There is a long tradition in formal epistemology and in the psychology of reasoning to investigate indicative conditionals. In psychology, the propositional calculus was taken for granted to be the normative standard of reference. Experimental tasks, evaluation of the participants’ responses and psychological model building, were inspired by the semantics of the material conditional. Recent empirical work on indicative conditionals focuses on uncertainty. Consequently, the normative standard of reference has changed. I argue why neither logic nor standard probability theory provide (...) appropriate rationality norms for uncertain conditionals. I advocate coherence based probability logic as an appropriate framework for investigating uncertain conditionals. Detailed proofs of the probabilistic non-informativeness of a paradox of the material conditional illustrate the approach from a formal point of view. I survey selected data on human reasoning about uncertain conditionals which additionally support the plausibility of the approach from an empirical point of view. (shrink)

We present a unified approach for investigating rational reasoning about basic argument forms involving indicative conditionals, counterfactuals, and basic quantified statements within coherence-based probability logic. After introducing the rationality framework, we present an interactive view on the relation between normative and empirical work. Then, we report a new experiment which shows that people interpret indicative conditionals and counterfactuals by coherent conditional probability assertions and negate conditionals by negating their consequents. The data support the conditional probability interpretation of conditionals and the (...) narrow-scope reading of the negation of conditionals. Finally, we argue that coherent conditional probabilities are important for probabilistic analyses of conditionals, nonmonotonic reasoning, quantified statements, and paradoxes. (shrink)

A range of formal models of human reasoning have been proposed in a number of fields such as philosophy, logic, artificial intelligence, computer science, psychology, cognitive science, etc.: various logics, probabilistic systems, belief revision systems, neural networks, among others. Now, it seems reasonable to require that formal models of human reasoning be empirically adequate if they are to be viewed as models of the phenomena in question. How are formal models of human reasoning typically put to empirical test? One way (...) to do so is to isolate a number of key principles of the system, and design experiments to gauge the extent to which participants do or do not follow them in reasoning tasks. Another way is to take relevant existing results and check whether a particular formal model predicts these results. The present investigation provides an illustration of the second kind of empirical testing by comparing two formal models for reasoning – namely the non-monotonic logic known as preferential logic; and a particular version of belief revision theories, screened belief revision – against the reasoning phenomenon known as belief bias in the psychology of reasoning literature: human reasoners typically seek to maintain the beliefs they already hold, and conversely to reject contradicting incoming information. The conclusion of our analysis will be that screened belief revision is more empirically adequate with respect to belief bias than preferential logic and non-monotonic logics in general, as what participants seem to be doing is above all a form of belief management on the basis of background knowledge. The upshot is thus that, while it may offer valuable insights into the nature of human reasoning, preferential logic is ultimately inadequate as a formal model of the phenomena in question. (shrink)

There are numerous formal systems that allow inference of new conditionals based on a conditional knowledge base. Many of these systems have been analysed theoretically and some have been tested against human reasoning in psychological studies, but experiments evaluating the performance of such systems are rare. In this article, we extend the experiments in [19] in order to evaluate the inferential properties of c-representations in comparison to the well-known Systems P and Z. Since it is known that System Z and (...) c-representations mainly differ in the sorts of inheritance inferences they allow, we discuss subclass inheritance and present experimental data for this type of inference in particular. (shrink)

We investigate a lattice of conditional logics described by a Kripke type semantics, which was suggested by Chellas and Segerberg – Chellas–Segerberg (CS) semantics – plus 30 further principles. We (i) present a non-trivial frame-based completeness result, (ii) a translation procedure which gives one corresponding trivial frame conditions for arbitrary formula schemata, and (iii) non-trivial frame conditions in CS semantics which correspond to the 30 principles.

While there is now considerable experimental evidence that, on the one hand, participants assign to the indicative conditional as probability the conditional probability of consequent given antecedent and, on the other, they assign to the indicative conditional the “defective truth-table” in which a conditional with false antecedent is deemed neither true nor false, these findings do not in themselves establish which multi-premise inferences involving conditionals participants endorse. A natural extension of the truth-table semantics pronounces as valid numerous inference patterns that (...) do seem to be part of ordinary usage. However, coupled with something the probability account gives us—namely that when conditional-free ϕ entails conditional-free ψ, “if ϕ then ψ” is a trivial, uninformative truth—we have enough logic to derive the paradoxes of material implication. It thus becomes a matter of some urgency to determine which inference patterns involving indicative conditionals participants do endorse. Only thus will we be able to arrive at a realistic, systematic semantics for the indicative conditional. (shrink)

The term "non-monotonic logic" covers a family of formal frameworks devised to capture and represent defeasible inference , i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called "non-monotonic" because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base itself. This is (...) in contrast to classical (first-order) logic, whose inferences, being deductively valid, can never be "undone" by new information. (shrink)

We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the formif not-A,thenA, should not hold, since the conditional’s antecedentnot-Acontradicts its consequentA. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event$${A| \overline{A}}$$A|A¯is$${p(A| \overline{A})=0}$$p(A|A¯)=0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms (...) of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles:Aristotle’s theses,Aristotle’s Second Thesis,Abelard’s First Principle, andBoethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (likenon-symmetry). Both approaches satisfyminimalrequirements for a connexive logic. Finally, we compare both approaches conceptually. (shrink)

We present an interdisciplinary approach to argumentation combining logical, probabilistic, and psychological perspectives. We investigate logical attack principles which relate attacks among claims with logical form. For example, we consider the principle that an argument that attacks another argument claiming A triggers the existence of an attack on an argument featuring the stronger claim A ∧ B. We formulate a number of such principles pertaining to conjunctive, disjunctive, negated, and implicational claims. Some of these attack principles seem to be prima (...) facie more plausible than others. To support this intuition, we suggest an interpretation of these principles in terms of coherent conditional probabilities. This interpretation is naturally generalized from qualitative to quantitative principles. Specifically, we use our probabilistic semantics to evaluate the rationality of principles which govern the strength of argumentative attacks. In order to complement our theoretical analysis with an empirical perspective, we present an experiment with students of the TU Vienna ( n = 139) which explores the psychological plausibility of selected attack principles. We also discuss how our qualitative attack principles relate to well-known types of logical argumentation frameworks. Finally, we briefly discuss how our approach relates to the computational argumentation literature. (shrink)

Everyday life reasoning and argumentation is defeasible and uncertain. I present a probability logic framework to rationally reconstruct everyday life reasoning and argumentation. Coherence in the sense of de Finetti is used as the basic rationality norm. I discuss two basic classes of approaches to construct measures of argument strength. The first class imposes a probabilistic relation between the premises and the conclusion. The second class imposes a deductive relation. I argue for the second class, as the first class is (...) problematic if the arguments involve conditionals. I present a measure of argument strength that allows for dealing explicitly with uncertain conditionals in the premise set. (shrink)

In this paper we exploit the notions of conjoined and iterated conditionals, which are defined in the setting of coherence by means of suitable conditional random quantities with values in the interval [0,1]. We examine the iterated conditional (B|K)|(A|H), by showing that A|H p-entails B|K if and only if (B|K)|(A|H) = 1. Then, we show that a p-consistent family F={E1|H1, E2|H2} p-entails a conditional event E3|H3 if and only if E3|H3= 1, or (E3|H3)|QC(S) = 1 for some nonempty subset S (...) of F, where QC(S) is the quasi conjunction of the conditional events in S. Then, we examine the inference rules And, Cut, Cautious Monotonicity, and Or of System P and other well known inference rules (Modus Ponens, Modus Tollens, Bayes). We also show that QC(F)|C(F) = 1, where C(F) is the conjunction of the conditional events in F. We characterize p-entailment by showing that F p-entails E3|H3 if and only if (E3|H3)|C(F) = 1. Finally, we examine Denial of the antecedent and Affirmation of the consequent, where the p-entailment of (E3|H3) from F does not hold, by showing that (E3|H3)|C(F) is not equal to 1. (shrink)

This position paper advocates combining formal epistemology and the new paradigm psychology of reasoning in the studies of conditionals and reasoning with uncertainty. The new paradigm psychology of reasoning is characterized by the use of probability theory as a rationality framework instead of classical logic, used by more traditional approaches to the psychology of reasoning. This paper presents a new interdisciplinary research program which involves both formal and experimental work. To illustrate the program, the paper discusses recent work on the (...) paradoxes of the material conditional, nonmonotonic reasoning, and Adams’ Thesis. It also identifies the issue of updating on conditionals as an area which seems to call for a combined formal and empirical approach. (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)

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study selected probabilistic (...) versions of classical categorical syllogisms and construct a new version of the square of opposition in terms of defaults and negated defaults. (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)

We take coherence based probability logic as the basic reference theory to model human deductive reasoning. The conditional and probabilistic argument forms are explored. We give a brief overview of recent developments of combining logic and probability in psychology. A study on conditional inferences illustrates our approach. First steps towards a process model of conditional inferences conclude the paper.

We discuss O&C's probabilistic approach from a probability logical point of view. Specifically, we comment on subjective probability, the indispensability of logic, the Ramsey test, the consequence relation, human nonmonotonic reasoning, intervals, generalized quantifiers, and rational analysis.

Although we endorse the primacy of uncertainty in reasoning, we argue that a probabilistic framework cannot model the fundamental skill of proof administration. Furthermore, we are skeptical about the assumption that standard probability calculus is the appropriate formalism to represent human uncertainty. There are other models up to this task, so let us not repeat the excesses of the past.

We generalize Ripley’s results on the transitivity of consequence relation, without assuming a logic to be monotonic. Following Gabbay, we assume nonmonotonic consequence relation to be inclusive and cautious monotonic, and figure out the implications between different forms of transitivity of logical consequence. Weaker frameworks without inclusiveness or cautious monotonicity are also discussed. The paper may provide basis for the study of both non-transitive logics and nonmonotonic ones.

ABSTRACTThe new paradigm in the psychology of reasoning redirects the investigation of deduction conceptually and methodologically because the premises and the conclusion of the inferences are assumed to be uncertain. A probabilistic counterpart of the concept of logical validity and a method to assess whether individuals comply with it must be defined. Conceptually, we used de Finetti's coherence as a normative framework to assess individuals' performance. Methodologically, we presented inference schemas whose premises had various levels of probability that contained non-numerical (...) expressions and, as a control, sure levels. Depending on the inference schemas, from 60% to 80% of the participants produced coherent conclusions when the premises were uncertain. The data also show that except for schemas involving conjunction, performance was consistently lower with certain than uncertain premises, the rate of conjunction fallacy was consistently low (not exceeding 20%,.. (shrink)

The current special issue focuses on logical and probabilistic approaches to reasoning in uncertain environments, both from a formal, conceptual and argumentative perspective as well as an empirical point of view. In the present introduction we give an overview of the types of problems addressed by the individual contributions of the special issue, based on fundamental distinctions employed in this area. We furthermore describe some of the general features of the special issue.

A system of intermediate quantifiers (“Most S are P”, “m/n S are P”) is proposed for evaluating the rationality of human syllogistic reasoning. Some relations between intermediate quantifiers and probabilistic interpretations are discussed. The paper concludes by the generalization of the atmosphere, matching and conversion hypothesis to syllogisms with intermediate quantifiers. Since our experiments are currently still running, most of the paper is theoretical and intended to stimulate psychological studies.

Argumentation is a very fertile area of research in Artificial Intelligence, and various semantics have been developed to predict when an argument can be accepted, depending on the abstract structure of its defeaters and defenders. When these semantics make conflicting predictions, theoretical arbitration typically relies on ad hoc examples and normative intuition about what prediction ought to be the correct one. We advocate a complementary, descriptive-experimental method, based on the collection of behavioral data about the way human reasoners handle these (...) critical cases. We report two studies applying this method to the case of reinstatement (both in its simple and floating forms). Results speak for the cognitive plausibility of reinstatement and yet show that it does not yield the full expected recovery of the attacked argument. Furthermore, results show that floating reinstatement yields comparable effects to that of simple reinstatement, thus arguing in favor of preferred argumentation semantics, rather than grounded argumentation semantics. Besides their theoretical value for validating and inspiring argumentation semantics, these results have applied value for developing artificial agents meant to argue with human users. (shrink)

The knowledge representation and reasoning of both humans and artificial systems often involves conditionals. A conditional connects a consequence which holds given a precondition. It can be easily recognized in natural languages with certain key words, like “if” in English. A vast amount of literature in both fields, both artificial intelligence and psychology, deals with the questions of how such conditionals can be best represented and how these conditionals can model human reasoning. On the other hand, findings in the psychology (...) of reasoning, such as those in the Suppression Task, have led to a paradigm shift from the monotonicity assumptions in human inferences towards nonmonotonic reasoning. Nonmonotonic reasoning is sensitive for information change, that is, inferences are drawn cautiously such that retraction of previous information is not required with the addition of new information. While many formalisms of nonmonotonic reasoning have been proposed in the field of Artificial Intelligence, their capability to model properties of human reasoning has not yet been extensively investigated. In this paper, we analyzed systematically from both a formal and an empirical perspective the power of formal nonmonotonic systems to model possible explicit defeaters, as in the Suppression Task, and more implicit conditional rules that trigger nonmonotonic reasoning by the keywords in such rules. The results indicated that the classical evaluation for the correctness of inferences has to be extended in the three major aspects regarding the inference system, the knowledge base, and possible assumed exceptions for the rule. (shrink)

There are several approaches implementing reasoning based on conditional knowledge bases, one of the most popular being System Z (Pearl, Proceedings of the 3rd conference on theoretical aspects of reasoning about knowledge, TARK ’90, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, pp. 121–135, 1990). We look at ranking functions (Spohn, The Laws of Belief: Ranking Theory and Its Philosophical Applications, Oxford University Press, Oxford, 2012) in general, conditional structures and c-representations (Kern-Isberner, Conditionals in Nonmonotonic Reasoning and Belief Revision: Considering (...) Conditionals as Agents, vol. 2087 of LNCS, Springer, Berlin, 2001) in order to examine the reasoning strength of the different approaches by learning which of the known calculi of nonmonotonic reasoning (System P and R) and Direct Inference are applicable to these inference relations. Furthermore we use the recently proposed Enforcement-postulate (Kern-Isberner and Krümpelmann, Proceedings of the 22nd international joint conference on artificial intelligence, vol. 2, IJCAI’11, AAAI Press, pp. 937–942, 2011) to show dependencies between these approaches. (shrink)

This paper reviews the psychological investigation of reasoning with conditionals, putting an emphasis on recent work. In the first part, a few methodological remarks are presented. In the second part, the main theories of deductive reasoning (mental rules, mental models, and the probabilistic approach) are considered in turn; their content is summarised and the semantics they assume for if and the way they explain formal conditional reasoning are discussed, in particular in the light of experimental work on the probability of (...) conditionals. The last part presents the recent shift of interest towards the study of conditional reasoning in context, that is, with large knowledge bases and uncertain premises. (shrink)

When humans reason, they are able to revise their beliefs in light of new information and abandon obsolete conclusions. Logicians argued, that in some cases, such reasonings appear to be non-monotonic. Thus, many different, seemingly non-monotonic systems were created to formally model such cases. The purpose of this article is to re-examine the definition of non-monotonicity and its implementation in non-monotonic logics and in examples of everyday human reasoning. We will argue that many non-monotonic logics employ some weakened versions of (...) the definitions of non-monotonicity, since in-between different steps of reasoning they either: a) allow previously accepted premises to be removed, or b) change the rules of inference. Of the two strategies, the second one seems downright absurd, since changing the rules of a given logic is a mere replacement of that logic with the rules of another. As a consequence we obtain two logics, whereas the definition of a non-monotonic logic is supposed to define one. The definition of non-monotonicity does not permit either of these cases, which means that such logics are monotonic. (shrink)