, the percentage of participants giving a conditional event response increased from around 40% at the beginning of the task to nearly 80% at the end, with most participants shifting from a conjunction interpretation. The shift was moderated by the order of shape and color in each conditional’s antecedent and consequent: participants were more likely to shift if the antecedent referred to a color. In Experiment 2 we collected response times: conditional event interpretations took longer to process than conjunction interpretations (mean difference 500 ms). We discuss implications of our results for mental models theory and probabilistic theories of reasoning. (shrink)

Two experiments were conducted to investigate the roles of covariation and of causality in people's readiness to believe a conditional. The experiments used a probabilistic truth-table task (Oberauer & Wilhelm, 2003) in which people estimated the probability of a conditional given information about the frequency distribution of truth-table cases. For one group of people, belief in the conditional was determined by the conditional probability of the consequent, given the antecedent, whereas for another group it (...) depended on the probability of the conjunction of antecedent and consequent. There was little evidence that covariation, expressed as the probabilistic contrast or as the pCI rule (White, 2003), influences belief in the conditional. The explicit presence of a causal link between antecedent and consequent in a context story had a weak positive effect on belief in a conditional when the frequency distribution of relevant cases was held constant. (shrink)

In this paper, a novel experimental task is developed for testing the highly influential, but experimentally underexplored, possible worlds account of conditionals (Stalnaker, 1968; Lewis, 1973). In Experiment 1, this new task is used to test both indicative and subjunctive conditionals. For indicative conditionals, five competing truth tables are compared, including the previously untested, multi-dimensional possible worlds semantics of Bradley (2012). In Experiment 2, these results are replicated and it is shown that they cannot be accounted for (...) by an alternative hypothesis proposed by our reviewers. In Experiment 3, individual variation in truth assignments of indicative conditionals is investigated via Bayesian mixture models that classify participants as following one of several competing models. As a novelty of this study, it is found that a possible worlds semantics of Lewis and Stalnaker is capable of accounting for participants’ aggregate truth value assignments in this task. Applied to indicative conditionals, we show across three experiments, that the theory both captures participants’ truth values at the aggregate level (Experiments 1 and 2) and that it makes up the largest subgroup in the analysis of individual variation in our experimental paradigm (Experiment 3). (shrink)

Conditionals are central to inference. Before people can draw inferences about a natural language conditional, they must interpret its meaning. We investigated interpretation of uncertain conditionals using a probabilistic truth table task, focussing on (i) conditional event, (ii) material conditional, and (iii) conjunction interpretations. The order of object (shape) and feature (color) in each conditional's antecedent and consequent was varied between participants. The conditional event was the dominant interpretation, followed by conjunction, and took longer to process (...) than conjunction (mean di erence 500 ms). Material conditional responses were rare. The proportion of conditional event responses increased from around 40% at the beginning of the task to nearly 80% at the end, with 55% of participants showing a qualitative shift of interpretation. Shifts to the conditional event occurred later in the feature-object order than in the object-feature order. We discuss the results in terms of insight and suggest implications for theories of interpretation. (shrink)

There is ample evidence that in classical truth table task experiments false antecedents are judged as ?irrelevant?. Instead of interpreting this in support of a suppositional representation of conditionals, Schroyens (2010a, 2010b) attributes it to the induction problem: the impossibility of establishing the truth of a universal claim on the basis of a single case. In the first experiment a truth table task with four options is administered and the correlation with intelligence is (...) inspected. It is observed that ?undetermined? is chosen in one third of the judgements and ?irrelevant? in another third. A positive correlation is revealed between intelligence and the number of ?irrelevant? and ?undetermined? judgements. The data do not exclude that a part of the ?irrelevant? judgements in classical truth table task experiments might be caused by the induction problem. In the second experiment participants are presented with a simplified four-option truth table task and asked for a justification of their judgements. These justifications show the induction problem is not the reason for choosing the ?irrelevant? or ?undetermined? option, which is supportive for a suppositional representation of conditionals. (shrink)

Conditionals are central to inference. Before people can draw inferences about a natural language conditional, they must interpret its meaning. We investigated interpretation of uncertain conditionals using a probabilistic truth table task, focussing on (i) conditional event, (ii) material conditional, and (iii) conjunction interpretations. The order of object (shape) and feature (color) in each conditional's antecedent and consequent was varied between participants. The conditional event was the dominant interpretation, followed by conjunction, and took longer to process (...) than conjunction (mean di erence 500 ms). Material conditional responses were rare. The proportion of conditional event responses increased from around 40% at the beginning of the task to nearly 80% at the end, with 55% of participants showing a qualitative shift of interpretation. Shifts to the conditional event occurred later in the feature-object order than in the object-feature order. We discuss the results in terms of insight and suggest implications for theories of interpretation. (shrink)

ABSTRACTTo build a process model of the understanding of conditionals we extract a common core of three semantics of if-then sentences: the conditional event interpretation in the coherencebased probability logic, the discourse processingtheory of Hans Kamp, and the game-theoretical approach of Jaakko Hintikka. The empirical part reports three experiments in which each participant assessed the probability of 52 if-then sentencesin a truth table task. Each experiment included a second task: An n-back task relating the interpretation (...) of conditionals to working memory, a Bayesian bookbag and poker chip task relating the interpretation of conditionals to probability updating, and a probabilistic modus ponens task relating the interpretation of conditionals to a classical inference task. Data analysis shows that the way in which the conditionals are interpreted correlates with each of the supplementary tasks. The results are discussed within the process model proposed in the introduction. (shrink)

This dissertation is devoted to empirically contrasting the Suppositional Theory of conditionals, which holds that indicative conditionals serve the purpose of engaging in hypothetical thought, and Inferentialism, which holds that indicative conditionals express reason relations. Throughout a series of experiments, probabilistic and truth-conditional variants of Inferentialism are investigated using new stimulus materials, which manipulate previously overlooked relevance conditions. These studies are some of the first published studies to directly investigate the central claims of Inferentialism empirically. In contrast, the (...) Suppositional Theory of conditionals has an impressive track record through more than a decade of intensive testing. The evidence for the Suppositional Theory encompasses three sources. Firstly, direct investigations of the probability of indicative conditionals, which substantiate “the Equation” (P(if A, then C) = P(C|A)). Secondly, the pattern of results known as “the defective truth table” effect, which corroborates the de Finetti truth table. And thirdly, indirect evidence from the uncertain and-to-if inference task. Through four studies each of these sources of evidence are scrutinized anew under the application of novel stimulus materials that factorially combine all permutations of prior and relevance levels of two conjoined sentences. The results indicate that the Equation only holds under positive relevance (P(C|A) – P(C|¬A) > 0) for indicative conditionals. In the case of irrelevance (P(C|A) – P(C|¬A) = 0), or negative relevance (P(C|A) – P(C|¬A) < 0), the strong relationship between P(if A, then C) and P(C|A) is disrupted. This finding suggests that participants tend to view natural language conditionals as defective under irrelevance and negative relevance (Chapter 2). Furthermore, most of the participants turn out only to be probabilistically coherent above chance levels for the uncertain and-to-if inference in the positive relevance condition, when applying the Equation (Chapter 3). Finally, the results on the truth table task indicate that the de Finetti truth table is at most descriptive for about a third of the participants (Chapter 4). Conversely, strong evidence for a probabilistic implementation of Inferentialism could be obtained from assessments of P(if A, then C) across relevance levels (Chapter 2) and the participants’ performance on the uncertain-and-to-if inference task (Chapter 3). Yet the results from the truth table task suggest that these findings could not be extended to truth-conditional Inferentialism (Chapter 4). On the contrary, strong dissociations could be found between the presence of an effect of the reason relation reading on the probability and acceptability evaluations of indicative conditionals (and connate sentences), and the lack of an effect of the reason relation reading on the truth evaluation of the same sentences. A bird’s eye view on these surprising results is taken in the final chapter and it is discussed which perspectives these results open up for future research. (shrink)

In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle A, with probability one, the degree of the one-query tautologies with respect to A is strictly higher than the degree of A.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to (...) the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let c be a positive real number. We consider a concept of truth-table reduction whose norm is at most c times size of input, where for a relativized propositional formula F, the size of F denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that c is small enough. How small c can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all c < 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65–76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes. (shrink)

Using a latent variable modelling strategy we study individual differences in patterns of answers to the selection task and to the truth table task. Specifically we investigate the prediction of mental model theory according to which the individual tendency to select the false consequent card (in the selection task) is negatively correlated with the tendency to judge the false antecedent cases as irrelevant (in the truth table task). We fit a psychometric model (...) to two large samples ( N = 486, twice), and find no evidence for this negative correlation. We examine which of the assumptions of the model theory must be amended to accommodate our findings. (shrink)

Psychological research on people’s understanding of natural language connectives has traditionally used truth table tasks, in which participants evaluate the truth or falsity of a compound sentence given the truth or falsity of its components in the framework of propositional logic. One perplexing result concerned the indicative conditional if A then C which was often evaluated as true when A and C are true, false when A is true and C is false but irrelevant“ (devoid of (...) value) when A is false (whatever the value of C). This was called the “psychological defective table of the conditional.” Here we show that far from being anomalous the “defective” table pattern reveals a coherent semantics for the basic connectives of natural language in a trivalent framework. This was done by establishing participants’ truth tables for negation, conjunction, disjunction, conditional, and biconditional, when they were presented with statements that could be certainly true, certainly false, or neither. We review systems of three-valued tables from logic, linguistics, foundations of quantum mechanics, philosophical logic, and artificial intelligence, to see whether one of these systems adequately describes people’s interpretations of natural language connectives. We find that de Finetti’s (1936/1995) three-valued system is the best approximation to participants’ truth tables. (shrink)

In this study we investigate the influence of reason-relation readings of indicative conditionals and ‘and’/‘but’/‘therefore’ sentences on various cognitive assessments. According to the Frege-Grice tradition, a dissociation is expected. Specifically, differences in the reason-relation reading of these sentences should affect participants’ evaluations of their acceptability but not of their truth value. In two experiments we tested this assumption by introducing a relevance manipulation into the truth-table task as well as in other tasks assessing the participants’ acceptability (...) and probability evaluations. Across the two experiments a strong dissociation was found. The reason-relation reading of all four sentences strongly affected their probability and acceptability evaluations, but hardly affected their respective truth evaluations. Implications of this result for recent work on indicative conditionals are discussed. (shrink)

Given a basic conditional of the form, If A then C, individuals usually list three cases as possible: A and C, not‐A and not‐C, not‐A and C. This result corroborates the theory of mental models. By contrast, individuals often judge that the conditional is true only in the case of A and C, and that cases of not‐A are irrelevant to its truth or falsity. This result corroborates other theories of conditionals. To resolve the discrepancy, we devised two new (...) tasks: the “collective” truth task, in which participants judged whether sets of assertions about a specific individual, such as: If A then C, not‐A, C, could all be true at the same time; and one in which participants judged the truth of conditional predictions about specific future events. The results consistently matched the three possibilities, thereby corroborating the model theory. They also showed a massive violation of the probability calculus in estimates of the probabilities of the four cases in the partition of conditionals (A and C, A and not‐C, not‐A and C, and not‐A and not‐C), which summed to over 200%. (shrink)

Four experiments investigated the effects of probability manipulations on the indicative four card selection task (Wason, 1966, 1968). All looked at the effects of high and low probability antecedents (p) and consequents (q) on participants' data selections when determining the truth or falsity of a conditional rule, if p then q . Experiments 1 and 2 also manipulated believability. In Experiment 1, 128 participants performed the task using rules with varied contents pretested for probability of occurrence. (...) class='Hi'>Probabilistic effects were observed which were partly consistent with some probabilistic accounts but not with non-probabilistic approaches to selection task performance. No effects of believability were observed, a finding replicated in Experiment 2 which used 80 participants with standardised and familiar contents. Some effects in this experiment appeared inconsistent with existing probabilistic approaches. To avoid possible effects of content, Experiments 3 (48 participants) and 4 (20 participants) used abstract material. Both experiments revealed probabilistic effects. In the Discussion we examine the compatibility of these results with the various models of selection task performance. (shrink)

This work is in two parts. The main aim of part 1 is a systematic examination of deductive, probabilistic, inductive and purely inductive dependence relations within the framework of Kolmogorov probability semantics. The main aim of part 2 is a systematic comparison of (in all) 20 different relations of probabilistic (in)dependence within the framework of Popper probability semantics (for Kolmogorov probability semantics does not allow such a comparison). Added to this comparison is an examination of (in all) 15 (...) purely inductive dependence relations. ————Part 1 leads in an axiomatic step-by-step development from the elementary classical truth value semantics of a sentential-logical language, called ‘L’, (chapter 1) to the elementary Kolmogorov probability semantics of L (chapter 2), which is then extended to four axiomatic semantical theories of dependence relations between the formulae of L. First the elementary Kolmogorov probability semantics of L is extended to a theory, called ‘Kdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 3). Then Kdd is extended to a theory, called ‘Kpd1’, of the degree to which formulae of L probabilistically depend on each other in regard to a given probability distribution on the set of all formulae of L (chapter 4). Kpd1, in its turn, gets extended to a theory, called ‘Kpd2’, of the relations of probabilistic dependence and independence, relativized to unary Kolmogorov probability functions defined on L (chapter 5). Then Kpd2 is extended to a theory, called ‘Kid’, of the relations of inductive dependence and inductive independence, again relativized to unary Kolmogorov probability functions defined on L (chapter 6). Finally, Kid is extended to a theory, called ‘Kpid’, of the relations of purely inductive positive and negative dependence, relativized to unary Kolmogorov probability functions defined on L (chapter 7). ——Chapter 1, which deals with the familiar notions of truth value functions, tautologies, consequence relations and relations of logical opposition, is naturally the shortest chapter of part 1.——In chapter 2, the elementary classical semantics of L is extended to the elementary Kolmogorov probability semantics of L, i.e. to an axiomatic theory of unary and of binary Kolmogorov probability functions defined on the set of formulae of L. Because of the elementary character of this theory, chapter 2 is also rather short.——Chapter 3 introduces the first theory on dependence relations, to wit: Kdd, the theory of deductive (in)dependence between formulae of L. I follow here the well-known idea of Popper and Miller, who have used it in a famous discussion on the nature of probabilistic support for their arguments that probabilistic support is deductive, not inductive. I develop Kdd in the form of about 100 theorems, making ample use of the fact that deductive independence is nothing but subcontrary opposition, and close with a remark on the fundamental difference between deductive and logical dependence—two relations the ideas of which are all too easily mixed up.——Chapters 4 and 5 deal extensively with the traditional ideas of probabilistic (in)dependence, applied to formulae rather than to events. As always, I proceed axiomatically in a step-by-step process under systematic viewpoints and obtain about 300 theorems in this way. In the formulation of the theorems, I took special care to state clearly and expressly so-called tacit assumptions, especially those concerning the probability values of the formulae said to be dependent on each other. These assumptions are usually missing in the literature, due either to economy of writing or to sloppiness of thinking. Presumably, both chapters contain little that is new, their value lying more in the systematic grouping and organic development of the theorems than in the newness of these.——In chapter 6, I extend the axiomatic theory about probabilistic (in)dependence which has been elaborated in chapter 5, to an axiomatic theory of inductive (in)dependence by requiring of the relation of inductive (in)dependence that it be probabilistic (in)dependence, but not also logical implication or logical opposition. I point out the differences between probabilistic and inductive (in)dependence by means of some 60 theorems and close my examination of inductive (in)dependence by considering its relationship to the notion of support in the philosophy of science.——Finally, in chapter 7, the last of part 1, I take the step from inductive dependence to what I call ‘purely inductive dependence’ by combining the idea of inductive dependence with that of deductive independence in a way which is suggested by writings of Popper and Miller. I arrive at two noteworthy theorems. Firstly, there is indeed no purely inductive support. But secondly, and perhaps amazingly, countersupport is purely inductive.————Whereas the probabilistic framework of part 1 of the present work is Kolmogorov probability semantics, the framework of part 2 is Popper probability semantics, which is not only worth examining as a fascinating alternative to orthodox Kolmogorov probability semantics, but also allows us to examine dependence relations more deeply, than Kolmogorov probability semantics does. Part 2 leads—again in an axiomatic step-by-step development—from the basic Popper probability semantics of L, called ‘Pb’, (chapter 8) via a probabilistic theory of logical attributes, called ‘Ps’, (chapter 9) to four axiomatic semantical theories of dependence relations between the formulae of L. First, Ps is extended to a theory, called ‘Pdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 10). Then Pdd is extended to a theory, called ‘Ppd’, of (in all) 20 relations of probabilistic (in)dependence, relativized to binary Popper probability functions defined on L (chapter 11). Ppd, in its turn, is extended to a theory, called ‘Pid’, of (in all) 10 relations of inductive dependence, again relativized to binary Popper probability functions defined on L (first part of chapter 12). Finally, Pid is extended to a theory, called ‘Ppid’, of (in all) 15 relations of purely inductive positive and negative dependence, relativized to binary Popper probability functions defined on L (second part of chapter 12).——Chapter 8, the first chapter of part 2 of the present work, is entirely preparatory. It introduces the axioms and about 180 theorems (150 of them together with their proofs) of basic Popper probability semantics in order to set this kind of semantics under way.——Then, in chapter 9, basic Popper probability semantics is extended to a probabilistic theory of logical properties of and relations between the formulae of L. Although I think that the way I did this extension is of some interest in itself, the main task of chapter 9 is again a preparatory one: to yield the indispensable lemmata (about 90 in number) for the theorems concerning probabilistic dependence relations in chapter 11 and concerning inductive dependence relations in chapter 12.——Chapter 10 brings the extension of Ps to the theory Pdd of deductive (in)dependence. Only half a dozen theorems are noted here for later use in the Pdd-extensions Ppd and Ppid. In view of the over 100 theorems already gained on this topic in the Kolmogorovian framework (cf. chapter 3), a similar extensive elaboration of Pdd would have been superfluous.——Chapter 11 is the most important one of part 2. It consists of a systematic comparison of 20 probabilistic (in)dependence concepts by means of about 230 theorems, obtained within the axiomatic theory Ppd, which is built up as an extension of Pdd. The main points of comparison were: differences in logical strength; reflexivity and symmetry; behaviour under the condition that the probability values of the formulae in question are extreme. It turned out that each of the examined concepts violates a strong and straightforward version of the intuitive requirement that probabilistic dependence should go with logical dependence. Whereas the corresponding chapter 5 in part 1 of the present work may not have led to new theorems, chapter 11 yields dozens of them in the process of comparison of concepts of dependence and independence which had—as far as I know—never before been treated in a single theoretical framework. With Popper probability semantics, this framework has become available, and here I have simply made full use of it.——In chapter 12, I extend the theory Ppd of probabilistic (in)dependence to the theories Pid and Ppid of inductive and purely inductive dependence, in a way very similar to that in which I have extended the theory Kpd2 to the theories Kid and Kpid in chapters 6 and 7. The first main result of Kpid (roughly: there is no purely inductive support) could be repeated for four of the five purely inductive positive dependence relations considered in chapter 12, whereas the second main result of Kpid (roughly: there is purely inductive countersupport ) could be repeated for each of the five examined purely inductive negative dependence relations. Chapter 12 closes with a brief recapitulation and critical discussion of the main results. (shrink)

Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of mathematical claims. In (...) this paper, I argue that none of the epistemic objectives of mathematicians that are currently on the table provide a satisfactory explanation of this rejection of probabilistic proofs. (shrink)

Changing the information format from probabilities into frequencies as well as employing appropriate visualizations such as tree diagrams or 2 × 2 tables are important tools that can facilitate people’s statistical reasoning. Previous studies have shown that despite their widespread use in statistical textbooks, both of those visualization types are only of restricted help when they are provided with probabilities, but that they can foster insight when presented with frequencies instead. In the present study, we attempt to replicate this effect (...) and also examine, by the method of eye tracking, why probabilistic 2 × 2 tables and tree diagrams do not facilitate reasoning with regard to Bayesian inferences (i.e., determining what errors occur and whether they can be explained by scan paths), and why the same visualizations are of great help to an individual when they are combined with frequencies. All ten inferences of N = 24 participants were based solely on tree diagrams or 2 × 2 tables that presented either the famous “mammography context” or an “economics context” (without additional textual wording). We first asked participants for marginal, conjoint, and (non-inverted) conditional probabilities (or frequencies), followed by related Bayesian tasks. While solution rates were higher for natural frequency questions as compared to probability versions, eye-tracking analyses indeed yielded noticeable differences regarding eye movements between correct and incorrect solutions. For instance, heat maps (aggregated scan paths) of distinct results differed remarkably, thereby making correct and faulty strategies visible in the line of theoretical classifications. Moreover, the inherent structure of 2 × 2 tables seems to help participants avoid certain Bayesian mistakes (e.g., “Fisherian” error) while tree diagrams seem to help steer them away from others (e.g., “joint occurrence”). We will discuss resulting educational consequences at the end of the paper. (shrink)

Over the past decade, sequence learning has gradually become a central paradigm through which to study implicit learning. In this chapter, we start by briefly summarizing the results obtained with different variants of the sequence learning paradigm. We distinguish three subparadigms in terms of whether the stimulus material is generated either by following a fixed and repeating sequence (e.g., Nissen & Bullemer, 1987), by relying on a complex set of rules from which one can produce several alternative deterministic sequences (e.g., (...) Lewicki, Hill & Bizot, 1988; Stadler, 1989), or by following the output of a probabilistic set of rules such as instantiated by noisy finite-state grammars (Cleeremans & McClelland, 1991; Jiménez, Mendéz & Cleeremans, 1996). Next, we focus on the processes involved in sequence representation and acquisition. We suggest that the sensitivity to the sequential structure observed in the probabilistic subparadigm can only be a result of the acquisition of a representation of the statistical constraints of the material, and that this sensitivity emerges through the operation of mechanisms that are well instantiated by connectionist models such as the Simple Recurrent Network (Elman, 1990; Cleeremans, 1993b). We present new simulation work meant to explore to what extent the model can also account for specific data obtained in a paradigmatic instance of deterministic, rule-based sequence learning task: Lewicki et al. (1988)'s situation. Finally, we report on the results of an experiment that compares learning on otherwise similar deterministic and probabilistic structures, and we show that learning of both types of structures is equivalent only under conditions that maximally hinder explicit acquisition. Taken together, these simulation and experimental data lend support to the claim that implicit learning in all three sequence learning subparadigms can amount to a form of statistical sequence learning. They also suggest that distinguishing among several theories of sequence representation and acquisition may require us to analize the data in great detail. Hopefully, however, some truth can be found in such details.. (shrink)

ABSTRACTStudies examining the interpretation that is given to if–then statementstypically use what are referred to as basic conditionals, which give contextless relations between two unrelated concrete terms. However, there is some evidence that basic conditionals require a more abstract form of representation. In order to examine this, we presented participants with truth-table tasks involving either basic conditionals or conditionals referring to imaginary categories, and standard conditional inference tasks with abstract and familiar premises. As expected, fewer typical defective conditional (...) interpretations were given to basic conditionals. In addition, partial correlations showed a unique relationship between the interpretation of basic conditionals and abstract inferential reasoning. Results suggest that people process basic conditionals as a form of abstract reasoning, and that the interpretation of conditionals... (shrink)

Matching bias in conditional reasoning consists of a tendency to select as relevant cases whose lexical content matches that referred to in the conditional statement, regardless of the presence of negatives. Evans demonstrated that use of explicit rather than implicit negative cases markedly reduced the matching bias effect on the conditional truth table task. In apparent contrast, recent studies of explicit negation on the Wason selection task have failed to find evidence of logical facilitation. Experiment 1 (...) of the present study strongly replicated the Evans findings and extended them to three forms of conditional statement. Experiments 2 and 3 showed further that the use of explicit negatives removed completely the matching bias effect on the Wason selection task. However, consistent with other recent studies, this elimination of bias didnotlead to facilitation of correct responding. The findings are interpreted as providing evidence that matching bias reflects a linguistically cued relevance effect. (shrink)

The knowledge norm of assertion is mainly in competition with a high probability or rational credibility norm. The argument for the knowledge norm that I offer turns on cases in which a hearer responds to a speaker's assertion by asserting another sentence that would lower the probability of the speaker's assertion, were its probability less than one. In cases like this, though with qualifications, is the hearer's contribution a challenge to the speaker's assertion or complementary to it? My answer is (...) the latter, and that only the knowledge norm yields that answer.The cases that I rely on follow from an elementary probability relation, though one that is inconsistent with the still influential relevance criterion for confirmation and evidence . Assume, for illustrative purposes, that p is in your belief corpus, as a consequence of your believing ∨ ∨ . 1 You learn that ∼ & ∼ , which would lessen the probability of p, were its probability less than one, 2 since it eliminates two rows of the truth-table in which p holds. What conclusion do you reach about p?Rather than withdrawing p, you acquire the belief that p & ∼ s & ∼ r, even though Formula where pr would be the subjective or epistemic probability of p on a probabilistic view of the knowledge norm before learning ∼ & ∼ . Instead of withdrawal …. (shrink)

Four experiments are reported which investigated the types of truth tables that people associate with conditional sentences and the kinds of inferences that they will draw from them. The present studies differed from most previous ones in using different types of content in the conditionals, for example promises and warnings. It was found that the type of content had a strong and consistent effect on both truth tables and inferences. It is suggested that this is because in real (...) life conditionals make probabilistic assertions, and that the strength of the probabilistic link is determined by the situation in which the conditional occurs. The implications of these findings for current theories of reasoning are considered and it is concluded that none of them is entirely satisfactory. It is suggested that more linguistically based theories may prove more successful. (shrink)

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of (...) a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

Qualitative comparative analysis (QCA) opens up two new forms of knowledge: (1) knowing about alternative pathways to one outcome (equifinality) and (2) perceiving nuances of necessary cause and sufficient cause. Several misunderstandings of QCA occur in the article by Lucas and Szatrowski (this volume, p. 1). First, there are minor problems with expressions. Second, there are differences between their philosophy of science (arguments 1, 2, and 3 below) and a realist approach. Third, they misinterpret what was meant by sufficient and (...) necessary cause (arguments 4 and 5 below). The minor problems with expressions arise in sections 2.2 and 3. In section 2.2, the authors define consistency, but here they miss out on two key points. First, they neglect to say that this particular measure of consistency is a measure of sufficiency of cause, not of necessary cause. Second, they ignore the way that the consistency level exists for each possible configuration of characteristics. A configuration can be shown in one row of a truth table, with n1 = 28, n2 = 17 within a larger sample of 155, and so on (see Byrne 2009:262). Using QCA’s definition of “consistency,” we can discern more about the pattern of causality than one might initially expect. This learning is possible whether N overall is 155, or just 39, or thousands of cases. The essential task is to simplify and rank the list of configurations (Ragin 2001, 2006). (shrink)

We present an information-theoretic interpretation of quantum formalism based on a Bayesian framework and devoid of any extra axiom or principle. Quantum information is construed as a technique for analyzing a logical system subject to classical constraints, based on a question-and-answer procedure. The problem is posed from a particular batch of queries while the constraints are represented by the truth table of a set of Boolean functions. The Bayesian inference technique consists in assigning a probability distribution within a (...) real-valued probability space to the joint set of queries in order to satisfy the constraints. The initial query batch is not unique and alternative batches can be considered at will. They are enabled mechanically from the initial batch, quite simply by transcribing the probability space into an auxiliary Hilbert space. It turns out that this sole procedure leads to exactly rediscover the standard quantum information theory and thus provides an information-theoretic rationale to its technical rules. In this framework, the great challenges of quantum mechanics become simple platitudes: Why is the theory probabilistic? Why is the theory linear? Where does the Hilbert space come from? In addition, most of the paradoxes, such as uncertainty principle, entanglement, contextuality, nonsignaling correlation, measurement problem, etc., become straightforward features. In the end, our major conclusion is that quantum information is nothing but classical information processed by a mature form of Bayesian inference technique and, as such, consubstantial with Aristotelian logic. (shrink)

We use the method of fixed points to describe a form of probabilistic truth approximation which we illustrate by means of three examples. We then contrast this form of probabilistic truth approximation with another, more familiar kind, where no fixed points are used. In probabilistic truth approximation with fixed points the events are dependent on one another, but in the second kind they are independent. The first form exhibits a phenomenon that we call ‘fading (...) origins’, the second one is subject to a phenomenon known as ‘the washing out of the prior’. We explain that the two phenomena may seem very similar, but are in fact quite different. (shrink)

We study the weak truth table degree spectra of first-order relational structures. We prove a dichotomy among the possible wtt degree spectra along the lines of Knight’s upward-closure theorem for Turing degree spectra. We prove new results contrasting the wtt degree spectra of finite- and infinite-signature structures. We show that, as a method of defining classes of reals, the wtt degree spectrum is, except for some trivial cases, strictly more expressive than the Turing degree spectrum.

In this work, we discuss the failure of the principle of truth functionality in the quantum formalism. By exploiting this failure, we import the formalism of N-matrix theory and non-deterministic semantics to the foundations of quantum mechanics. This is done by describing quantum states as particular valuations associated with infinite non-deterministic truth tables. This allows us to introduce a natural interpretation of quantum states in terms of a non-deterministic semantics. We also provide a similar construction for arbitrary (...) class='Hi'>probabilistic theories based in orthomodular lattices, allowing to study post-quantum models using logical techniques. (shrink)

The relativized propositional calculus is a system of Boolean formulas with query symbols. A formula in this system is called a one-query formula if the number of occurrences of query symbols is just one. If a one-query formula is a tautology with respect to a given oracle A then it is called a one-query tautology with respect to A. By extending works of Ambos-Spies (1986) and us (2002), we investigate the measure of the class of all oracles A such that (...) the set of all one-query tautologies with respect to A does not p-btt-reduce to A, where p-btt denotes polynomial-time bounded-truth-table. We show that certain Dowd-type generic oracles all belong to the class, and hence measure of the class is one. (shrink)

After Karl Popper’s original work, several approaches were developed to provide a sound explication of the notion of verisimilitude. With few exceptions, these contributions have assumed that the truth to be approximated is deterministic. This collection of ten papers addresses the more general problem of approaching probabilistic truths. They include attempts to find appropriate measures for the closeness to probabilistic truth and to evaluate claims about such distances on the basis of empirical evidence. The papers employ (...) multiple analytical approaches, and connect the research to related issues in the philosophy of science. (shrink)

Developing a suggestion of Wittgenstein, I provide an account of truth tables as formulas of a formal language. I define the syntax and semantics of TPL (the language of Tabular Propositional Logic), and develop its proof theory. Single formulas of TPL, and finite groups of formulas with the same top row and TF matrix (depiction of possible valuations), are able to serve as their own proofs with respect to metalogical properties of interest. The situation is different, however, for groups (...) of formulas whose top rows differ. For them I provide (i) a tree-style system of ‘row tree proofs’, which is shown to be sound and complete, and (ii) an alternative, re-writing strategy. (shrink)

Let us assume that you are entrusted by UNESCO with an important task. You are asked to devise a universal logical language, a Begriffsschrift in Frege's sense, which is to serve the purposes of science, business and everyday life. What requirements should such a “conceptual notation” satisfy? There are undoubtedly many relevant desiderata, but here I am focusing on one unmistakable one. In order to be a viable lingua universalis, your language must in any case be capable of representing (...) any possible configuration of dependence and independence between different variables. For if such a configuration is possible in principle, there is no guarantee that it might not one day show up among the natural, human or social phenomena we have to study.But how are dependencies and independencies between variables expressed in our familiar logical notation? Every logician worth his or her truth-table knows the answer. Dependencies between two variables are expressed by dependencies between the quantifiers to which they are bound. For instance, inthe variable y depends on x, while inz depends on x but not on y, while u depends on both x and y.But how is the dependence of a quantifier on another one expressed in familiar logical languages? Obviously by occurring in its scope, indicated by the pair of parentheses following it. But the nesting of scopes is a transitive and antisymmetrical relation which allows branching only in one direction. Hence other kinds of structures of dependence and independence between variables are not representable in the received logical notation. Such previously inexpressible structures form the subject matter of what has been referred to as independence-friendly logic. (shrink)

We propose a new model for forming and revising beliefs about unknown probabilities. To go beyond what is known with certainty and represent the agent’s beliefs about probability, we consider a plausibility map, associating to each possible distribution a plausibility ranking. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. We consider two forms of conditioning or belief update, corresponding to the acquisition of two types of information: learning observable evidence obtained (...) by repeated sampling from the unknown distribution; and learning higher-order information about the distribution. The first changes only the plausibility map, but leaves the given set of possible distributions essentially unchanged; the second rules out some distributions, thus shrinking the set of possibilities, without changing their plausibility ordering.. We look at stability of beliefs under either of these types of learning, defining two related notions, as well as a measure of the verisimilitude of a given plausibility model. We prove a number of convergence results, showing how our agent’s beliefs track the true probability after repeated sampling, and how she eventually gains in a sense knowledge of that true probability. Finally, we sketch the contours of a dynamic doxastic logic for statistical learning. (shrink)

Matching bias refers to the non-normative performance that occurs when elements mentioned in a rule do not correspond with those in a test item. One aim of the present work is to capture matching bias via reaction times as participants carry out truth-table evaluation tasks. Experiment 1 requires participants to verify conditional rules, and Experiment 2 to falsify them as the paradigm employs four types of conditional sentences that systematically rotate negatives in the antecedent and consequent; and presents (...) predominantly cases having true antecedents. These experiments reveal that mismatching is linked to higher rates of incorrect responses and slower evaluation times. A second aim is to investigate the way not is processed. We compare a narrow view of negations, which argues that negation only denies information, to a search for alternatives view, which says that negations function to prime appropriate alternatives. Findings from both experiments support a narrow reading view. (shrink)

In Probabilistic Knowledge Sarah Moss proposes that our credences and subjective probability judgments (SPJs) can constitute knowledge. Mossean probabilistic knowledge is grounded in probabilistic beliefs that are justified, true, and unGettiered. In this paper I aim to address and solve two challenges that arise in the vicinity of the factivity condition for probabilistic knowledge: the factivity challenge and the challenge from probabilistic arguments from ignorance (probabilistic AIs). I argue that while Moss's deflationary solution to (...) the factivity challenge formally works, it leaves us ill-equipped to handle probabilistic AIs. An account of probabilistic knowledge that cannot overcome probabilistic AIs makes knowledge of thoroughly probabilistic contents a rare and unstable phenomenon, at best, or, at worst, impossible. I hold that establishing a metaphysically enriched account of probabilistic truth is therefore mandatory. I go on to develop a truth-conditional approach on probabilistic factivity that is relativistic in its nature and centers on objective chances. I show that while the approach is still compatible with Moss's overall semantics for probabilistic knowledge, it provides us with a simple but forceful answer to probabilistic AIs. (shrink)

There is a new Bayesian, or probabilistic, paradigm in the psychology of reasoning, with new psychological accounts of the indicative conditional of natural language. In psychological experiments in this new paradigm, people judge that the probability of the indicative conditional, P(if A then C), is the conditional probability of C given A, P(C | A). In other experiments, participants respond with what has been called the 'de- fective' truth table: they judge that if A then C is (...) true when A holds and C holds, is false when A holds and C does not, and is neither true nor false when A does not hold. We argue that these responses are not 'defective' in any negative sense, as many psychologists have implied. We point out that a number of normative researchers, including de Finetti, have proposed such a table for various coherent interpretations of the third value. We review the relevant general tables in the normative literature, in which there is a third value for A and C and the logically compound forms of the natural language conditional, negation, conjunction, disjunction, and the material conditional. We describe the results of an experiment on which of these tables best describes ordinary people's judgements when the third value is interpreted as indicating uncertainty. (shrink)

ABSTRACTDo truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. (...) Rather than any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions. (shrink)

Recently several papers have reported relevance effects on the cognitive assessments of indicative conditionals, which pose an explanatory challenge to the Suppositional Theory of conditionals advanced by David Over, which is influential in the psychology of reasoning. Some of these results concern the “Equation” (P(if A, then C) = P(C|A)), others the de Finetti truth table, and yet others the uncertain and-to-inference task. The purpose of this chapter is to take a Birdseye view on the debate and (...) investigate some of the open theoretical issues posed by the empirical results. Central among these is whether to count these effects as belonging to pragmatics or semantics. (shrink)

In my article, I present a new version of a probabilistic truth prescribing semantics for natural language indicative conditionals. The proposed truth conditions can be paraphrased as follows: an indicative conditional is true if the corresponding conditional probability is high and the antecedent is positively probabilistically relevant for the consequent or the probability of the antecedent of the conditional equals 0. In the paper, the truth conditions are defended and some of the logical properties of the (...) proposed semantics are described. (shrink)

The basic problem of a theory of truth approximation is defining when a theory is “close to the truth” about some relevant domain. Existing accounts of truthlikeness or verisimilitude address this problem, but are usually limited to the problem of approaching a “deterministic” truth by means of deterministic theories. A general theory of truth approximation, however, should arguably cover also cases where either the relevant theories, or “the truth”, or both, are “probabilistic” in nature. (...) As a step forward in this direction, we first present a general characterization of both deterministic and probabilistic truth approximation; then, we introduce a new account of verisimilitude which provides a simple formal framework to deal with such issue in a unified way. The connections of our account with some other proposals in the literature are also briefly discussed. (shrink)

We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey, Griffiths and LaForte.