Reasoning From Quantified Modal Premises.

Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given.

If is Easier Than Or in the GRE Yingrui Yang (yangyri@rpi.edu) Department of Cognitive Sciences, Rensselaer Polytechnic Institute Carnegie Hall, Troy, NY 12180 USA Philip N. Johnson-Laird (phil@princeton.edu) Psychology Department, Princeton University Princeton, NJ 08544 USA In logic, a conditional, such as: “If the trend continues then a decline will occur” is equivalent to a disjunction: “Either the trend stops, i.e., doesn't continue, or a decline will occur”. Both assertions are compatible with the following three possibilities, where ¬ denotes negation: Trend Decline ¬ Trend Decline ¬ Trend ¬ Decline The equivalence may break down as a result of the specific content or context of assertions (Johnson-Laird and Byrne, 2002). But, where the two assertions are equivalent, the conditional has the mental models: Trend Decline in which the first model represents the possibility in which the antecedent is true, and the second wholly implicit model represents the possibilities in which the consequent is false. The disjunction has the mental models: ¬ Trend Decline ¬ Trend Decline It follows that reasoning should be easier with the conditional than with the disjunction. We conducted three experiments to test this prediction using logical reasoning problems from the Graduate Record Examination (the GRE, devised by Educational Testing Services, Princeton. In Experiment 1, 20 participants carried out either a conditional version or a disjunctive version of 8 GRE problems, e.g.: the conditional version: Because the number of surgeons is growing faster than the number of operations and because noninvasive medical therapies are increasingly replacing surgery, the average annual number of operations per surgeon has fallen by one- fourth in recent years. It can be concluded that, if these trends continue, a dangerous decline in the level of surgical skill will occur. The argument is based on which of the following assumptions? (A) Surgeons now spend a large percentage of their time performing noninvasive medical procedures. (B) A surgeon’s skill cannot be properly maintained unless the surgeon performs operations with a certain minimum frequency. Option (B) is the correct answer. The disjunctive version included instead the following final assertion: It can be concluded that, either these trends stop, or a dangerous decline in the level of surgical skill will occur. The participants had to select the correct response from the pair of assertions, which were the correct conclusion and the most frequently chosen foil (according to ETS). The accuracy of responses did not differ, but the participants were reliably faster to solve the conditional problems (mean 1.76 min.) than the disjunctive problems (mean 2.06 min.). Experiment 2 was a replication but in which the two response options were conditionals (for the conditional problems) and disjunctions (for the disjunctive problems). The participants were reliably more accurate and faster with the conditional problems (73% correct, 0.8 min.) than with the disjunctive problems (61% correct, 1.17 minutes). The use of a sentence containing a given connective in both the text and the two response options evidently amplified the difference between conditionals and disjunctions. Experiment 3 used conditional and disjunctive texts with conditional and disjunctive response options in all four combinations. The results showed that the nature of the response options was decisive. The 40 participants were faster and more accurate with problems that had conditional responses than with problems that had disjunctive response options. We conclude that the model theory's predictions about the different representations of conditionals and disjunctions extend to realistic problems based on the GRE. Theories based on formal rules of inference (e.g., Braine and O'Brien, 1998; Rips, 1994) make no predictions about this difference. The research was a part of the project, eWriter, which was supported by a grant from ETS and the GRE Board to S. Bringsjor, Y. Yang, P.N. Johnson-Laird, and M. Bauer. References Braine, M.D.S., and O’Brien, D.P., Eds. (1998) Mental Logic, Mahwah, NJ: Erlbaum. Johnson-Laird, P.N., and Byrne, R.M.J. (2002) Conditionals: A theory of meaning, pragmatics, and inference. Psychological Review, in press. Rips, L.J. (1994). The Psychology of Proof. Cambridge, MA: MIT Press.

There has been a significant interest in modal logics with intersection, prominent examples including epistemic and doxastic logics with distributed knowledge, propositional dynamic logic with intersection, and description logics with concept intersection. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 used, e.g., in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logic. A standard proof method for the S5 case uses an “unraveling-folding” technique to achieve a treelike model to deal with the problem of undefinability. This method, however, is not easily adapted to other logics, due to its reliance on S5 in a number of steps. In this paper we demonstrate a simpler and more general proof technique by building a treelike canonical model directly, which avoids the complications in the processes of unraveling and folding. We illustrate the technique by showing completeness of the normal modal logics K, D, T, B, S4 and S5 extended with intersection modalities. Furthermore, these treelike canonical models are compatible with Fischer-Ladner-style closures, and we combine the methods to show the completeness of the mentioned logics further extended with transitive closure of union modalities known from PDL or epistemic logic. Some of these completeness results are new.

Reasoning about preferences is a fundamental component of rationality, and therefore central in AI and computational social choice. Most logic-based frameworks for representing and reasoning about preferences assume that preferences are arbitrarily given, typically as a ranking of a set of alternatives or using utilities, with little concern about how preferences are formed or where they come from. Recent work in rational choice theory, however, has devoted attention to giving more internal structure to the notion of rationality, focusing more on the faculties of individual agents, such as their mood, mindset, and motivating reasons. In this paper we develop a modal logic for reasoning about preferences that depend on a set of motivationally salient properties, based on recent work on reasons behind preferences by Dietrich and List. The main result is that we show how the problem of reasoning in this logic can be translated to reasoning in a standard modal logic (KT with universal modality), and consequently that reasoning systems and algorithms developed for modal logic (with universal modality) can be employed for reasoning about reason-based preferences.

In this paper I give conditions under which a matrix characterisation of validity is correct for first order logics where quantifications are restricted by statements from a theory. Unfortunately the usual definition of path closure in a matrix is unsuitable and a less pleasant definition must be used. I derive the matrix theorem from syntactic analysis of a suitable tableau system, but by choosing a tableau system for restricted quantification I generalise Wallen's earlier work on modal logics. The tableau system is only correct if a new condition I call “alphabetical monotonicity” holds. I sketch how the result can be applied to a wide range of logics such as first order variants of many standard modal logics, including non-serial modal logics.

There are two typical ways to formalize reasoning over time. The first is in a modal logic where time is characterized implicitly through tense operators. The second addresses time explicitly in a first-order theory, where the reasoning is itself essentially first order. In this paper we shall introduce an amalgamated temporal logic which combines the tense operators of the modal approach with the referential aspects of the first-order approach.The modal operators of the new logic, called IQ-C, include the familiar tense operators plus a few new ones; together they allow complex temporal propositions to be represented in a compact form. Unlike standard modal temporal logics, IQ-C contains terms which refer to times; this allows the logic to characterize certain temporal properties and operators which cannot be axiomatized or defined by conventional modal operators. IQ-C is expressively complete over any connected flow of time, and has several other interesting theoretical properties, some of which will be developed in this paper.Most interval logics take one of two approaches to temporal assertions; either they analyze assertions for intervals in terms of assertions at points, or they treat intervals as primitive, allowing assertions for intervals but not at points. IQ-C allows assertions both for intervals and at points, and treats them as independent; this yields a special degree of semantic discrimination and at the same time simplifies the representation of several temporal properties. In this paper we shall outline the IQ-C framework, specifying its special features and defining its semantics. We shall also extend a method of skolemization, which is designed to support a resolution based approach to the proof theory. This will be sketched in some detail, with a view to providing a preliminary account of the basic proof theory. A complete specification of the deductive theory, together with relevant proofs, will be developed elsewhere.KeywordsIntervalsPointsTemporal Representation and ExpressivenessHomogeneitySkolemizationand Modal Theorem Proving

In standard modal logic, □ ≡ ∼◊ ∼ and ◊ ≡ ∼□∼. I will, first, examine why in tense-logic, Arthur Prior thinks that ∼ ◊ ∼ is weaker than □ and ∼ □ ∼ is weaker than ◊. I will, then, examine whether there are similar motivations in modal logic to take ∼ ◊ ∼ to be weaker than □ and ∼ □ ∼ to be weaker than ◊. The upshot will be that, just as certain metaphysical views within the philosophy of time (e.g., Presentism) motivate one to deny the standard tense equivalences, certain metaphysical views within the metaphysics of modality (e.g., Contingentism, nonmodalism) motivate one to deny the standard modal equivalences.

Human reasoning involving quantified statements is one area in which findings from cognitive psychology and linguistic pragmatics complement each other. I will show how mental models theory provides a promising account of the mechanisms underlying peoples’ performance in three types of reasoning tasks involving quantified premises and conclusions. I will further suggest that relevance theory can help to explain the way in which mental models are employed in the reasoning processes. Conversely, mental models theory suggests that human reasoning typically does not involve deductive rules, which in turn entails a modification to the nature of the deductive processes proposed by relevance theory. The mechanism proposed by mental models theory also helps to clarify the nature of the relevance theory distinction between conceptual and procedural information.

We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic \(\mathsf {K}\), a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent calculi for the considered non-normal modal logics.KeywordsModal LogicInference RuleModal RuleLinear LogicSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Semantics of Entailment

When solving a simple probabilistic problem, people tend to build an incomplete mental representation. We observe this pattern in responses to probabilistic problems over a set of premises using the conjunction, disjunction, and conditional propositional connectives. The mental model theory of extensional reasoning explains this bias towards underestimating the number of possibilities: In reckoning with different interpretations of the premises (logical rules, mental model theoretical, and, specific to conditional premises, conjunction and biconditional interpretation) the mental model theory accounts for the majority of observations. Different interpretations of a premise result in a build-up of mental models that are often incomplete. These mental models are processed using either an extensional strategy relying on proportions amongst models, or a conflict monitoring strategy. The consequence of considering too few possibilities is an erroneous probability estimate akin to that faced by decision makers who fail to generate and consider all alternatives, a characteristic of bounded rationality. We compare our results to the results published by Johnson-Laird, Legrenzi, Girotto, Legrenzi, and Caverni [Johnson-Laird, P., Legrenzi, P., Girotto, V., Legrenzi, M., & Caverni, J. (1999). Naive probability: A mental model theory of extensional reasoning. Psychological Review, 106, 62-88. doi: 10. 1037/0033-295X.106.1.62], and we observe lower performance levels than those in the original article.

This study is situated within the scope of critical analysis. It examines Moroccan citizens’ cognitive reception of political discourse through a linguistic and socio-cognitive perspective. Focusing on three major political parties representing the country’s principal ideological poles—the CU, PJD, and SUPF—the research investigates how citizens interpret political messages and the cognitive mechanisms underlying their perceptions. Guided by Van Dijk’s (2008) mental model theory, the study explores how participants construct understanding based on personal experiences, cultural knowledge, and socially shared representations. The Data collected via a cognitively structured questionnaire reveal that participants frequently rely on pre-existing mental models shaped by negative episodic memories, social images, and culturally ingrained perceptions. These cognitive patterns result in a persistent misalignment between politicians’ intended discourse and the public’s interpretations. The findings also indicate that participants judge political discourse more by the parties’ perceived social roles than by their actual ideologies, which contributes to fixed perceptions and widespread political disengagement among the Moroccan public. Accordingly, the study highlights the importance of political literacy as a mechanism to align citizens’ mental models with parties’ cognitive frameworks, thereby fostering a more coherent social understanding of Moroccan political messaging.

Summary form only given. The importance of mental models in complex control tasks is discussed, and the lattice theory of mental models is described. Plans for a series of experiments using a simulation developed with the Lab View programming language to validate the lattice theory of mental models are reported. >

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \(\exists x(x=\alpha )\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account.

In the last two decades, modal and description logics have provided a theoretical framework for important applications in many areas of computer science. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In this chapter we show how efficient Boolean reasoning techniques have been imported, used and integrated into reasoning tools for modal and description logics. To this extent, we focus on modal logics, and in particular mainly on K(m). In particular, we provide some background in modal logics; we describe a basic theoretical framework and we present and analyze the basic tableau-based and DPLL-based techniques; we describe optimizations and extensions of the DPLL-based procedures; we introduce the automata-theoretic/OBDD-based approach; finally, we present the eager approach.