The phenomenon of multi-used stock photography in the news contradicts the photojournalism professional values of truthful and emotional depictions. This reality echoes other false images increasingly appearing in the media, including deepfakes and artificial intelligence. In the present study, a two (stock and staff photo) by two (positive and negative valence) quasi-experiment is conducted. The dependent variables include: 1) credibility; 2) self-reported arousal level; 3) emotional valence perceptions; 4) fixation duration; and 5) fixation count. Participants viewed staff photos as more (...) credible in both positive and negative valences. Negative photos were perceived as slightly less credible. Stock and staff photos were similarly arousing, received similar valence perceptions, and had similar gaze fixations and durations. Overall, participants perceived the genuine emotion portrayed truthfully in photojournalism as opposed to the faux emotion acted out in stock photos. (shrink)

As the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how one (...) can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens. (shrink)

We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still "partly" true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic - or (...) any fragment thereof under consideration - conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last ('congruentiality') desideratum in the two four-valued logics we end up giving our most extended consideration to. (shrink)

In this article, the epistemological interpretation of the relationship between concepts of relativism, beliefs, and probability ensures a defense of two theses, namely, (i) epistemic relativism refers to attitudes that depend on the repetition and anchoring of probabilistic beliefs, and (ii) Popper’s propensity interpretation of probability discloses the connections between relativity, probability, and collective subjectivity. The propensity interpretation brings a framework for describing the role of collective subjectivity in epistemic systems. This approach, as an acceptable epistemological stance, is related to (...) the variability of anchored collective beliefs. The repetition and anchoring of subjective probabilistic beliefs becomes a relative reference frame for evaluating the elements of knowledge. With this in mind, each particular way of considering probability indicates a manner of distinguishing epistemic systems. In epistemic systems, the statements are collective points of view expressed in the form ‘p is justified when belief in p is probably reliable.’ The statements can be accurate in a particular epistemic system but inaccurate in other systems. As a result, by avoiding the ‘all statements must be true or false’ requirement, each proponent of a certain epistemic system can build a probabilistic picture of the world. In this picture, not every statement needs to be ‘true’ when and if the statement expresses a collective point of view that has a propensity to be reliable and justified. (shrink)

The aim of this paper is to determine whether and to what extent Woodward’s interventionist theory of causation is variable relative. In an influential review, Strevens has accused Woodward’s account of a damaging form of variable relativity, according to which obviously false causal claims can be made true by choosing a depleted variable set. Following McCain, I show that Strevens’ objection doesn’t succeed. However, Woodward also wants to avoid another kind of variable relativity, according to which (...) it can be true that X is a cause of Y in one set of background conditions, but false in another. I show that Woodward’s account is problematically overpermissive, unless there are restrictions on the values that certain variables can take. I formulate a modified account that makes these restrictions explicit, then use it to argue that Woodward’s attempt to avoid relativity to background conditions is misguided. On the best interpretation of the interventionist theory, causal claims are assessed relative to a particular kind of variable set. Thus, I conclude that the theory should be understood as variable relative, in a specific, unproblematic sense. (shrink)

A new technique for proving realisability results is presented, and is illustrated in detail for the simple case of arithmetic minus induction. CL is a Gentzen formulation of classical logic. CPQ is CL minus the Cut Rule. The basic proof theory and model theory of CPQ and CL is developed. For the semantics presented CPQ is a paraconsistent logic, i.e. there are non-trivial CPQ models in which some sentences are both true and false. Two systems of arithmetic minus induction are (...) introduced, CL-A and CPQ-A based on CL and CPQ, respectively. The realisability theorem for CPQ-A is proved: It is shown constructively that to each theorem A of CPQ-A there is a formula A *, a so-called “realised disjunctive form of A ”, such that variables bound by essentially existential quantifiers in A * can be written as recursive functions of free variables and variables bound by essentially universal quantifiers. Realisability is then applied to prove the consistency of CL-A, making use of certain finite non-trivial inconsistent models of CPQ-A. (shrink)

We prove constructively that for any propositional formula in Conjunctive Normal Form, we can either find a satisfying assignment of true and false to its variables, or a refutation of showing that it is unsatisfiable. This refutation is a resolution proof of ¬. From the formalization of our proof in Coq, we extract Robinson’s famous resolution algorithm as a Haskell program correct by construction. The account is an example of the genre of highly readable formalized mathematics.

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of "and", "or", (...) "not" and "conditioning" are then defined on these conditional events with variable conditions. The probabilities of the disjunction and of the conjunction of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed. (shrink)

We investigate how the selection of assurance topics and the format of their communication influence the credibility perception of sustainability report readers. This is important because misleading communication may discredit ethical sustainability assurance practices. Based on signaling theory and using an experimental approach, we are the first to examine false credibility signals in the context of sustainability assurance. We find that two variables related to sustainability assurance, reference explicitness and assurance depth, jointly influence the assurance signal and the perceived credibility (...) of a sustainability report. Our findings indicate that readers are not at risk of false signaling but can make incorrect interpretations of the assurance signal and might respond negatively to well-intentioned signals. The main implications of our findings are that firms should refrain from increasing reference explicitness and should select only the most material topics. Taken together, our results provide new insights on the unethical practice of false signaling and provide an example of an incorrect signal interpretation by readers. (shrink)

In this dissertation I provide a novel logic of the ordinary-language conditional. First, however, I endeavor to make clearer and more precise just what the objects of the study of the conditional are, as a lack of clarity as to what counts as an instance of a given category of conditional has resulted in deep and significant confusions in subsequent analysis. I motivate for a factual/counterfactual distinction, though not at the level of particular instances of the conditional. Instead, I argue (...) that each individual instance of the conditional may be interpreted either factually or counterfactually, rather than these instances dividing into distinct types. I examine the classic Oswald–Kennedy pair of sentences, typically taken to be the quintessential example of how conditionals must be split into two different categories, to show that they in fact do not demonstrate this. I then present my account of the logic underlying the ordinary-language conditional, the system C3, and a justification of the form it takes. This logic provides distinct interpretations of the conditional as it concerns what is factual or what is counterfactual, respectively. The factual interpretation is true when both antecedent and consequent are true at the actual world, false when the antecedent is true and consequent false, and not truth-apt otherwise. The counterfactual interpretation incorporates a ceteris paribus clause, ensuring that it is not falsified by extraordinary, unforeseeable occurrences, and is true when, at all worlds at which the antecedent is ceteris paribus true, the consequent too is true. It is false when, at any of these worlds the consequent is false; and not truth-apt otherwise. I go on to examine alternative theories of the conditional—from the suppositionalist approach to evaluating the putative indicative conditional, to Stalnaker’s combined indicative and counterfactual account, and Lewis’s analysis of the counterfactual; among a number of others—and offer comparison to my own theory. Finally, I look at various challenges to my account of the conditional. Being a strict conditional (albeit variably so), that of C3 is open to the objection that it fails to match ordinary speakers’ intuitions as regards its truth-value assignments to, for instance, conditionals with necessarily true consequents. I address these so- called paradoxes of the strict conditional, and also discussion the question of truth-preservation in the C3 system in the case of such inferences as those relying on transitivity and that of modus ponens. I maintain that these inferences are indeed truth-preserving under certain, specifiable, conditions, and I close by offering possible avenues for further research. (shrink)

The science of operations research is one of the modern sciences that have made a great revolution in all areas of life through the methods provided by it, suitable and appropriate to solve most of the problems that were facing researchers, scholars and those interested in the development of societies, and the most beneficiaries of this science were companies and institutions that are looking for scientific methods that help them manage their work so that they achieve the greatest profit and (...) the lowest cost, and one of the important methods that have been used in the management of companies we offer in this research two methods, Dynamic programming method. This method has been used in many practical matters and helped decision-makers in companies to achieve a maximum profit and less cost by formulating the reality of the state of the company and the data provided by decision-makers with a dynamic mathematical model that is solved using methods of solving dynamic models and we will provide in this research an example of this through the issue of choosing the optimal investment for the budget of a company so that it achieves a maximum profit, and the method of programming with integers: the method that provided these companies with solutions with integer values suitable for the nature of its work, through the use of the binary integer in the formulation of the appropriate mathematical model on the one hand, and on the other hand, the use of the binary integer variable helped to convert some nonlinear models that lead to some practical problems into linear models, and it should be noted here that in the previous two methods there is something indeterminable because we must make a decision in choosing or not choosing something, but the optimal solution that we will get remains A specific value because we are building the mathematical model for any realistic issue through the data provided by those responsible for the work and these data are calculated quantities and therefore they are uncertain values because their validity depends on the circumstances surrounding the work environment, they may be exposed to increase or decrease, and therefore the optimal solution on which the company will base its decision is suitable for specific values and any change in them can cause the company an uncalculated loss, so in this research we will use the concepts of neutrosophic science, the branch of science founded by the American scientist Florentin Smarandache in 1995 based on his belief that there is no absolute truth, a science that is interested in the study of ideas and concepts that are neither true nor false, but just in between, and we will take the data (calculated quantities) neutrosophic values that are specified or unspecified values are any set close to the calculated quantities, then the resulting mathematical model is a neutrosophic model and the optimal solution has neutrosophic values and thanks to the indefinite uncertainty that these values have, companies from the development of appropriate plans for all circumstances and thus achieve the greatest profit and the lowest cost, and we will clarify the above through two issues, the issue of optimal designation of a warehouse site, which we will formulate the mathematical model of using the neutrosophic integer programming method - and the issue of capital budget, which we will present in two different forms, we use in the first form the neutrosophic integer programming method and in the second the neutrosophic dynamic programming method. (shrink)

Counterfactuals are inherently ambiguous in the sense that the same counterfactual may be true under one mode of counterfactualization but false under the other. Many have regarded the ambiguity of counterfactuals as consisting in the distinction between forward-tracking and backtracking counterfactuals. This is incorrect since the ambiguity persists even in cases not involving backtracking counterfactualization. In this paper, I argue that causal modeling semantics has the resources enough for accounting for the ambiguity of counterfactuals. Specifically, we need to distinguish two (...) types of causal manipulation, which I call “intervention” and “extrapolation” respectively. To intervene in a causal model M is to change M’s structural equations in some specific ways, while to extrapolate M is to change the value assignment of M’s variables in some specific ways. I argue that intervention and extrapolation offer a natural explanation for the ambiguity of counterfactuals. (shrink)

In order to block controversial predictions of 2D semantics (The Nesting Problem), Chalmers and Rabern (2014) propose adding an additional constriction called “the liveness constraint” in definitions of epistemic modals. Without this constraint, all scenario‐world pairs counterfactual to a scenario‐world pair considered as actual in a 2D matrix for a contingent a priori proposition ϕ appear problematic for 2D semantics. This is because, although it is false that ϕ in such pairs, it is a priori true that ϕ. I consider (...) two versions of 2D semantics, those with unstructured and structured intensions. I argue that the unstructured version of 2D semantics is over‐simplified and due to this, it has controversial predictions for counterfactual scenario‐world pairs – the definition of epistemic conceivability with the liveness constraint causes some contingent propositions to appear inconceivable in counterfactual scenario‐world pairs, while some other contingent propositions remain conceivable. In turn, the structured version of the 2D account of propositions has no explanation of how it is possible to possess purely qualitative knowledge without its being dependent on a particular individual or on a particular actual world. (shrink)

Contextualists and pragmatists agree that knowledge-denying sentences are contextually variable, in the sense that a knowledge-denying sentence might semantically express a false proposition in one context and a true proposition in another context, without any change in the properties traditionally viewed as necessary for knowledge. Minimalists deny both pragmatism and contextualism, and maintain that knowledge-denying sentences are not contextually variable. To defend their view from cases like DeRose and Stanley's high stakes bank case, minimalists like Patrick Rysiew, Jessica (...) Brown, and Wayne Davis forward ‘warranted assertability maneuvers.’ The basic idea is that some knowledge-denying sentence seems contextually variable because we mistake what a speaker pragmatically conveys by uttering that sentence for what she literally says by uttering that sentence. In this paper, I raise problems for the warranted assertability maneuvers of Rysiew, Brown, and Davis, and then present a warranted assertability maneuver that should succeed if any warranted assertability maneuver will succeed. I then show how my warranted assertability maneuver fails, and how the problem with my warranted assertability maneuver generalizes to pragmatic responses in general. The upshot of my argument is that, in order to defend their view from cases like DeRose and Stanley's high stakes bank case, minimalists must prioritize the epistemological question whether the subjects in those cases know over linguistic questions about the pragmatics of various knowledge-denying sentences. (shrink)

The recent debate between liberals and their communitarian critics has reached a false plateau, with liberals conceding more than they should. After explicating the central communitarian thesis, the four ways that thesis could be understood, and the corresponding four senses of "independence," I argue that communitarians are right that liberalism requires a view of the self as 'unencumbered,' but I defend that view as superior to the alternatives. This allows me to defend true moral impartiality and universality as well as (...) the full normative primacy of the individual. I also present three variable factors of the 'inner citadel' model of the self and argue that communitarianism paradoxically presupposes a variant of that model which causes it internal contradictions. My discussion of these issues leads to a brief list of mandates for acceptable conceptions of personhood. After listing these, I briefly indicate implications of my analysis for the broader political and ethical realm. (shrink)

The paper focuses on the problematic of selfinterpretation and the correspondence between selfimage and the reality. Its presupposition is, that a human being is not committed to a single way of life, fully determined by his/her biological characteristics or by the dictate of his/her social background; first of all, he/she is not dependent on single ways of the interpretation of the world or of his/her selfinterpretation. It is his/her ability to accept certain characteristics as his/her own that makes him/her a (...) human being. There is usually a distinction between his/her own seeing himself/herself, which does not necessarily, leads to a failure. The strategies of survival are various and they need not to be rooted in true interpretations in all their aspects. In conclusion the author argues, that the critical thinking making its way into everyday life as well as into the public sphere offers the possibility of the variability of the ways of life and of a consensual coexistence on the basis of manifold interpretative structures and models. Enforcing a uniform view as a presupposition of human cooperation is replaced by an approach, respecting the legitimacy of the pluralistic image of the world and of the humans, for which this approach makes no obstacle in cooperative working of society. (shrink)

Various laws and theories in the natural and social sciences are presented with a view to discerning the syntactic and semantic characteristics of many idealizations in science. Three different kinds of idealizations are discussed: ideal conditions, ideal cases, and idealized theories. An ideal condition is a formula in which state variables occur, whose existential closure is false, and for which there is another formula that can be constructed out of the original formula such that the existential closure of the new (...) formula is true. An ideal case is a statement which is logically equivalent to a universal conditional which has an ideal condition as its antecedent. And an idealized theory is a set of false universal conditional statements. Alternative syntactic and semantic analyses are viewed and criticized. (shrink)

This paper is about the differences between probabilities and beliefs and why reasoning should not always conform to probability laws. Probability is defined in terms of urn models from which probability laws can be derived. This means that probabilities are expressed in rational numbers, they suppose the existence of veridical representations and, when viewed as parts of a probability model, they are determined by a restricted set of variables. Moreover, probabilities are subjective, in that they apply to classes of events (...) that have been deemed (by someone) to be equivalent, rather than to unique events. Beliefs on the other hand are multifaceted, interconnected with all other beliefs, and inexpressible in their entirety. It will be argued that there are not sufficient rational numbers to characterise beliefs by probabilities and that the idea of a veridical set of beliefs is questionable. The concept of a complete probability model based on Fisher's notion of identifiable subsets is outlined. It is argued that to be complete a model must be known to be true. This can never be the case because whatever a person supposes to be true must be potentially modifiable in the light of new information. Thus to infer that an individual's probability estimate is biased it is necessary not only to show that the estimate differs from that given by a probability model, but also to assume that this model is complete, and completeness is not empirically verifiable. It follows that probability models and Bayes theorem are not necessarily appropriate standards for people's probability judgements. The quality of a probability model depends on how reasonable it is to treat some existing uncertainty as if it were equivalent to that in a particular urn model and this cannot be determined empirically. Bias can be demonstrated in estimates of proportions of finite populations such as in the false consensus effect. However the modification of beliefs by ad hoc methods like Tversky and Kahneman's heuristics can be justified, even though this results in biased judgements. This is because of pragmatic factors such as the cost of obtaining and taking account of additional information which are not included even in a complete probability model. Finally, an analogy is drawn between probability models and geometric figures. Both idealisations are useful but qualitatively inadequate characterisations of nature. A difference between the two is that the size of any error can be limited in the case of the geometric figure in a way that is not possible in a probability model. (shrink)

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on in our field—a (...) book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical terms, (...) since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)

A fundamental problem in game theory is the possibility of reaching equilibrium outcomes with undesirable properties, e.g., inefficiency. The economics literature abounds with models that attempt to modify games in order to avoid such undesirable properties, for example through the use of subsidies and taxation, or by allowing players to undergo a bargaining phase before their decision. In this paper, we consider the effect of such transformations in Boolean games with costs, where players control propositional variables that they can set (...) to true or false, and are primarily motivated to seek the satisfaction of some goal formula, while secondarily motivated to minimise the costs of their actions. We adopt preparation sets as our basic solution concept. A preparation set is a set of outcomes that contains for every player at least one best response to every outcome in the set. Prep sets are well-suited to the analysis of Boolean games, because we can naturally represent prep sets as propositional formulas, which in turn allows us to refer to prep formulas. The preference structure of Boolean games with costs makes it possible to distinguish between hard and soft prep sets. The hard prep sets of a game are sets of valuations that would be prep sets in that game no matter what the cost function of the game was. The properties defined by hard prep sets typically relate to goal-seeking behaviour, and as such these properties cannot be eliminated from games by, for example, taxation or subsidies. In contrast, soft prep sets can be eliminated by an appropriate system of incentives. Besides considering what can happen in a game by unrestricted manipulation of players’ cost function, we also investigate several mechanisms that allow groups of players to form coalitions and eliminate undesirable outcomes from the game, even when taxes or subsidies are not a possibility. (shrink)

Misunderstanding occurs between speakers when they disagree about the meaning of words in use. In the case of truth-values, Frege took these to be referents of sentences which consist of classes of accepted (i.e. “true”) or rejected (i.e. “false”) sentences. From this usual depiction of truth and falsity, a general algebraic framework is proposed to systematize the use of truth-values from a dialogical point of view of logic. A special attention will be paid to two radically opposed pseudo-speakers: Heraclites and (...) Nagarjuna, according to whom truth-values respectively refer to everything or nothing. Finally, dialectical (pseudo-Hegelian) negation will be rendered as a very special function: an ontological object-forming operator, similar to the arithmetic successor-forming operator Sn+1(x) on integers and consistent within our generalized theory of truth-values as variable referents. (shrink)

Once Zheng Xuan, a man of Han dynasty, made notations of “Yiwei”, he said: “The word ‘change’ contains three meanings: the first is simplifying, the second is transformation, and the third is unchanging ”, thus called to “three changes”. The wording “three changes” is able to be the different explanations of “Zhouyi”, and also can be understand to three meanings of the word “change” in “Zhouyi”. Everywhere in the nature, and in nature science, there are incalculable examples about “three changes”. (...) The process of “simplifying”, it is that nature science obtains the principles by means of summing up the multitudinous complex spontaneous phenomena. There is “the fundamental of thinking economy”, and “Aucum razor” in the west, then they emphasize to wipe off the miscellaneous and keep up the simple, and to eliminate the false and retain the true. Therefore, it can say where there is no “simplifying”, there is no nature science. “Transformation” means movement and change. This is the essentiality of the nature. From the biggest such as the universe to the smallest such as cells, all these are the outcomes of evolution, and are changing continueously. Physics researching from static objects to dynamic objects, and mathematics calculation from the constant to the variable, all of these studies are closing up to the essentiality of the natural “transformation” nearer and nearer. If there is no “transformation”, nature science world stop, and the nature world be deathly stillness. “Unchanging” means that the static state exists in the dynamic state, and the constant exists in the variation. The various theorems and laws, such as the theorem of constant of light velocity and the law of conservation of energy, are the summation and accumulation of the regularities of which the relative stabilization is kept in changeable movement. Various kinds of equation are all linked by equal-sign, and based on the condition recognizing which the two sides is equal constantly. Therefore the calculation of the relations between matter characteristics can be set up. Otherwise, not only the nature world be disorder and unsystematic, but also the science world have no laws to go by. (shrink)

Tarski’s relation of satisfaction has been mentioned or discussed a good deal recently, but not often, it seems, with full understanding. Many misconceptions concerning it abound throughout the literature. The relation, it will be recalled, is one holding between an infinite sequence of objects and a sentential function containing an arbitrary number of free variables. A sentence is then any sentential function containing no free variables, and a true sentence is, by definition, one satisfied by all infinite sequences, a false (...) sentence being satisfied by none. (shrink)

The recent intense interest in conspiratorial thinking is fuelled by the perception that belief in conspiracies is highly irrational. However, there have been few studies that have examined the associations of conspiracy belief with a comprehensive battery of rational thinking tasks that tap both epistemic and instrumental rationality. The Comprehensive Assessment of Rational Thinking (CART) provides an opportunity to do just that because one of the subtests on the CART assesses the tendency to believe false conspiracies. That subtest is in (...) the part of the CART that measures the presence of contaminated mindware—stored declarative knowledge that embodies poorly justified beliefs. Converging analyses (N = 747) using the 18 subtests and four thinking dispositions measured on the CART indicated that three variables were key predictors of conspiratorial thinking: superstitious thinking, actively open-minded thinking, and probabilistic reasoning. Theoretical consideration of these best predictors, and of the variables that predict the endorsement of true conspiracies, led us to rethink the classification of conspiracy belief as contaminated mindware and move instead towards a conception of conspiratorial thinking as a cognitive style. (shrink)

Purpose While psychology, sociology and communications studies hypothesise a range of independent variables that might impact on individuals’ acceptance or rejection of rumours, almost all studies of the phenomenon have taken place in environments featuring notable, and sometimes very deep, partisan divisions, making it almost impossible to isolate the impact of partisan influences on views on different rumour subjects. This study aims to remove the possibility of partisan influences on readers of internet rumours by testing the impact of independent demographic (...) variables in China, a one-party state with no overt partisan divisions. The study provides an opportunity to strip away the influence of ideology and see whether this factor may have coloured previous studies on susceptibility to believe rumours. Design/methodology/approach An empirical study was used to examine belief in false and true online rumours in a non-partisan environment. A large sample group was presented with rumours across four subject areas and respondents’ conclusions and demographic information was then subject to logistic regression analysis to identify relationships between factors and ability to identify the veracity of online rumours. Findings Unexpectedly, the regression analysis revealed no statistically significant nexus between many independent demographic variables and patterns of believing or disbelieving rumours. In other cases, a statistically significant relationship was revealed, but only to a limited degree. The results suggest that once the role of partisanship in explaining the proliferation of and belief in false rumours and the ability to identify true ones is removed from consideration, no other independent variables enjoy convincing links with rumour belief. Originality/value The study tests in China, a jurisdiction featuring a non-partisan environment, the impact of independent variables on media users’ belief in a wide range of rumours. (shrink)

From a formal point of view, the nihilistic propositional calculi, called in this text „npc”, are such propositional calculi that include three unary propositional conectives: T, F, ~, and one binary propositional connective ≡. They read respectively: true, false, untrue that, if and only if. Their theorems are, mong other things, such expressions as: Tp ≡ p, Fp ≡ ~p, with p being a propositional variable. The four npc are presented in this work, i.e. on two-valued, two three-valued, and (...) one four-valued. Moreover, two of them are also paraconsistent calculi. Npc are constructed by means of the axiomatic method. Following the presentation of npc axioms, the four so called n-algebras are introduced. The npc axioms are proved to be adequate to appropriate n-algebras, i.e. sets of theorems and tautologies of each npc are identical. (shrink)

Recent empirical work on folk moral objectivism has attempted to examine the extent to which folk morality presumes that moral judgments are objectively true or false. Some researchers report findings that they take to indicate folk commitment to objectivism (Goodwin & Darley, 2008, 2010, 2012; Nichols & Folds-Bennett, 2003; Wainryb et al., 2004), while others report findings that may reveal a more variable commitment to objectivism (Beebe, 2014; Beebe et al., 2015; Beebe & Sackris, 2016; Sarkissian, et al., 2011; (...) Wright, 2018; Wright, Grandjean, & McWhite, 2013; Wright, McWhite, & Grandjean, 2014). However, the various probes that have been used to examine folk moral objectivism almost always fail to be good direct measures of objectivism. Some critics (Beebe, 2015; Pölzler, 2017, 2018) have suggested that the problems with existing probes are serious enough that they should be viewed as largely incapable of shedding any light on folk metaethical commitments. Building upon the work of Justin Khoo and Joshua Knobe (2018), I argue that many of the existing probes can be seen as good measures of the extent to which people think that the truth of one moral judgment excludes the possibility that a judgment made by a disagreeing party is also true and that the best explanation of the findings obtained using these measures is significant folk support for indexical moral relativism—the view that the content of moral judgments is context-sensitive. If my thesis is correct, many contemporary moral philosophers are deeply mistaken about the metaethical contours of folk morality in one very important respect. (shrink)

The purpose of this article is to provide a non‐contradictory interpretation of sentences such as “Smith's murderer might not have murdered Smith”. An anti‐actualist, two‐dimensional framework including partial functions provides the basis for my solution. I argue for two claims. The modal profile of the proposition expressed by “The F might not have been an F” is complex: at any world where there is a unique F the proposition is true; at any world without a unique F the proposition has (...) no truth‐value; hence, at no world is it false. It remains an open semantic and epistemological question which of the first two kinds of world the actual world is. The semantic method should be based on explicit intensionalization in lieu of actualism. Actualism accords a privileged role to the actual world. Explicit intensionalization places all possible worlds, including the actual one, on an equal footing. Syntactically, a lambda‐bound world variable replaces the actual‐world constant or operator, while the other world variable is existentially bound. (shrink)

In medicine and the social sciences, researchers often measure only a handful of variables simultaneously. The underlying assumption behind this methodology is that combining the results of dozens of smaller studies can, in principle, yield as much information as one large study, in which dozens of variables are measured simultaneously. Mayo-Wilson :864–874, 2011, Br J Philos Sci 65:213–249, 2013. https://doi.org/10.1093/bjps/axs030) shows that assumption is false when causal theories are inferred from observational data. This paper extends Mayo-Wilson’s results to cases in (...) which experimental data is available. I prove several new theorems that show that, as the number of variables under investigation grows, experiments do not improve, in the worst-case, one’s ability to identify the true causal model if one can measure only a few variables at a time. However, stronger statistical assumptions significantly aid causal discovery in piecemeal inquiry, even if such assumptions are unhelpful when all variables can be measured simultaneously. (shrink)

Taking as a point of departure a recently published collection of representative contributions from various philosophers who claim to ?proceed from ordinary language?, this article examines ordinary language philosophy in the light of some of the claims made by these philosophers. The claims are criticized mainly for failing to account for the variability of the use of terms in respect both of depth of intention and special contexts. These factors are such as to render the claims in question false when (...) interpreted as significant assertions about language, but trivial when they are interpreted as reasonable and true. (shrink)

In his 1903 Principles of Mathematics, Russell holds that “it is a characteristic of the terms of a proposition”—that is, its “logical subjects”—“that any one of them may be replaced by any other entity without our ceasing to have a proposition”. Hence, in PoM, Russell holds that from the proposition ‘Socrates is human’, we can obtain the propositions ‘Humanity is human’ and ‘The class of humans is human’, replacing Socrates by the property of humanity and the class of humans, respectively. (...) Hence also, in PoM, Russell accepts the doctrine of the unrestricted variable: if we replace a logical subject of a proposition by a variable to yield a propositional function, the range of that variable includes absolutely every entity. For absolutely any entity can be taken as a value of that variable to yield a proposition, true or false. The doctrine of the unrestricted variable is thus incompatible with any type-theoretic metaphysics, according to which only entities of a certain restricted type can replace a logical subject of a given proposition so as to yield another proposition. (shrink)

The notion of epistemic standards has gained prominence in the literature on the semantics of knowledge ascriptions. Defenders of Epistemic Contextualism claim that in certain scenarios the truth value of a knowledge-ascribing sentence of the form “S knows p (at t)”—where S is an epistemic subject and p is a proposition S is said to know at time t—can change even if S, p and t are assigned constant values. This sort of variability, contextualists claim, is due to the epistemic (...) standards governing the context in which the knowledge ascription is uttered. While a specific knowledge ascription may be true when uttered in a context with “low” epistemic standards, it may be false when uttered in a context with “high” epistemic standards. The reason for this, as far as contextualists are concerned, is the context sensitivity of the verb “knows”. In standard semantics an expression is said to be context sensitive if and only if it expresses different contents (or intensions) relative to different contexts of utterance. Thus, as epistemic standards influence the content of “knows”, they play a crucial role in contextualist semantics. In this paper, I examine different conceptions of epistemic standards and argue that all but one lead to counterintuitive consequences. The conception which avoids these consequences, however, has the downside of seriously restricting the talk of “high” or “low” standards that is so frequent in discussions on the semantics of knowledge ascriptions. (shrink)

The issue of materialism has recently been raised again. Mr. Putnam argues against philosophical behaviorism [4]. Such a position holds, as he construes it, that statements like ‘Jones is angry’ can be analyzed in solely behavioral terms. When one argues against philosophical behaviorism, he might be expected to distinguish this metaphysical position from behavior science. Putnam, however, does not make the distinction. Consequently he argues against both. I shall first state the distinction between these two different things, namely, philosophical behaviorism (...) and behavior science, as I see it. The behavior scientist adopts the thesis that in principle it is possible to predict future behavior on the basis of data concerning environmental, behavioral, and physiological variables. All three of these he considers in physical terms. The behavior scientist thus speaks about physical objects and properties of such. Talking in such terms, he believes that it is in principle possible to coordinate to statements asserting that person X has or is in state of mind Y another statement, employing only the above mentioned physical terms, such that either both are true or both are false. The reasons for the behavior scientist's program are the well known quandaries involved in the observation of other people's minds and the need for intersubjective verification in science. One can further distinguish between a narrower and a broader view of behavior science. The former restricts itself to environmental and behavioral variables at what some call the macro level; the latter includes, or even concentrates upon, physiological variables. As scientists neither the behaviorist nor the physiologist asks or answers philosophical questions, either epistemological or ontological, about minds, bodies, and mental contents. (shrink)

In his 1903 Principles of Mathematics, Russell holds that “it is a characteristic of the terms of a proposition”—that is, its “logical subjects”—“that any one of them may be replaced by any other entity without our ceasing to have a proposition”. Hence, in PoM, Russell holds that from the proposition ‘Socrates is human’, we can obtain the propositions ‘Humanity is human’ and ‘The class of humans is human’, replacing Socrates by the property of humanity and the class of humans, respectively. (...) Hence also, in PoM, Russell accepts the doctrine of the unrestricted variable: if we replace a logical subject of a proposition by a variable to yield a propositional function, the range of that variable includes absolutely every entity. For absolutely any entity can be taken as a value of that variable to yield a proposition, true or false. The doctrine of the unrestricted variable is thus incompatible with any type-theoretic metaphysics, according to which only entities of a certain restricted type can replace a logical subject of a given proposition so as to yield another proposition. (shrink)

Down through the ages, logic has adopted many strange and awkward technical terms: assertoric, prove, proof, model, constant, variable, particular, major, minor, and so on. But truth-value is a not a typical example. Every proposition, even if false, no matter how worthless, has a truth-value:even “one plus two equals four” and “one is not one”. In fact, every two false propositions have the same truth-value—no matter how different they might be, even if one is self-contradictory and one is consistent. (...) It is not such a big surprise that every true proposition, no matter how worthless, has a truth-value. Every proposition, whether valuable or worthless, has a truth-value.But it is a little strange that every two true propositions, no matter how different they might be, have the same truth-value. The Pythagorean Theorem has the same truth-value as the proposition that one is one. It is a stretch to think of truth-values as values in any of the normal senses of the word ‘value’. (shrink)

In the first part of the article true expressions and false expressions are presented that served A. W. Burks to build a formal system of logic of causal propositions along with considering their later use in the system built in 1963. In the second part of the article the possibility of conducting an axiomatic proof of the theorem on adding a superfluous condition is analyzed. The consequences are also shown of substituting a negation of the same prepositional variable for (...) the propositional variable and conditions are analyzed of substitution for propositional variables and possibilities of avoiding those difficulties. (shrink)

The two Heisenberg Uncertainties (UR) entail an incompatibility between the two pairs of conjugated variables E, t and p, q. But incompatibility comes in two kinds, exclusive of one another. There is incompatibility defineable as: (p → − q) & (q→ − p) or defineable as [(p →− q) & (q →− p)] ↔ r. The former kind is unconditional, the latter conditional. The former, in accordance, is fact independent, and thus a matter of logic, the latter fact dependent, and (...) thus a matter of fact. The two types are therefore diametrically opposed. In spite of this, however, the existing derivations of the Uncertainties are shown here to entail both types of incompatibility simultaneously. Δ E Δ t ≥ h is known to derive from the quantum relation E = hν plus the Fourier relation Δ ν Δ t ≥ 1. And the Fourier relation assigns a logical incompatibility between Δ ν = 0, Δ t = 0. (Defining a repetitive phenomenon at an instant t → 0 is a self contradictory notion.) An incompatibility, therefore, which is fact independent and unconditional. How can one reconcile this with the fact that Δ EΔ t exists if and only if h > 0, which latter supposition is a factual truth, entailing that a Δ E = 0, Δ t = 0 incompatibility should itself be fact dependent? Are we to say that E and t are unconditionally incompatible (via Δ ν Δ t ≥ 1) on condition that E = hν is at all true? Hence, as presently standing, the UR express a self-contradicting type of incompatibility. To circumvent this undesirable result, I reinterpret E = hν as relating the energy with a period. Though only one such period. And not with frequency literally. (It is false that E = ν. It is true that E = ν times the quantum.) In this way, the literal concept of frequency does not enter as before, rendering Δ ν Δ t ≥ 1 inapplicable. So the above noted contradiction disappears. Nevertheless, the Uncertainties are derived. If energy is only to be defined over a period, momentum only over a distance (formerly a wavelength) resulting during such period, thus yielding quantized action of dimensions Et = pq, then energies will become indefinite at instants, momenta indefinite at points, leading, as demanded, to (symmetric!) Δ E Δ t = Δ p Δ q ≥ h’s. (shrink)

1. Lemmon’s stratification. By a “tense logic” I mean a system with the following features: it contains sentential variables which stand for sentences which in some cases are true at some times and false at others; it contains the usual truth-functions, whose truth-conditions are given the obvious modifications, e.g. Np is true when and only when p is false, Kpq is true when and only when both its conjuncts are; and it contains two additional functions which may be interpreted as (...) ‘It will be the case that p’ and ‘It has been the case that p’, the former being true when and only when the plain p will be true later on and the latter when and only when the plain p has been true at some earlier time. (shrink)

The two Heisenberg Uncertainties (UR) entail an incompatibility between the two pairs of conjugated variables E, t and p, q. But incompatibility comes in two kinds, exclusive of one another. There is incompatibility defineable as: (p → -q) & (q → -p) or defineable as [(p → -q) & (q → -p)] ↔ r. The former kind is unconditional, the latter conditional. The former, in accordance, is fact independent, and thus a matter of logic, the latter fact dependent, and thus (...) a matter of fact. The two types are therefore diametrically opposed. In spite of this, however, the existing derivations of the Uncertainties are shown here to entail both types of incompatibility simultaneously. ΔEΔt ≥ h is known to derive from the quantum relation E = hv plus the Fourier relation ΔvΔt ≥ 1. And the Fourier relation assigns a logical incompatibility between Δv = 0, Δt = 0. (Defining a repetitive phenomenon at an instant t → 0 is a self contradictory notion.) An incompatibility, therefore, which is fact independent and unconditional. How can one reconcile this with the fact that ΔEΔt exists if and only if h > 0, which latter supposition is a factual truth, entailing that a ΔE = 0, Δt = 0 incompatibility should itself be fact dependent? Are we to say that E and t are unconditionally incompatible (via ΔvΔt ≥ 1) on condition that E = hv is at all true? Hence, as presently standing, the UR express a self-contradicting type of incompatibility. To circumvent this undesirable result, I reinterpret E = hv as relating the energy with a period. Though only one such period. And not with frequency literally. (It is false that E = v. It is true that E = v times the quantum.) In this way, the literal concept of frequency does not enter as before, rendering ΔvΔt ≥ 1 inapplicable. So the above noted contradiction disappears. Nevertheless, the Uncertainties are derived. If energy is only to be defined over a period, momentum only over a distance (formerly a wavelength) resulting during such period, thus yielding quantized action of dimensions Et = pq, then energies will become indefinite at instants, momenta indefinite at points, leading, as demanded, to (symmetric!) ΔEΔt = ΔpΔq ≥ h's. (shrink)

Here is a view at least much like Lewis’s “Humean supervenience,” and in any case highly influential—in that some endorse it, and many more worry that it is true. All truths about the world are fixed by the pattern of instantiation, by individual points in space-time, of the “perfectly natural properties” posited by end-of-inquiry physics. In part, this view denies independent variability: the world could not have been different from how it actually is, in the ways depicted by common sense (...) and the special sciences, without differing in the punctiform instantiation of fundamental physical properties. In part, it makes an ontological claim: what it is for one of the objects recognized by common sense or special sciences to be there in the world, bearing the properties attributed by a true description, is “nothing over and above” the obtaining of fundamental physical properties at points, and fundamental physical relations among points. I argue that this view is untenable. I concede that for every true claim in familiar discourses, there is a state of affairs at the level of fundamental microphysics that is the truth-maker—some state of affairs sufficient for truth in the familiar claim. The problem is that the view needs to posit not just truth-makers at the level of microphysics, but truth-conditions—states of affairs the obtaining of which is required for truth in any familiar claim, and the failure of which renders the familiar claim false. That is, the view must posit necessary conditions, at the level of microparticles, for truth in familiar claims. This it cannot plausibly do. (shrink)

The first problem which Gochet takes up in this important book is whether the proposition is necessary to logical syntax. Gochet is intent upon following out the nominalistic enterprise of desolving [[sic]] the ontological status of the proposition as much as possible. He notes that Quine’s schematic letters can replace the propositional variables, and thus the first transference is made from semantics to syntax, the first important loosening of ontological commitments. Tarski’s thesis that sentences are true or false, and not (...) propositions, is the leading idea in the analysis and in the nominalistic criterion, in general. Throughout, the specific problem is to determine when the use of the notion of proposition is ontologically compromising. Here Quine’s criterion of ontological commitment is the analytic tool. Gochet distinguishes a syntactic version and a semantic version. Gochet refines this criterion by means of several objections of Ayer and Church and making some technical innovations proposed by Prior and Marcus-Barcan. Thus, along with the law of parsimony, he now has a powerful means for detecting the ontological commitments of certain modes of speech. (shrink)

This paper uses a non-distributive system of Boolean fractions (a|b), where a and b are 2-valued propositions or events, to express uncertain conditional propositions and conditional events. These Boolean fractions, 'a if b' or 'a given b', ordered pairs of events, which did not exist for the founders of quantum logic, can better represent uncertain conditional information just as integer fractions can better represent partial distances on a number line. Since the indeterminacy of some pairs of quantum events is due (...) to the mutual inconsistency of their experimental conditions, this algebra of conditionals can express indeterminacy. In fact, this system is able to express the crucial quantum concepts of orthogonality, simultaneous verifiability, compatibility, and the superposition of quantum events, all without resorting to Hilbert space. A conditional (a|b) is said to be "inapplicable" (or "undefined") in those instances or models for which b is false. Otherwise the conditional takes the truth-value of proposition a. Thus the system is technically 3-valued, but the 3rd value has nothing to do with a state of ignorance, nor to some half-truth. People already routinely put statements into three categories: true, false, or inapplicable. As such, this system applies to macroscopic as well as microscopic events. Two conditional propositions turn out to be simultaneously verifiable just in case the truth of one implies the applicability of the other. Furthermore, two conditional propositions (a|b) and (c|d) reside in a common Boolean sub-algebra of the non-distributive system of conditional propositions just in case b = d, their conditions are equivalent. Since all aspects of quantum mechanics can be represented with this near classical logic, there is no need to adopt Hilbert space logic as ordinary logic, just a need perhaps to adopt propositional fractions to do logic, just as we long ago adopted integer fractions to do arithmetic. The algebra of Boolean fractions is a natural, near-Boolean extension of Boolean algebra adequate to express quantum logic. While this paper explains one group of quantum anomalies, it nevertheless leaves no less mysterious the 'influence-at-a-distance', quantum entanglement phenomena. A quantum realist must still embrace non-local influences to hold that "hidden variables" are the measured properties of particles. But that seems easier than imaging wave-particle duality and instant collapse, as offered by proponents of the standard interpretation of quantum mechanics. (shrink)

We are much better equipped to let the facts reveal themselves to us instead of blinding ourselves to them or stubbornly trying to force them into preconceived molds. We no longer embarrass ourselves in front of our students, for example, by insisting that “Some Xs are Y” means the same as “Some X is Y”, and lamely adding “for purposes of logic” whenever there is pushback. Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, (...) contextualism, and pluralism. Besides the new spirit there have been quiet developments in logic and its history and philosophy that could radically improve logic teaching. One rather conspicuous example is that the process of refining logical terminology has been productive. Future logic students will no longer be burdened by obscure terminology and they will be able to read, think, talk, and write about logic in a more careful and more rewarding manner. Closely related is increased use and study of variable-enhanced natural language as in “Every proposition x that implies some proposition y that is false also implies some proposition z that is true”. Another welcome development is the culmination of the slow demise of logicism. No longer is the teacher blocked from using examples from arithmetic and algebra fearing that the students had been indoctrinated into thinking that every mathematical truth was a tautology and that every mathematical falsehood was a contradiction. A fifth welcome development is the separation of laws of logic from so-called logical truths, i.e., tautologies. Now we can teach the logical independence of the laws of excluded middle and non-contradiction without fear that students had been indoctrinated into thinking that every logical law was a tautology and that every falsehood of logic was a contradiction. This separation permits the logic teacher to apply logic in the clarification of laws of logic. This lecture expands the above points, which apply equally well in first, second, and third courses, i.e. in “critical thinking”, “deductive logic”, and “symbolic logic”. (shrink)

Standard conditionals $\varphi > \psi$, by which I roughly mean variably strict conditionals à la Stalnaker and Lewis, are trivially true for impossible antecedents. This article investigates three modifications in a doxastic setting. For the neutral conditional, all impossible-antecedent conditionals are false, for the doxastic conditional they are only true if the consequent is absolutely necessary, and for the metaphysical conditional only if the consequent is ‘model-implied’ by the antecedent. I motivate these conditionals logically, and also doxastically by properties of (...) conditional belief and belief revision. For this I show that the Lewisian hierarchy of conditional logics can be reproduced within ranking semantics, provided we slightly stretch the notion of a ranking function. Given this, acceptance of a conditional can be interpreted as a conditional belief. The epistemic and the neutral conditional deviate from Lewis’ weakest system $V$, in that ID or even CN are dropped, and new axioms appear. The logic of the metaphysical conditional is completely axiomatised by $V$ to which we add the known Kripke axioms T5 for the outer modality. Related completeness results for variations of the ranking semantics are obtained as corollaries. (shrink)

Idealized scientific representations result from employing claims that we take to be false. It is not surprising, then, that idealizations are a prime example of allegedly inconsistent scientific representations. I argue that the claim that an idealization requires inconsistent beliefs is often incorrect and that it turns out that a more mathematical perspective allows us to understand how the idealization can be interpreted consistently. The main example discussed is the claim that models of ocean waves typically involve the false assumption (...) that the ocean is infinitely deep. While it is true that the variable associated with depth is often taken to infinity in the representation of ocean waves, I explain how this mathematical transformation of the original equations does not require the belief that the ocean being modeled is infinitely deep. More generally, as a mathematical representation is manipulated, some of its components are decoupled from their original physical interpretation. (shrink)

A refutation mechanism is introduced into logic programming, dual to the usual proof mechanism; then negation is treated via refutation. A four-valued logic is appropriate for the semantics: true, false, neither, both. Inconsistent programs are allowed, but inconsistencies remain localized. The four-valued logic is a well-known one, due to Belnap, and is the simplest example of Ginsberg’s bilattice notion. An efficient implementation based on semantic tableaux is sketched; it reduces to SLD resolution when negations are not involved. The resulting system (...) can give reasonable answers to queries that involve both negation and free variables. Also it gives the same results as Prolog when there are no negations. Finally, an implementation in Prolog is given. (shrink)

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.