Potential Counterexamples Research Articles

Cubic graphs that cannot be covered with four perfect matchings

A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is 3-edge-colourable, the rest of cubic graphs fall into two classes: those that can be covered with four perfect matchings, and those that need at least five. Cubic graphs that require more than four perfect matchings to cover their edges are particularly interesting as potential counterexamples to several profound and long-standing conjectures including the celebrated cycle double cover conjecture. However, so far they have been extremely difficult to find.In this paper we build a theory that describes coverings with four perfect matchings as flows whose flow values represent points and outflow patterns represent lines of a configuration of ten points and six lines spanned by four points of the 3-dimensional projective space P3(F2) in general position. This theory provides powerful tools for investigation of graphs that do not admit such a cover and offers a great variety of methods for their construction. As an illustrative example we produce a rich family of snarks (nontrivial cubic graphs with no 3-edge-colouring) that cannot be covered with four perfect matchings. The family contains all previously known graphs with this property.

Contra intuições

This paper intends to offer a critical evaluation of the role played by intuitions in the mainstream methodology in traditional analytic epistemology. The criticism directed to what I call methodology of evaluation by intuitions shows that epistemologists often idealize the reliability of intuitions when assessing a theory against potential counterexamples, ignoring that the disposition to answer intuitively to a case of concept attribution is bounded to the familiar dimension in which the relevant conceptual competence emerges. Thus, the reliability of intuitions does not necessarily extend to far-fetched cases. Given that the assessment by our intuitions is not sufficient to decide for or against an epistemological theory, I offer a sketch for a pragmatic and naturalized conception of theoretical decision-making in epistemology.Key-words: Intuitions, Analytic epistemology, Conceptual abilities, Naturalism, Pragmatism.

Searching for a universal constraint on the possible denotations of clause-embedding predicates

We propose a new universal constraint on the relationship between the meaning of a clause-embedding predicate when it takes an interrogative complement and its meaning when it takes a declarative complement. According to this proposal, every clause-embedding predicate V satisfies a constraint that we refer to as P-to-Q Entailment. That is, for any exhaustivity-neutral interrogative complement Q, if there is an answer p to Q such that "x Vs p" is true, then it follows that "x Vs Q" is true as well. We discuss empirical advantages of this proposal over existing proposals and explore potential counterexamples to P-to-Q Entailment.

Wine experts' recognition of wine odors is not verbally mediated.

Experts have better memory for items within their domain of expertise. Critically, this does not depend on more efficient use of language. However, this conclusion is based mainly on findings from experts in visual and auditory domains. Olfactory experts constitute an interesting potential counterexample since language has been implicated to be critically involved in odor memory in previous studies. We examined the role language plays in odor recognition memory for wine experts, who typically display better wine odor memory than novices and who are also able to name odors better than lay people. This suggests wine experts' superior recognition memory for odors may be verbally mediated. In 2 experiments, recognition memory for wine odors, wine-related odors, and common odors was tested in wine experts and novices. The use of language was manipulated in Experiment 1 with an overt naming versus no-naming condition, and in Experiment 2, with a verbal interference task inhibiting covert verbalization. Across the two experiments the results showed wine experts have better recognition memory for wines, but not for wine-related or common odors, indicating their memory advantage is expertise specific. Critically, this effect was not verbally mediated, as there was no relationship between experts' ability to name wines and their memory for them. Likewise, directly inhibiting online use of verbalization did not affect memory for wine odors in experts. In sum, once expertise has been acquired, language does not play a causal role in recognition memory for odors. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Reasoning strategies explain individual differences in social reasoning.

The dual-strategy model of reasoning suggests that when people reason they can either use (a) a statistical strategy which generates an estimation of conclusion likelihood using a rapid form of associative processing or (b) a counterexample strategy which identifies potential counterexamples to a conclusion using a more conscious working memory intensive process. Previous results suggest that strategy use is a strong individual difference that represents a broad distinction in the way that information is processed that goes beyond deductive reasoning. In 3 studies, we examined if this model could predict individual differences in the processing of social information by examining socially relevant cognitive biases. Study 1 found that strategy use predicted the extent of the self-serving bias. Study 2 found that strategy use predicted use of racist stereotypes even when need for closure was accounted for. Study 3 found that an essentialist prime resulted in a higher level of gender bias among statistical reasoners but that this prime had no effect on counterexample reasoners. These results indicate that the processing distinction between the 2 reasoning strategies underlies individual differences in social biases such as stereotypes, sexism, and racism. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Justified Group Belief, Group Knowledge and Being in a Position to Know

AbstractJennifer Lackey has recently presented a new and lucid analysis of the notion of justified group belief, i.e. a set of individually necessary and jointly sufficient conditions for a group to justifiedly believe some proposition. In this paper, however, I argue that the analysans she proposes is too narrow: one of the conditions she takes to be necessary for justified group belief is not necessary. To substantiate this claim, I present a potential counterexample to Lackey's analysis where a group knows and thus justifiedly believes some proposition but there is no single group member who actually believes that proposition. I close by defending the example against the objection that the group in question does not know but is at most in a position to know the target proposition.

Information and Diagrammatic Reasoning: An Inferentialist Reading

In current philosophy of information, different authors have been supporting the veridicality thesis (VT). According to this thesis, an epistemically-oriented concept of information must have truth as one of its necessary conditions. Two challenges can be raised against VT. First, some philosophers object that veridicalists erroneously ignore the informativeness of false messages. Secondly, it is not clear whether VT can adequately explain the information considered in hypothetical reasoning. In this sense, logical diagrams offer an interesting case of analysis: by manipulating a logical diagram we can verify that a certain conclusion follows from a set of premises, but it cannot help us to determine the actual truth-value of a given set of propositions. Focusing on the latter challenge, in this paper I claim that logical diagrams set out potential counterexamples to VT and, consequently, pose a real challenge to this thesis. First, a veridicalist analysis of logical diagrams requires the assumption of metatheoretical properties which are not satisfied by some logical systems (and, consequently, are not satisfied by some systems of logical diagrams). So, VT does not fit well as a general framework for a theory of logical diagrammatic information. Secondly, based on semantic inferentialism, one can propose a normative interpretation of the inferential content of logical diagrams not exposed to the problems faced by VT. Moreover, there are several reasons to believe that veridicalism cannot accommodate such a normative interpretation. In other words, normativism represents a real (though still underexplored) alternative to veridicality. Due to these reasons, I conclude that, until further research, we should adopt a more parsimonious standpoint and say that logical diagrams provide inferential information simply.

Reasoning strategies predict use of very fast logical reasoning.

The dual strategy model proposes that people use one of two potential ways of processing information when making inferences. The statistical strategy generates a rapid probabilistic estimate based on associative access to a wide array of information, while the counterexample strategy uses a more focused representation, allowing for a search for potential counterexamples. In the following studies, we explore the hypothesis that individual differences in strategy use are related to the ability to make rapid intuitive logical judgments. In Study 1, we show that this is the case for rapid judgments requiring a distinction between simple logical form and for a novel form of judgment, the ability to identify inferences that are not linked to their premises (non sequiturs). In Study 2, we show that strategy use is related to the ability to make the kinds of rapid logical judgments previously examined over and above contributions of working memory capacity. Study 3 shows that strategy use explains individual variability in rapid logical responding with belief-biased inferences over and above the contribution of IQ. The results of Studies 2 and 3 indicate that under severe time constraint cognitive capacity is a very poor predictor of reasoning, while strategy use becomes a stronger predictor. These results extend the notion that people can make rapid intuitive "logical" judgments while highlighting the importance of strategy use as a key individual difference variable.

Explanation or Modeling: a Reply to Kellen and Klauer

In Wason’s “selection” task, individuals often overlook potential counterexamples in selecting evidence to test hypotheses. Our recent meta-analysis of 228 experiments corroborated the main predictions of the task’s original theory, which aimed to explain the testing of hypotheses. Our meta-analysis also eliminated all but 1 of the 15 later theories. The one survivor was the inference-guessing theory of Klauer et al., but it uses more free parameters to model the data. Kellen and Klauer (this issue) dissent. They defend the goal of a model of the frequencies of all 16 possible selections in Wason’s task, including “guesses” that occur less often than chance, such as not selecting any evidence. But an explanation of hypothesis testing is not much advanced by modeling such guesses with independent free parameters. The task’s original theory implies that individuals tend to choose items of evidence that are dependent on one another, and the inference-guessing theory concurs for those selections that are inferred. Kellen and Klauer argue against correlations as a way to assess dependencies. But our meta-analysis did not use them; it used Shannon’s measure of information to establish dependencies. Their modeling goal has led them to defend a “purposely vague” theory. Our explanatory goal has led us to defend a “purposely clear” algorithm and to retrieve long-standing evidence that refutes the inference-guessing theory. Individuals can be rational in testing a hypothesis: in repeated tests, they search for some examples of it, and then exhaustively for counterexamples.

Intrinsicality, Independence and Grounding

AbstractThis paper investigates the plausibility of Witmer, Butchard and Trogdon’s proposal to distinguish intrinsic properties from extrinsic ones in terms of independence from accompaniment and grounding. I argue that the proposed criterion is not adequate to determine intrinsicality, since according to it some intuitively extrinsic properties turn out to be intrinsic. I suggest and evaluate two responses: first, one could characterize a conception of independence which is specific to the individual instantiating the property; and second, one could justify two assumptions about properties which entail that counterexample properties of the kinds I present do not exist, most importantly that there are no fundamental properties which are instantiated in an intrinsic fashion by some individuals and an extrinsic fashion by others. Although the latter seems prima facie plausible, I present some potential counterexamples to it from current physical theory. I conclude that the grounding- and independence-based criterion of intrinsicality can be defended from my objections, although the cost of doing so might make it more attractive to characterize intrinsicality in terms of independence, or in terms of grounding, alone.

Can an infinitely long object fit in an expanding universe?

AbstractDoes space stretch its contents as the universe expands? Usually, we say the answer is no—the stretching of space is not like the stretching of a rubber sheet that might drag things with it. In this paper, we explore a potential counterexample—namely, we show that it is impossible to make an arbitrarily long object in an expanding universe, because it is impossible to hold the distant end of the object ‘stationary’ with respect to us (as defined in the Friedmann–Lemaître–Robertson–Walker metric). We show that this does not mean that expanding space has a force associated with it, rather, some fictitious forces arise due to our choice of reference frame. By choosing our usual time slice (where all comoving observers agree on the age of the universe), we choose a global frame that does not correspond to the frame of any inertial observer. As a result, simple relativistic velocity transforms generate an apparent acceleration, even where no force exists. This effect is similar to the fictitious forces that arise in describing objects in rotating reference frames, as in the case of the Coriolis effect.

Computing bounds on product graph pebbling numbers

Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number π(G) is the smallest number such that, for any distribution of π(G) pebbles to the vertices of G and choice of root vertex r of G, there exists a sequence of pebbling moves that places a pebble on r. Computing π(G) is provably difficult, and recent methods for bounding π(G) have proved computationally intractable, even for moderately sized graphs.Graham conjectured that π(G□H)≤π(G)π(H), where G□H is the Cartesian product of G and H (1989). While the conjecture has been verified for specific families of graphs, in general it remains open. This study combines the focus of developing a computationally tractable, IP-based method for generating good bounds on π(G□H), with the goal of shedding light on Graham's conjecture. We provide computational results for a variety of Cartesian-product graphs, including some that are known to satisfy Graham's conjecture and some that are not. Our approach leads to a sizable improvement on the best known bound for π(L□L), where L is the Lemke graph, and L□L is among the smallest known potential counterexamples to Graham's conjecture.

On the Empirical Scope and Theoretical Status of the OC-NC Generalization

The article discusses the OC-NC generalization ( Landau 2004 , 2015 ) that establishes a relationship between the obligatory control nature of a construction and its semantic (tensedness/attitude) and formal (morphological agreement) properties. I discuss three counterexamples attested in languages of the Caucasus and demonstrate that complements of desiderative verbs in these languages violate predictions of the generalization: agreeing tensed complements instantiate obligatory control, or uninflected tensed complements are noncontrolled. I also give references to other potential counterexamples and discuss issues raised by the empirical evidence, ultimately arguing that no such typological universal holds.

Beliefs and Blameworthiness

In this paper, I analyze epistemic blameworthiness. After presenting Michael Bergmann’s definition of epistemic blameworthiness, I argue that his definition is problematic because it does not have a control condition. I conclude by offering an improved definition of epistemic blameworthiness and defending this definition against potential counterexamples.

Reasoning strategy modulates gender differences in performance on a spatial rotation task.

The dual-strategy model of reasoning has proposed that individual differences in reasoning can be understood as due to two general ways of processing information: an analytic, counterexample strategy that examines information for explicit potential counterexamples and an intuitive, statistical strategy that uses associative access to generate a likelihood estimate of putative conclusions. Previous studies have examined this model in the context of basic conditional reasoning tasks. However, the distinctions that underlie the dual-strategy model can be seen as a basic description of more general differences in information processing. A recent study examining interactions between gender and strategy use in processing of negative emotions found that gender differences were modulated by strategy, with the general advantage of females concentrated within statistical reasoners. Two studies were performed to extend this analysis to performance on a mental rotation task for which there also exist clear gender differences. The initial study presented rotation tasks with unlimited time. Results show that males perform better on more difficult rotation tasks than females, with the difference concentrated among statistical reasoners. The second study replicated this using a restricted time (4 s) to make each judgement and showed an increase in the effect of both gender and strategy. This provides additional evidence that the dual-strategy model captures an important individual difference in the general way that information is processed.

Is epistemic circularity a fallacy?

The author uses a series of potential counterexamples to argue against attempts by Bergmann and Plantinga to articulate a distinction between malignant and benign epistemic circularity and, more radically, to argue that epistemic circularity per se is no fallacy, and the concept of epistemic circularity plays no role in the explanation of why some instances of epistemic circularity are irrational. The author contrasts an inferential framework, in which circularity is a problem, with an equilibrium framework, in which the concept of circularity plays no useful role and argues that defeasible reasoning can only be understood in an equilibrium, not an inferential, framework. The author uses an example of reasoning about the reliability of one’s own memory to explain how seemingly malignant epistemic circularity can be rational in an equilibrium framework. The author discusses the relevance of this conclusion to two contemporary issues: (1) the cogency of Plantinga’s Evolutionary Argument against Naturalism and (2) the evolutionary naturalists’ (e.g., Street) challenges to non-naturalist moral realism—and, indeed, to all forms of non-naturalist normative realism.

Sound black-box checking in the LearnLib

In black-box checking (BBC) incremental hypotheses on the behavior of a system are learned in the form of finite automata, using information from a given set of requirements, specified in Linear-time Temporal Logic (LTL). The LTL formulae are checked on intermediate automata and potential counterexamples are validated on the actual system. Spurious counterexamples are used by the learner to refine these automata. We improve BBC in two directions. First, we improve checking lasso-like counterexamples by assuming a check for state equivalence. This provides a sound method without knowing an upper-bound on the number of states in the system. Second, we propose to check the safety portion of an LTL property first, by deriving simple counterexamples using monitors. We extended LearnLib’s system under learning API to make our methods accessible, using LTSmin as model checker under the hood. We illustrate how LearnLib’s most recent active learning algorithms can be used for BBC in practice. Using the RERS 2017 challenge, we provide experimental results on the performance of all LearnLib’s active learning algorithms when applied in a BBC setting. We will show that the novel incremental algorithms TTT and ADT perform the best. We also provide experiments on the efficiency of various BBC strategies.

Conjugacy search problem and the Andrews–Curtis conjecture

AbstractWe develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples{\operatorname{AK}(n)}. We devise a number of algorithms in an attempt to disprove the most interesting counterexample{\operatorname{AK}(3)}. That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to{\operatorname{AK}(3)}with relations of lengths up to 20 (previous record was 17).

Out on a limb? On multiple cognitive systems within the octopus nervous system

ABSTRACTThe idea that there can be only one cognitive system within any single given cognitive organism is an established albeit implicit one within cognitive science and related studies of the mind. The firm foothold of this notion is due largely to the immense corpus of empirical evidence for the correlation of a high level of cognitive sophistication with a centralized nervous system. However, it must be pointed out that these findings are sourced in large part from studies on vertebrates. This paper presents a potential counterexample to the notion that only one cognitive system can be realized within any single genuine cognitive organism. This counterexample is the octopus, an invertebrate with what initially appears to be a paradoxical combination of vertebrate-like cognitive and behavioral capacities and a functionally decentralized nervous system. The extensive relegation of sensorimotor processing and control responsibilities to the peripheral nervous system which controls the arms of the octopus raises principled reasons to believe that the octopus is an organism that may house multiple independent cognitive systems.

Knowing How One Knows

In this paper, I argue that knowledge is dimly luminous. That is: if a person knows that p, she knows how she knows that p. The argument depends on a safety-based account of propositional knowledge, which is salient in Williamson’s critique of the ‘KK’ principle. I combine that account with non-intellectualism about knowledge-how – according to which, if a person knows how to φ, then in nearly all (if not all) nearby possible worlds in which she φes in the same way as in the actual world, she only φes successfully. Thus, the possession of first-order propositional knowledge implies secondorder practical knowledge, and this can be iterated. Because of the assumed nonintellectualism about know-how, dim luminosity does not imply bright luminosity about knowledge, which is expressed by the traditional KK principle. I conclude by considering some potential counterexamples to the view that knowledge is dimly luminous.