In "Intuitionistic validity in -normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory , that of all -normal Kripke structures for which he gave an r.e. axiomatization. In the language of arithmetic and denote PA plus Open Induction or Open LNP, and are their intuitionistic deductive closures. We show is recursively axiomatizable and , while . If proves PEM but not totality of a classically provably total Diophantine function of , then and so . A (...) result due to Wehmeier then implies . We prove is not -conservative over . If , then is not closed under MR or Friedman's translation, so range (). Both and are closed under the negative translation. (shrink)

The paper generalizes Van McGee's well-known result that there are many maximal consistent sets of instances of Tarski's schema to a number of non-classical theories of truth. It is shown that if a non-classical theory rejects some classically valid principle in order to avoid the truth-theoretic paradoxes, then there will be many maximal non-trivial sets of instances of that principle that the non-classical theorist could in principle endorse. On the basis of this it is argued that (...) the idea of classical recapture, which plays such an important role for non-classical logicians, can only be pushed so far. (shrink)

Historians of modern medicine often divide their subject into two parts, separated by the bacteriological revolution of the late nineteenth century, when medicine supposedly became 'scientific' for the first time. The history of medical geography - to say nothing of other subjects - calls this common view into question. At least in the United States, students of medical geography, arguably the pre-eminent medical science in an age dominated by miasmatic theories of disease, readily adapted to the discovery of germs. (...) And although bacteriology quickly eclipsed medical geography in the world of medicine, place remained an important consideration in treating asthma (and allergies generally) throughout the post-bacteriological period. (shrink)

The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” (...) to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices. (shrink)

The purpose of this dissertation is to provide a defence of Aristotle's principle of contradiction against the critique made on it by Jan Lukasiewicz in an article he wrote in 1910 which was translated and published in the March 1971 number of The Review of Metaphysics. Lukasiewicz maintains in general that the law of contradiction has no logical worth. Specifically, he charges Aristotle with having several laws of contradiction instead of one as Aristotle claims; with attempting to prove (...) the law despite his claim that this is impossible and finally with failing in the very attempt to prove the law, or at least one of its formulations. In 20 chapters, each dealing with a respective section of Lukasiewicz's article, I attempt to show that all of Lukasiewicz's allegations are unfounded. My methodology is simply to follow Lukasiewicz through his 20 sections and compare what he says with what Aristotle actually wrote. If successful, this dissertation will show, in basic agreement with Professor Joseph Owens, that the best way to read Aristotle is on his own terms and not, as in the case of Lukasiewicz, via the latest developments in symbolic logic. I reiterate, with Aristotle, that the law of contradiction is the basic principle of being qua being and has logical worth precisely because of that fact. (shrink)

The study aimed to identify the creative environment of the technical colleges operating in Gaza Strip. The analytical descriptive method was used through a questionnaire which was randomly distributed to 289 employees of the technical colleges in Gaza Strip with a total number of (1168) employees and a response rate equal to (79.2%) of the sample study. The results confirmed the existence of a high degree of approval for the dimensions of the creative environment with a relative weight of (...) (75.19%) according to the perspective of the employees of the technical colleges in Gaza Strip. The results of the study showed a high level of creative environment (fluency, flexibility, originality, sensitivity to problems) in the technical colleges in Gaza Strip, where the field (fluency) ranked first and relative weight (76.86%), in the second place came the area (sense of problems) and relative weight (74.89%). The field of elasticity came in third place with a relative weight of (74.59%). Finally, the field of originality came in fourth and final rank with a relative weight of (74.41%). The results showed that there are differences between the technical colleges in the principles of the creative environment in all fields and the overall degree except for the field of flexibility. The most available colleges in these principles were the university college of applied sciences, and that was the least of the Faculty of Al-Aqsa Society. The results showed that there were differences according to the age variable in the areas of creative environment only in the sense of problems for age groups (less than 30 years) and (from 40 to 50 years). The researchers suggested a number of recommendations, the need to enhance the dimensions of the creative environment in technical colleges by working to improve the faculties of fluency, flexibility, originality and sensitivity to problems. The need to work on continuity and improve aspects of the creative environment of the colleges and the creation of new and innovative ways to support and develop and support these aspects combined. Necessity of technical colleges to continue to develop the creative environment (fluency, flexibility, originality, sensitivity to problems) for employees by engaging them in specialized training courses for creative thinking and problem solving. Developing work procedures with new innovative methods that will accomplish the various tasks quickly and accurately, and provide the effort, time and costs. The researchers urged more future studies that address the same variables of the current study in the field of creative environment and applied to other sectors. (shrink)

Georg Cantor argued that pure mathematics would be better-designated “free mathematics” since mathematical inquiry need not justify its discoveries through some extra-mental standard. Even so, he spent much of his later life addressing ancient and scholastic objections to his own transfinite number theory. Some philosophers have argued that Cantor need not have bothered. Thomas Aquinas at least, and perhaps Aristotle, would have consistently embraced developments in number theory, including the transfinite numbers. The author of this paper asks (...) whether the restriction of arithmetic to the natural numbers that is apparently assumed by Aristotle and Aquinas is necessary in the light of their stated principles. The author concludes that, while some texts from Aristotle and Thomas suggest that such discoveries as zero, rational, and real numbers, and even Cantor’s own transfinite numbers, are legitimate objects of scientific knowledge, a careful analysis shows that they are incompatible with the ultimate arithmetical principle, the unit. (shrink)

If, as is universally acknowledged, the proper goal of the mathematical sciences is the discovery of the least possible number of principles from which the universal laws of empirically given facts emerge with mathematical necessity, and thus the discovery of principles equivalent to those empirical facts, then it must appear as a duty of indubitable importance to reflect carefully on the principles that have already surfaced with some certainty in one area of the natural sciences and present them (...) in a form that really fulfills the equivalence requirement. (shrink)

The question of the limits of reason, not just within philosophy but also in the modern sciences, is arguably more important than ever given numerous recent commentaries on “life,” “reality,” meaning, purpose, pointlessness and so on, emanating not from philosophers or metaphysicians, but rather from physicists and biologists such as Steven Weinberg and Richard Dawkins. It will be argued that such commentaries concerning the “pointlessness” of the universe, or the purpose of “life,” and other such things, are flawed and unconvincing, (...) not least because they seem to overlook or forget a number of well known and significant philosophical contributions on the question of limits, particularly by Kant, but also by Hume, Russell and Sir A J Ayer. (shrink)

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties can (...) easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space. (shrink)

One of the main motivations for having a compositional semantics is the account of the productivity of natural languages. Formal languages are often part of the account of productivity, i.e., of how beings with finite capaci- ties are able to produce and understand a potentially infinite number of sen- tences, by offering a model of this process. This account of productivity con- sists in the generation of proofs in a formal system, that is taken to represent the way speakers (...) grasp the meaning of an indefinite number of sentences. The informational basis is restricted to what is represented in the lexicon. This constraint is considered as a requirement for the account of productivity, or at least of an important feature of productivity, namely, that we can grasp auto- matically the meaning of a huge number of complex expressions, far beyond what can be memorized. However, empirical results in psycholinguistics, and especially particular patterns of ERP, show that the brain integrates informa- tion of different sources very fast, without any felt effort on the part of the speaker. This shows that formal procedures do not explain productivity. How- ever, formal models are still useful in the account of how we get at the seman- tic value of a complex expression, once we have the meanings of its parts, even if there is no formal explanation of how we get at those meanings. A practice-oriented view of modeling gives an adequate interpretation of this re- sult: formal compositional semantics may be a useful model for some ex- planatory purposes concerning natural languages, without being a good model for dealing with other explananda. (shrink)

This note deals with the prepositional uniformity principlep-UP: p x N A (p, x) x N p A (p, x) ( species of all propositions) in intuitionistic mathematics.p-UP is implied by WC and KS. But there are interestingp-UP-cases which require weak KS resp. WC only. UP for number species follows fromp-UP by extended bar-induction (ranging over propositions) and suitable weak continuity. As corollaries we have the disjunction property and the existential definability w.r.t. concrete objects. Other consequences are: there is (...) no non-trivial countable partition of;id is the only injective function from to; there are no many-place injective prepositional functions; card () is incomparable with the cardinality of all metric spaces containing at least three elements. (shrink)

Philosophers have both produced as well as replied to a number of alleged "counter-examples" to the rule of substitution. Recently, Cartwright has urged that the standard reply to at least one of them is inadequate. The counter-example he singles out is:1). Giorgioni is so-called because of his size.2). Giorgiori = Barbarelli :3). Barbarelli is so-called because of his size.Cartwright argues that since 1) and 2) are true while 3) false, substitution has failed. It is argued in reply that, (...) contrary to Cartwright's claim, substitution does not even occur in the above argument. Rather, the meaning of the predicate "is so-called because of his size" changes from 1) to 3), rendering the invalidity the result of equivocation, not the failure of substitution. (shrink)

Timothy Williamson’s potentially most important contribution to epistemicism about vagueness lies in his arguments for the basic epistemicist claim that the alleged cut-off points of vague predicates are not knowable. His arguments for this are based on so-called ‘margin for error principles’. This paper argues that these principles fail to provide a good argument for the basic claim. Williamson has offered at least two kinds of margin for error principles applicable to vague predicates. A certain fallacy of equivocation seems (...) to underlie his justification for both kinds of principles. Besides, the margin for error principles of the first kind can be used in the derivation of unacceptable consequences, while the margin for error principles of the second kind can be shown to be compatible with the falsity of epistemicism, under a number of assumptions acceptable to the epistemicist. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting (...) the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper. (shrink)

The Pigeonhole Principle states that if n items are sorted into m categories and if n > m, then at least one category must contain more than one item. For instance, if 22 pigeons are put into 17 pigeonholes, at least one pigeonhole must contain more than one pigeon. This principle seems intuitive, yet when told about a city with 220,000 inhabitants none of whom has more than 170,000 hairs on their head, many people think that (...) it is merely likely that two inhabitants have the exact same number of hair. This failure to apply the Pigeonhole Principle might be due to the large numbers used, or to the cardinal rather than nominal presentation of these numbers. We show that performance improved both when the numbers are presented nominally, and when they are small, albeit less so. We discuss potential interpretations of these results in terms of intuition and reasoning. (shrink)

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new axiom (...) not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T $_0 \upharpoonright$ + UMID and T $_0 \upharpoonright$ + IND N + UMID have at least the strength of (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T $_0 \upharpoonright$ + UMID N and T $_0 \upharpoonright$ + IND N + UMID N have the same strength as (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

There are a number of candidate rationales for the settlement of Mars. These are considered in Sect. 10.1. At least one of them is economically plausible: its use as a base of operations for asteroid mining in the Main Belt. This rationale suggests that environmental protection on Mars needs to be considered in a broader context than that of the planet alone. More specifically, the authors argue in Sect. 10.2 that planetary environmental protection is partly a matter of (...) containment and so requires a framing principle. Section 10.3 supplies such a principle for protection, “the 1/8 principle”: while economic growth in space remains exponential and comparable to historical patterns on Earth, we should use no more than 1/8 of the available resources, in order to secure a reasonable ‘breaking distance’ before we reach a point of super-exploitation. Section 10.4 examines the difficulties of applying such a principle to Mars, the need for trade-offs and special weighting for special strategic resources such as uncollapsed lava tubes. Section 10.5 points out the ease with which the principle can be turned into policy options and its good matchup with human practices of valuing and use. The latter considerations may not be decisive, but they do count strongly in its favour. (shrink)

The article critically analyzes one of the central assumptions of Michael Walzer’s version of just war theory, as presented in his main work devoted to war ethics. As requested by the author of Just and Unjust Wars, the controversial nature of the principle of the moral equality of soldiers is revealed by discussing the actual course of events of a historical military conflict – namely, the outbreak of World War II, one of the main issues dealt with in Walzer’s (...) book. The exclusion of individual responsibility of people involved in a military operation which finds no moral justification seems to entail a number of consequences that put a heavy burden on Walzer’s overall model of military ethics. In order to ensure the necessary level of coherence of just war theory one may need to modify Walzer’s principle of the moral equality of soldiers – at least in relation to the participants of total war. (shrink)

One of the most striking features of the epistemological situation of Quantum Mechanics is the number of interpretations and the many schools of thought, with no consensus on the way to understand the theory. In this article, I introduce a distinction between orthodox interpretations and heterodox interpretations of Quantum Mechanics: the orthodox interpretations preserve all the quantum principles while the heterodox interpretations replace at least one of them. Then, I argue that we have strong empirical and epistemological reasons (...) to prefer orthodox interpretations to heterodox interpretations. The first argument is that all the experiments on the foundations of Quantum Mechanics give a high degree of corroboration to the quantum principles and, consequently, to the orthodox interpretations. The second argument is that the scientific progress needs a consensus: this consensus is impossible with the heterodox interpretations, while it is possible with the orthodox interpretations. Giving the preference to the orthodox interpretations is a reasonable position which could preserve both a consensus on quantum principles and a plurality of views on Quantum Mechanics. (shrink)

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications possesses a least fixed point. To be more precise, the new axiom not merely postulates the (...) existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T$_0$ + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T$_0$ + UMID in relating them to fragments of second order arithmetic based on $\Pi^1_2$ comprehension. [14] showed that T$_0 \upharpoonright$ + UMID and T$_0 \upharpoonright$ + IND$_\mathbb{N}$ + UMID have at least the strength of $\upharpoonright$ and, respectively. Here we are concerned with the exact reversals. Let UMID$_\mathbb{N}$ be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T$_0 \upharpoonright$ + UMID$_\mathbb{N}$ and T$_0 \upharpoonright$ + $IND_\mathbb{N}$ + UMID$_\mathbb{N}$ have the same strength as $ $\upharpoonright$ and, respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

This article develops an unconventional perspective on the utilitarianism of Bentham and Mill in at least four areas. First, it is shown that both authors conceived of utility as irreducibly multi-dimensional, and that Bentham in particular was very much aware of the ambiguity that multi-dimensionality imposes upon optimal choice under the greatest happiness principle. Secondly, I argue that any attribution of intrinsic worth to any form of human behaviour violates the first principles of Bentham's and Mill's utilitarianism, and (...) that this renders both authors immune to the claim by G. E. Moore that they committed a ‘naturalistic fallacy’. Thirdly, in light of these contentions, I find no flaw in Mill's ‘proof of utility’. Fourthly, I use the notion of intrapersonal utility weights to provide an interpretation of Mill's qualitative hedonism that is entirely consistent with his value monism. (shrink)

Contact between culturally distinct human groups in the contemporary ‘glocal’ -global and local- world is much greater than at any point in history. The challenge we face is the identification of the most convenient ways to organise the coexistence of different human language groups in order that we might promote their solidarity as members of the same culturally developed biological species. Processes of economic and political integration currently in motion are seeing increasing numbers of people seeking to become polyglots. Thus, (...) English is establishing itself as the usual world supra-language, although it coexists with other lingua francas that are widely used in certain parts of the globe. All this communicative reorganization of the human species may very well pose new problems and aggravate existing tensions as regards language and identity. It would seem that these processes comprise at least four major conceptual dimensions which must be taken into account above all else, as they are both widespread and, left unaddressed, may lead to significant social instability. These dimensions concern linguistic recognition, communicability, sustainability and integration. While accepting the utility of having an inter-national language, the keystone of the system is clearly that it must ensure the linguistic sustainability of each group. The basic principle is likely to be functional subsidiarity, i.e., whatever can be done by the local language should not be done by another one which is more global. As in the quote from Paracelsus --“the dose alone makes the poison”-- contact between languages is not ‘poisonous’ per se, but when the correct dose is exceeded it can prove harmful to the language whose position is weaker. A multilingual and communicated humanity is possible. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume and the Lockean Background: Induction and the Uniformity Principle David Owen Introduction What has come to be called Hume's problem of induction is special in many ways. It is arguably his most important and influential argument, especially when seen in its overall context of the more general argument about causaUty. It has come to be one of the great "standard problems" ofphilosophyandyetis,by most accounts, almost unique in (...) having no ancient precursor. Interestingly, the argument is frequently presented in terms that the author never used. Itis difficult for us not to think of the problem in terms of the contrast between deductive vs. inductive reasoning (or logic, or whatever). But this is a contrast nowhere to be found in Hume: he talks of demonstrative vs. probable (or moral) reasoning. Hume uses the word "induction" only twice in the Treatise1 and in neither case is it clear that the word is used toreportapieceofprobablereasoning.2One ofthefewoccurrences of"deduce" or "deducing" occurs in the second Enquiry, where he talks about "following the experimental method, and deducing general maximsfrom acomparisonofparticularinstances,"3hardlyaparadigm of what we would call deductive reasoning.4 Many modern studies of Hume that make a real attempt to understand what Hume meant by demonstration and probability, in the context of his argument about induction, tend towards the ahistorical,6 treating Hume's writings as contemporary texts. But failure to take into account some of the background concerning the nature of demonstrative and probable reason makes it unlikely, I think, that we can understand Hume's argument. Part ofthe problem is the failure to identify Hume's targets in that famous negative argument about induction (vague reference is sometimes made to the "rationalists") and part of the problem is identifying the role the uniformity principle plays in Hume's famous argumentand thelightitsheds on the conception ofprobablereasoning Hume was attacking. In this paper I want to argue that a careful look at Locke solves some ofthese problems: he has a well developed view ofdemonstrative and probable reasoning, his confidence in conformity as a ground of probability is a Ukely target of Hume's attack and his Volume XVIII Number 2 179 DAVID OWEN accountshedsmuchlightonwhatrole the principleofuniformitymight play in Hume's account.6 Of course, at a certain level of philosophical abstraction it is irrelevant whether we have understood what Hume "really meant." Theproblemofinductionhas emergedas a central onein epistemology, philosophical logic, and philosophy of science. It would be an interesting question in philosophical historiography to trace the development ofthe "problem" from Hume's time to our own but that is not my task here. I trust one needs no justification for taking an historical look at a famous problem rather than tackling the problem head-on in one's own terms. But if one is needed, here is mine: sometimes, by looking at a problem from a different, historical perspective, one sees a different problem or afamiliar problem in a new light. One's philosophical horizons are broadened. Furthermore, one may make an interesting historical discovery, or at least draw some interesting historical connections. I take it as uncontroversial that Locke, following Descartes, rejected a formal, logical account of demonstrative reasoning. Instead ofusing a formal, syllogistic account ofhow two propositions can entail a third, Locke concentrated on the relation two ideas bear one to another.7 Thus a demonstration, for Locke, getsits force, notfrom what we would call a deductive relation between one set of propositions (.the premises) and anotherproposition (the conclusion), butrather from the relation one idea bears to another, either directly, or via a chain of intermediate ideas. It is true that Locke frequently talks of inferring one proposition from another, but this is always cashed out in terms of relations ofideas via a chain ofintermediate ideas.8 It is also true that Locke occasionally uses the terms "deduce" and "deduction." But these are never used in our sense that has to do with formal deductive validity. Rather, they are simply used loosely in the informal sense of "argue" or "infer," and "argument" or "inference."9 In the rest of this paper, all uses ofthe word"deduction" andits cognates shouldbe taken as having the modern sense, unless flagged as having "Locke... (shrink)

Based on his theory of animalrights, Regan concludes that humans are morallyobligated to consume a vegetarian or vegandiet. When it was pointed out to him that evena vegan diet results in the loss of manyanimals of the field, he said that while thatmay be true, we are still obligated to consumea vegetarian/vegan diet because in total itwould cause the least harm to animals (LeastHarm Principle, or LHP) as compared to currentagriculture. But is that conclusion valid? Isit possible that (...) some other agriculturalproduction alternatives may result in leastharm to animals? An examination of thisquestion shows that the LHP may actually bebetter served using food production systemsthat include both plant-based agriculture and aforage-ruminant-based agriculture as comparedto a strict plant-based (vegan) system. Perhapswe are morally obligated to consume a dietcontaining both plants and ruminant(particularly cattle) animal products. (shrink)

This work deals with obstacles hindering a metaphysics of laws of nature in terms of dispositions, i.e., of fundamental properties that are causal powers. A recent analysis of the principle of least action has put into question the viability of dispositionalism in the case of classical mechanics, generally seen as the physical theory most easily amenable to a dispositional ontology. Here, a proper consideration of the framework role played by the least action principle within the classical (...) image of the world allows us to build a consistent metaphysics of dispositions as charges of interactions. In doing so we develop a general approach that opens the way towards an ontology of dispositions for fundamental physics also beyond classical mechanics. (shrink)

Albert Einstein once made the following remark about "the world of our sense experiences": "the fact that it is comprehensible is a miracle." (1936, p. 351) A few decades later, another physicist, Eugene Wigner, wondered about the unreasonable effectiveness of mathematics in the natural sciences, concluding his classic article thus: "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" (1960, p. 14). (...) At least three factors are involved in Einstein's and Wigner's miracles: the physical world, mathematics, and human cognition. One way to relate these factors is to ask how the universe could possibly be structured in such a way that mathematics would be applicable to it, and we would be able to understand that application. This is roughly Wigner's question. Alternatively, the way of the mathematical naturalist is to argue that we abstract certain properties from the world, perhaps using our bodies and physical tools, thereby articulating basic mathematical concepts, which we continue building into the complex formal structures of mathematics. John Stuart Mill, Penelope Maddy, and Rafael Nuñez teach this strategy of cognitive abstraction, in very different manners. But what if the very concepts and basic principles of mathematics were built into our cognitive structure itself? Given such a cognitive a priori mathematical endowment, would the miracles of the link between world and cognition (Einstein) and mathematics and world (Wigner) not vanish, or at least significantly diminish? This is the stance of Stanislas Deheane and Elizabeth Brannon's 2011 anthology, following a venerable rationalist tradition including Plato and Immanuel Kant. (shrink)

This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two-node PA-normal Kripke structure which does not force iΣ2. We prove i∀1 ⊬ i∃1, i∃1 ⊬ i∀1, iΠ2 ⊬ iΣ2 and iΣ2 ⊬ (...) iΠ2. We use Smorynski's operation Σ′ to show HA ⊬ lΠ1. (shrink)

If you’ve read the first five hundred pages of this book, you’ve read most of it (we assume that ‘most’ requires more than ‘more than half’). The set of natural numbers n such that the first n pages are most of this book is nonempty. Therefore, by the least number principle, it has a least member k. What is k? We do not know. We have no idea how to find out. The obstacle is something about (...) the term ‘most’. It is recognisably the same feature as the feature of ‘heap’ that prevents us from finding an answer to the question ‘How many grains make a heap?’ and the feature of many other expressions that prevents us from finding answers to similar questions involving them. Call this feature, whatever its underlying nature, vagueness. (shrink)

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res, as well as their tree-like versions, Res⁎. The contradictions we use are natural combinatorial principles: the Least number principle, LNPn and an ordered variant thereof, the Induction principle, IPn.LNPn is known to be easy for Resolution. We prove that its relativization is hard for Resolution, and more generally, the relativization of LNPn iterated d times provides a separation between Res (...) and Res. We prove the same result for the iterated relativization of IPn, where the tree-like variant Res⁎ is considered instead of Res.We go on to provide separations between the parameterized versions of Res and Res. Here we are able again to use the relativization of the LNPn, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res⁎ and Res⁎. Here, the relativization of IPn will not work as it is, and so we make a vectorizing amendment to it in order to address this shortcoming. (shrink)

ABSTRACT: It is argued that the equivalence, which is usually postulated to hold between infinite descent and transfinite induction in the foundations of arithmetic uses the law of excluded middle through the use of a double negation on the infinite set of natural numbers and therefore cannot be admitted in intuitionistic logic and mathematics, and a fortiori in more radical constructivist foundational schemes. Moreover it is shown that the infinite descent used in Dedekind-Peano arithmetic does not correspond to the infinite (...) descent of classical Fermatian arithmetic or number theory. However, from the point of view of classical logic, the principles of complete induction and transfinite induction, the least number principle and infinite descent are all equivalent. We find here a focal point for a foundational critique that aims for a a clarification of philosophical options in the foundations of logic and mathematics. (shrink)

If you’ve read the first five hundred pages of this book, you’ve read most of it (we assume that ‘most’ requires more than ‘more than half’). The set of natural numbers n such that the first n pages are most of this book is nonempty. Therefore, by the least number principle, it has a least member k. What is k? We do not know. We have no idea how to find out. The obstacle is something about (...) the term ‘most’. It is recognisably the same feature as the feature of ‘heap’ that prevents us from finding an answer to the question ‘How many grains make a heap?’ and the feature of many other expressions that prevents us from finding answers to similar questions involving them. Call this feature, whatever its underlying nature, vagueness. (shrink)

In the first chapter I have introduced Carnapian intensional logic against the background of Frege's and Quine's puzzles. The main body of the dissertation consists of two parts. In the first part I discussed Carnapian modal logic and arithmetic with descriptions. In the second chapter, I have described three Carnapian theories, CCL, CFL, and CNL. All three theories have three things in common. First, they are formulated in languages containing description terms. Second, they contain a system of modal logic. Third, (...) they do not contain the unrestricted classical substitution principle, but they do contain the classical substitution principle restricted to non-modal formulas and the Carnapian substitution principle, which says that two terms can be substituted salva veritate if they are necessarily coreferential. There are two major differences between the three theories. First, CCL and CFL allow universal instantiation with description terms, whereas CNL does not. Moreover, the quantificational theory of the CCL is classical, whereas the quantificational theory of CFL is a free logic. Another difference is that CCL and CFL contain different description principles. Most importantly, the description principle of CCL ensures that even improper descriptions have a denotation, whereas the description principle of CFL does not guarantee this. CNL does not have a description principle. In the third chapter, I have studied collapse arguments for CCL, CFL, and CNL. A collapse argument is an argument for the following statement: if p is true, then it is necessarily true. A crucial role in the proofs of these collapse results was played by so-called self-predication principles, which say that under certain conditions the predicate that expresses the descriptive condition can be combined by the description term formed out of that predicate with the result being a true sentence. In this chapter I have discussed a collapse argument for the extension of CCL with a self-predication principle, I have given a collapse argument for a similarly extended CFL, and most importantly, I have given a collapse argument for the extension of CNL with a self- predication principle. Finally, I have argued that the relevant self-predication principles are unsound under a Carnapian interpretation. In the fourth chapter, I have studied the extension of Peano Arithmetic with a Carnapian modal logic C, which is a dummy letter standing for either CCL or CFL. One can prove that the principle of the necessity of identity is a theorem of CPA. This implies that one gets a collapse result for CPA. The standard principle of weak induction was crucial for the proof. One can also prove that, if one assumes a particular self-predication principle, and if one assumes the principle of strong induction or, equivalently, the least-number principle, then one gets a partial collapse of de re modal truths in de dicto modal truths. I have argued that, if the box operator is interpreted as a metaphysical necessity operator, then Platonists would not be inimical to the collapse result. But if CPA is extended with a physical theory, then there is a threat that physical truths become physical necessiti es. It was shown that, under a Carnapian interpretation, the standard principle of weak induction is unsound, and that it can be replaced by a Carnapian principle of weak induction that is sound. The problem of logical and mathematical omniscience prevents ordinary Carnapian intensional logic from being taken seriously as a logic adequate for describing the principles of demonstrability. Yet many of the proof-theoretic results of the first part carry over to the part on Carnapian epistemic arithmetic with descriptions, since proof-theoretic results are independent of the informal reading of the operators. In the fifth chapter, I looked at extensions of arithmetic with a modal logic in which the box operator is interpreted as a demonstrability operator. A first extension in that sense is Shapiro s Epistemic Arithmetic. Shapiro himself offered the problem of mathematical omniscience as a reason why it is difficult to find a model theory for EA. Horsten attempted to provide a model theory via the detour of Modal-Epistemic Arithmetic. The attention of the reader was drawn to an incoherence in the model theory of. Two alternative solutions were presented and, after a short discussion of the problem of de re demonstrability one of those alternatives was chosen. The discussion of the problem of de re demonstrability made it clear that it would be interesting to study the epistemic properties of notation systems. Horsten himself provided a framework for this, viz. Carnapian Epistemic Arithmetic, and he started a systematic study of the epistemic properties of notation systems within that framework. However, he did not provide non-trivial but adequate models. To make a start with solving the problem of finding good models for CEA, I introduced Carnapian Modal-Epistemic Arithmetic In constructing CMEA I incorporated the lesson about the principle of weak induction learnt in the fourth chapter. In the sixth chapter, I gave a critical assessment of an argument concerning the limits of de re demonstrability about the natural numbers. The conclusion of the Description Argument is that it is undemonstrable that there is a natural number that has a certain property but of which it is undemonstrable that it has that property. A crucial step in the Description Argument involved a self-predication principle. Making good use of one of the results obtained in the third chapter, I proved a collapse result for the background theory against which the Description Argument was formulated. I concluded that either the either the Description Argument is sound but its conclusion is trivial, o r the Description Argument is unsound, or it is a cheapshot. As an appendix I included an article co-authored by prof. dr. Leon Horsten and me. The topic of the article is indirectly related to some other topics investigated in my dissertation. Also, it backs up one of the addition al theses I might be asked to publicly defend during my doctoral exam. T he topic of the appendix is the set of the so-called paradoxes of strict implication. Jonathan Lowe has argued that a particular variation on C.I. Lewis notion of strict implication avoids the paradoxes of strict implication. Pace Lowe, it is argued that Lowe s notion of implication does not achieve this aim. Moreover, a general argument is offered to the effect that no other variation on Lewis notion of constantly strict implication describes the logical behaviour of natural language conditional s in a satisfactory way. (shrink)

The least infringement principle has been widely endorsed by public health scholars. According to this principle, public health policies may infringe upon “general moral considerations” in order to achieve a public health goal, but if two policies provide the same public health benefit, then policymakers should choose the one that infringes least upon “general moral considerations.” General moral considerations can encompass a wide variety of goals, including fair distribution of burdens and benefits, protection of privacy and (...) confidentiality, and respect for autonomy.In this article, we argue that the Center for Disease Control and Prevention's 2006 HIV screening recommendations may violate the least infringement principle. This is a concern because, although not legally binding, the Recommendations appear to have already influenced state laws and will likely continue to do so as legislative proposals continue to be passed. At a minimum, therefore, the Recommendations have important implications for HIV screening within the United States. (shrink)

We study the quantifier complexity and the relative strength of some fragments of arithmetic axiomatized by induction and minimization schemes for Δn+1 formulas.

The present paper investigates why logical empiricists remained silent about one of the most philosophy-laden matters of theoretical physics of their day, the principle of least action (PLA). In the two decades around 1900, the PLA enjoyed a remarkable renaissance as a formal unification of mechanics, electrodynamics, thermodynamics, and relativity theory. Taking Ernst Mach's historico-critical stance, it could be liberated from much of its physico-theological dross. Variational calculus, the mathematical discipline on which the PLA was based, obtained a (...) new rigorous basis. These three developments prompted Max Planck to consider the PLA as formal embodiment of his convergent realist methodology. Typically rejecting ontological reductionism, David Hilbert took the PLA as the key concept in his axiomatizations of physical theories. It served one of the main goals of the axiomatic method: ''deepening the foundations.'' Although Moritz Schlick was a student of Planck's, and Hans Hahn and Philipp Frank enjoyed close ties to Gottingen, the PLA became a veritable Shibboleth to them. Rather than being worried by its historical connections with teleology and determinism, they erroneously identified Hilbert's axiomatic method tout court with Planck's metaphysical realism. Logical empiricists' strict containment policy against metaphysics required so strict a separation between physics and mathematics to exclude even those features of the PLA and the axiomatic method not tainted with metaphysics. (shrink)

The Oxford Companion to the Mind is a classic. Published in 1987, to huge acclaim, it immediately took its place as the indispensable guide to the mysteries - and idiosyncracies - of the human mind. In no other book can the reader find discussions of concepts such as language, memory, and intelligence, side by side with witty definitions of common human experiences such as the 'cocktail-party' and 'halo' effects, and the least effort principle. Richard Gregory again brings his (...) wit, wisdom, and expertise to bear on this most elusive of subjects. Research into the mind and brain has moved on in bounds in recent years, and interest in the subject has never been so high. There has been a shift in focus away from Freud's concept of the unconscious onto consciousness itself. The new edition of the Companion includes three 'mini symposia' - on consciousness, brain scanning, and artificial intelligence - with contributions from a number of specialists, and encompassing a range of approaches. Cultural as well as scientific in approach, this accessible book offers authoritative descriptions and analysis. With new entries on controversial topics such as artificial life, attachment theory, caffeine, cruetly, drama, extra-terrestrial intelligence, genetics of mental illness, imagination, lying, puzzles, and twins, this highly-anticipated second edition explores the most intriguing of subjects. (shrink)

Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.

Despite the importance of the variational principles of physics, there have been relatively few attempts to consider them for a realistic framework. In addition to the old teleological question, this paper continues the recent discussion regarding the modal involvement of the principle of least action and its relations with the Humean view of the laws of nature. The reality of possible paths in the principle of least action is examined from the perspectives of the contemporary metaphysics (...) of modality and Leibniz's concept of essences or possibles striving for existence. I elaborate a modal interpretation of the principle of least action that replaces a classical representation of a system's motion along a single history in the actual modality by simultaneous motions along an infinite set of all possible histories in the possible modality. This model is based on an intuition that deep ontological connections exist between the possible paths in the principle of least action and possible quantum histories in the Feynman path integral. I interpret the action as a physical measure of the essence of every possible history. Therefore only one actual history has the highest degree of the essence and minimal action. To address the issue of necessity, I assume that the principle of least action has a general physical necessity and lies between the laws of motion with a limited physical necessity and certain laws with a metaphysical necessity. (shrink)

We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we (...) give a new proof, which can be formalized in IΔ0 + Δ0EQ, of the fact that every prime of the form 4n + 1 is the sum of two squares. (shrink)

The principle of least action (PLA) has often been cited as a counterexample to the dominant mode of causal explanation in physics. In particular, PLA seems to involve an appeal to final causes or some other teleological ideology. However, Ben-Menahem (Causation in science, Princeton University Press, Princeton, 2018) argues that such implications no longer apply given that PLA can be recovered as limiting case from quantum theory. In this paper, I argue that the metaphysical implications of PLA-based explanations (...) are not undermined by its status as derivative. However, I contend that PLA functions as a diachronic constraint that licenses explanations by constraint (Lange, Because without cause: non-casual explanations in science and mathematics. Oxford University Press, Oxford, 2016). PLA-based explanations, on this account, are non-causal but also differ from familiar cases of teleological explanations in several respects. (shrink)

Epistemology has recently witnessed a number of efforts to rehabilitate rationalism, to defend the existence and importance of a priori knowledge or warrant construed as the product of rational insight or apprehension (Bealer 1987; Bigelow 1992; BonJour 1992, 1998; Burge 1998; Butchvarov 1970; Katz 1998; Plantinga 1993). This effort has sometimes been coupled with an attack on naturalistic epistemology, especially in BonJour 1994 and Katz 1998. Such coupling is not surprising, because naturalistic epistemology is often associated with thoroughgoing empiricism (...) and the rejection of the a priori. In this paper, however, I shall present a conception of naturalistic epistemology that is entirely compatible with a priori justification or knowledge. The resulting conception, I claim, gives us a better appreciation of the respective merits of the rational and the empirical, as well as a better understanding of how moderate epistemological naturalism comports, at least in principle, with moderate rationalism. This paper defends moderate rationalism; but it does not defend everything rationalists have often wanted, only what it is reasonable to grant them. (shrink)

Simon Evnine examines various epistemic aspects of what it is to be a person. Persons are defined as finite beings that have beliefs, including second-order beliefs about their own and others' beliefs, and are agents, capable of making long-term plans. It is argued that for any being meeting these conditions, a number of epistemic consequences obtain. First, all such beings must have certain logical concepts and be able to use them in certain ways. Secondly, there are at least (...) two principles governing belief that it is rational for persons to satisfy and are such that nothing can be a person at all unless it satisfies them to a large extent. These principles are that one believe the conjunction of one's beliefs and that one treat one's future beliefs as, by and large, better than one's current beliefs. Thirdly, persons both occupy epistemic points of view on the world and show up within those views. This makes it impossible for them to be completely objective about their own beliefs. Ideals of rationality that require such objectivity, while not necessarily wrong, are intrinsically problematic for persons. This "aspectual dualism" is characteristic of treatments of persons in the Kantian tradition. In sum, these epistemic consequences support a traditional view of the nature of persons, one in opposition to much recent theorizing. (shrink)

My aim is to argue for the incompatibility of one of the central principles of physics, namely the principle of least action (PLA), with the increasingly popular view that the world is, ultimately, merely something like a con- glomerate of objects and irreducible dispositions. First, I argue that the essentialist implications many suppose this view has are not compatible with the PLA. Second, I argue that, irrespective of whether this view has any essentialist implications, it is not compatible (...) with the kind of explanation that the PLA affords. (shrink)

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated (...) by J. Paris is whether $I\Delta_1$ proves the corresponding least element principle for decidable predicates, $L\Delta_1$ (or, equivalently. the $\Sigma_1-collection$ principle $B\Sigma_1$ ). We reduce this question to a purely computation-theoretic one. (shrink)

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind (...) is called an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

The paper investigates whether it is plausible to hold the late stage demented criminally responsible for past actions. The concern is based on the fact that policy makers in the United States and in Britain are starting to wonder what to do with prison inmates in the later stages of dementia who do not remember their crimes anymore. The problem has to be expected to become more urgent as the population ages and the number of dementia patients increases. This (...) paper argues that the late-stage demented should not be punished for past crimes. Applicable theories of punishment, especially theories with an appropriate expressivist, or communicative element, fail to justify the imprisonment of the late-stage demented. Further imprisonment would require a capacity for comprehension on the part of the punished, and, under certain narrowly specified conditions, even a capacity to be at least in principle capable of recalling the crime again. (shrink)