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)

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)

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)

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 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)

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)

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 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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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 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)

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)

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)

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)

Logic is traditionally considered to be a purely syntactic discipline, at least in principle. However, prof. David Isles has shown that this ideal is not yet met in traditional logic. Semantic residue is present in the assumption that the domain of a variable should be fixed in advance of a derivation, and also in the notion that a numerical notation must refer to a number rather than be considered a mathematical object in and of itself. Based on (...) his work, the central question of this thesis is what kind of logic, if any, results from removing this semantic residue from traditional logic.We differ from traditional logic in two significant ways. The first is that the assumption that a numerical notation must refer to a number is denied. Numerical notations are considered as mathematical objects in their own right, related to each other by means of rewrite rules. The traditional notion of reference is then replaced by the notion of reduction (by means of the rewrite rules) to a normal form. Two numerical notations that reduce to the same normal form would traditionally be considered identical, as they would refer to the same number, and hence they would be interchangeable salva veritate. In the new system, called Buridan-Volpin (BV), the numerical notations themselves are the elements of the domains of variables, and two numerical notations that reduce to the same normal form need not be interchangeable salva veritate, except when they are syntactically identical (i.e., have the same Gödel number).The second is that we do away with the assumption that the domains of variables need to be fixed in advance of a derivation. Instead we focus on what is needed to guarantee preservation of truth in every step of a derivation. These conditions on the domains of the variables, accumulated in the course of a derivation, are combined in a reference grammar. Whereas traditionally a derivation is considered valid when the conclusion follows from the premisses by way of the derivation rules (and possibly axioms), in the BV system a derivation must meet the extra condition that no inconsistency occurs within the reference grammar. For if the reference grammar were to give rise to inconsistency (i.e., it would be impossible to assign domains to all the variables without breaking at least one of the conditions placed on them in the reference grammar), there is no longer a guarantee that truth has been preserved in every step of the derivation, and hence the truth of the conclusion is not guaranteed by the derivation.In Chapter 2 the BV system is introduced in some formal detail. Chapter 3 gives some examples of derivations, notably totality of addition, multiplication and exponentiation, as well as a lemma needed for the proof of Euclid’s Theorem. These examples, taken from prof. Isles’ First-Order Reasoning and Primitive Recursive Natural Number Notations, show that there is a real proof-theoretical difference between traditional logic and the BV system. Here we also find the first major point of departure between myself and prof. Isles, centered on the notion of inheritance of conditions in the reference grammar by way of lemmata. These different points of view are best illustrated in the sections on the totality of exponentiation and on Euclid’s Lemma: prof. Isles maintains that the proof of totality of exponentiation is not BV valid, while I maintain that it is. But I do agree with him that the traditional proof of Euclid’s Lemma is not BV valid. Chapter 6 also expands the arguments for my choice in this matter.Now that it has been shown that there is a difference between traditional logic and BV, the properties of BV need to be examined. In Chapter 4 we give a proof of Cut-elimination for BV minus induction and the subformula property for BV, which allows us to prove the consistency of BV minus induction. We also expand on the reasons for excluding induction. In Chapter 5 we consider in detail the proof of a finite analog to the Löwenheim–Skolem theorem given by prof. Isles in his article with the same title. He proves that under certain conditions it is always possible, given the existence of a (possibly uncountable) model for a derivation, to give a finite model for this derivation. The system he considers deviates from BV as considered in this thesis in two significant ways: it does not contain the induction rule and the domains contain numbers instead of numerical notations. We then go on to show that it is possible to extend the result to include induction, in the sense that the existence of a possibly uncountable model for a derivation guarantees the existence of a model that is at most countable. We also consider the complications that arise from taking numerical expressions instead of numbers as the elements of domains.Finally, in Chapter 6 we consider the philosophical consequences of the BV system, informed by the formal results from the previous chapters. In particular we discuss the relation between reduction and reference, the status of reference grammars, the notion of induction and its function within BV, and some brief considerations on the consequences of the BV system for the discussion regarding nominalism and realism with regard to mathematical objects. The object of the chapter is twofold. On the one hand applying the formal results to philosophical questions, on the other hand arguing that BV is not just a theoretically acceptable alternative to traditional logic, but is in fact deserving of further development and research into its properties. The latter will probably appeal most to those of a nominalist and/or finitist bent.Abstract prepared by Harrie de Swart and Sven Storms.E-mail:[email protected]:https://research.tilburguniversity.edu/files/6170 1294/Storms_The_Buridan_Volpin_04_05_2022.pdf. (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)

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

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)

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)

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)

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)

Understanding the nature of errors in a simple, rule‐based task—programming a VCR—required analyzing the interactions among human cognition, the artifact, and the task. This analysis was guided by least‐effort principles and yielded a control structure that combined a rule hierarchy task‐to‐device with display‐based difference‐reduction. A model based on this analysis was used to trace action protocols collected from participants as they programmed a simulated VCR. Trials that ended without success (the show was not correctly programmed) were interrogated to yield (...) insights regarding problems in acquiring the control structure. For successful trials (the show was correctly programmed), steps that the model would make were categorized as matches to the model; steps that the model would not make were violations of the model. The model was able to trace the vast majority of correct keystrokes and yielded a business‐as‐usual account of the detection and correction of errors. Violations of the model fell into one of two fundamental categories. The model provided insights into certain subcategories of errors; whereas, regularities within other subcategories of error suggested limitations to the model. Although errors were rare when compared to the total number of correct actions, they were important. Errors were made on 4% of the keypresses that, if not detected, would have prevented two‐thirds of the shows from being successfully recorded. A misprogrammed show is a minor annoyance to the user. However, devices with the approximate complexity of a VCR are ubiquitous and have found their way into emergency rooms, airplane cockpits, power plants, and so on. Errors of ignorance may be reduced by training; however, errors in the routine performance of skilled users can only be reduced by design. (shrink)

According to infinitism, all justification comes from an infinite series of reasons. Peter Klein defends infinitism as the correct solution to the regress problem by rejecting two alternative solutions: foundationalism and coherentism. I focus on Klein's argument against foundationalism, which relies on the premise that there is no justification without meta-justification. This premise is incompatible with dogmatic foundationalism as defended by Michael Huemer and Time Pryor. It does not, however, conflict with non-dogmatic foundationalism. Whereas dogmatic foundationalism rejects the need for (...) any form of meta-justification, non-dogmatic foundationalism merely rejects Laurence BonJour's claim that meta-justification must come from beliefs. Unlike its dogmatic counterpart, non-dogmatic foundationalism can allow for basic beliefs to receive meta-justification from non-doxastic sources such as experiences and memories. Construed thus, non-dogmatic foundationalism is compatible with Klein's principle that there is no justification without meta-justification. I conclude that Klein's rejection of foundationalism. fails. Nevertheless, I agree with Klein that when in response to a skeptical challenge we engage in the activity of defending our beliefs, the number of reasons we can give is at least in principle infinite. I argue that this type of infinity is benign because, when we continue to give reasons, we will eventually merely repeat previously stated reasons. Consequently, I reject Klein's claim that the more reasons we give the more we increase the justification of our beliefs. (shrink)

This article explores the strategies available for defending Aquinas’s view of existence in the context of contemporary analytic philosophy. The rival view of existence prevalent among contemporary analytic philosophers is subject to serious objections. At the same time, the main contemporary analytic objections to Aquinas’s view can be adequately answered. The widespread use of “exist(s)” to ascribe existence to individuals and objects provides good reason to think that such use makes sense, and analogies like those of Aquinas can help to (...) illustrate the commonsense meaning of “exist(s)” to those who doubt it. The truth of Aquinas’s view of existence as a real property of individuals or objects can be defended at least in principle by appealing to its descriptive and explanatory usefulness. In addition to Aquinas’s own uses of it, there are a number of areas in contemporary analytic metaphysics where his view of existence could likely be of use, such as contemporary essentialism, substance-attribute ontology, and the revival of ontological pluralism. (shrink)

We begin with the following quotation from Karp [1964]: My interest in infinitary logic dates back to a February day in 1956 when I remarked to my thesis supervisor, Professor Leon Henkin, that a particularly vexing problem would be so simple if only I could write a formula which would say x = 0 or x = 1 or x = 2 etc. To my surprise, he replied, "Well, go ahead." 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— such as that in the above quotation—would be naturally identified as infinite sets. A "language" of this kind is called an infinitary language: in this article we 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 we shall see 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. In §1 we lay down the basic syntax and semantics of infinitary languages and demonstrate their expressive power by means of examples. §2 is devoted to those infinitary languages which permit only finite quantifier sequences: these languages turn out to be relatively well-behaved. In §3 we discuss the compactness problem for infinitary languages and its.. (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)

There is a long tradition in the philosophy of the social sciences that emphasizes the meaningfulness of human action. This tradition doubts or even negates the possibility of causal explanations of human action precisely on the basis that human actions have meaning. This paper provides an argument in favour of methodological naturalism in the social sciences. It grants the main argument of the Interpretivists, i.e. that human actions are meaningful, but it shows how a transformation of a “nexus of meaning” (...) into a “causal nexus” can take place, proposing the “successful transformation argument”. Based on previous work presented in my Naturalistic Hermeneutics, Cambridge University Press, 2005 the paper discusses four approaches that describe the “nexus of meaning” connected with a human action based on the motives, the intentions, the reasons, and the rationality of the action respectively. From the standpoint of eachapproach a causal nexus can arise, namely, if the respective motives, intentions, reasons, or the human rationality that are manifest in the different nexuses of meaning exhibit certain invariances. These approaches can be formulated as theories, which explain human action every time that the nexuses of meaning – described with the diverse conceptual apparatuses – can be transformed into causal nexuses. It is shown that explanations based on motives, intentions, reasons, or rationality are possible, as are explanations of human action based on, at least in principle, an unlimited number of other theories. (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)

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)

If there is one constant, uninvited guest in the typical public school classroom—or indeed in any setting in which children gather in numbers—it is conflict. The transcripts from which I draw in this reflection on how young children think together about conflict reflect two four-part sets of conversations with two second grades in a small school of roughly 300 students in a predominantly middle to upper middle class suburban town in a heavily populated metropolitan area in the northeastern U.S. Most (...) of the examples of conflict which the second graders chose to offer were located in their lives with friends or acquaintances or siblings, or incidents among adults that they had witnessed. There seemed to be a level of tolerance, even expectation and affirmation of these small conflicts in their lives; they had already become relatively “natural” occurrences for them. Large sections from four of the transcripts are included here, with commentary. In the first, conflict is represented by the group as a competition, either between two people or two possibilities only one of which can be fulfilled . There is disagreement as to whether interpersonal conflict can be avoided. The second section revolves around the reorganizational or reconstructive potential of conflict. The third section takes up the question of whether we can say that there is conflict within nature beyond just living things — i.e. whether conflict can be considered a metaphysical or at least ontological principle. Transcript and analysis of arguments are accompanied by reflections on the differing social atmospheres of the two classes, their possible relationships to the discussion styles of the two, and on the possibility of a form of pedagogy which allows for the self-organizing character of group life and the role of conflict in the dialectics of development. (shrink)

Philip Pettit has argued that more robust harms are worse than less robust ones, other things equal, and thinks that appealing to this presumption can help us rationalise the appeal of a number of widely-held moral principles. In this paper, I challenge this view. I argue against the presumption and suggest that, even if it were correct, it could not give much support to the moral principles that Pettit discusses. I also claim, however, that Pettit has the resources at (...) his disposal to explain the attraction of the principles in another way, and lay out how such an explanation would proceed. As moral heuristics, at least, these principles can be grounded on the need to maintain social norms necessary to guarantee individuals’ security. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Cambridge Companion to Galileo ed. by Peter MachamerMartin CurdPeter Machamer, editor. The Cambridge Companion to Galileo. Cambridge: Cambridge University Press, 1998. Pp xii + 462. Cloth, $59.95. Paper, $19.95.The contributions fall into three main areas: Galileo’s work on mechanics, his defense of Copernicus, and his relationship with the church. The relative number of pages devoted to these topics is unusual: the ratio is roughly 3 to (...) 1 to 3.The papers on Galileo’s mechanics and physics are by William Wallace, Peter Machamer, Rivka Feldhay, Wallace Hooper, and Pietro Redondi. Wallace concentrates on unpublished manuscripts from Galileo’s early period, arguing that they reveal Galileo’s intense involvement with Aristotelian philosophy. As in his other writings, Wallace emphasizes the importance of the demonstrative regressus argument for Galileo’s thought, especially for Galileo’s later claim in the Dialogue to have demonstrated the earth’s motion from the phenomenon of the tides. Machamer offers an “Archimedean” [End Page 364] interpretation of Galileo, stressing the Galileo’s use of simple machines, especially the balance, as a model of intelligibility throughout his scientific work. According to Machamer, these machines functioned as something like a Kuhnian exemplar for Galileo, providing the element of necessity and picturability that Galileo deemed essential for scientific explanation. Feldhay explores the role of mathematics in Galileo’s science, arguing that Galileo began to diverge from the Jesuit-Aristotelian tradition once he gave up trying to understand the causes of accelerated motion and focused instead on its geometrical analysis in terms of degrees of velocity. In a useful discussion, Hooper follows the development of Galileo’s inertial mechanics, concluding that even in the Discourses (1638) it is unclear whether Galileo abandoned his pre-Newtonian notion of impetus as the sustaining cause of motion. Redondi examines the relationship of Galileo’s physics to his theology. While Redondi’s conclusion is clear—“Galileo’s physics required God”—his argument is hard to follow, ranging as it does over Galileo’s views on time, atomism, macroscopic and microscopic physics, and cosmogony.Only two articles are directly about Galileo and the Copernican theory but those two pieces, by William Shea and Noel Swerdlow, are models of clarity. Swerdlow focuses on Galileo’s telescopic discoveries 1610–13 while Shea covers the entire period up to the Dialogue of 1632. Unfortunately, this involves repetition of the material about the telescopic discoveries, including some diagrams and illustrations. Tighter editorial control might have avoided this duplication and allowed Shea to devote more time to the fascinating arguments in the Dialogue. What we have is fine, but more would have been better.On Galileo and the church, there are contributions from Ernan McMullin, Richard Blackwell and Marcello Pera. McMullin argues that the seeds of Galileo’s downfall in 1633 were sown in his Letter to the Grand Duchess Christina (1615) in which Galileo endorsed a theory of scriptural interpretation remarkably similar to that contained in Augustine’s commentary on Genesis. McMullin finds a tension in that theory between regarding Scripture and natural science as mutually irrelevant while at the same time giving science priority over the literal reading of Scripture if and only if a demonstration has been provided. Once the demonstration principle was adopted, Galileo’s task of defending the earth’s motion against biblical literalism in the Dialogue became hopeless. Blackwell asks whether there could be another Galileo case in light of the Vatican’s admission of error in 1992. His answer is “Yes.” Blackwell reminds us that while in 1616 it was the Copernican theory that been tried and found heretical, in 1633 the sole issue was Galileo’s obedience to church authority. In Blackwell’s view, there remains the potential for conflict between the church’s “logic of centralized authority” and the democratic, pluralistic methods that give legitimacy to science. Pera, too, concludes that the possibility of conflict remains as long as the church insists that there are truths of faith, necessary for salvation, and those doctrines also have a factual content amenable, at least in principle, to scientific investigation.The book concludes with two pieces: a survey by Michael Segre of the evolution of Galileo’s... (shrink)

Bu çalışma, Türkiye’deki Şer’i yönetişim aktörlerinden olan banka içi danışma komitelerinin Şer’i yönetişimdeki yeri ve önemini keşfetmeyi amaçlamaktadır. Bu minvalde nitel araştırma desenlerinden olan durum çalışması tercih edilmiş ve 5 danışma komitesi üyesi ve 2 danışma komitesi başkanıyla yarı yapılandırılmış mülakatlar gerçekleştirilmiştir. Mülakatlar neticesinde toplam 625 dakikalık ses kayıtları elde edilerek çözümlenmiş ve 85 sayfalık metin elde edilmiştir. Elde edilen metinler ve görüşme notları ATLAS.ti programı aracılığıyla analiz edilerek BİDK, Şer’i yönetişim ve Merkezi Danışma Kurulu olmak üzere 3 ana tema (...) ve bunlara bağlı 26 alt temaya ulaşılmıştır. İlgili temalar BİDK’nin Türkiye’deki Şer’i yönetişimdeki yeri ve önemini keşfetmek açısından zengin bir içerik sunmaktadır. Türkiye’deki Şer’i yönetişimi ele alan çalışmaların literatürde sınırlı olması ve bu çalışmanın hem saha çalışmasına hem de Türkiye’deki Şer’i yönetişimde yer alan, katılım bankacılığının tarihine eş bir ömre sahip BİDK’ları konu etmesi literatürde özgün bir yere sahip olacağına işaret etmektedir. Islamic finance has developed the Shariah governance mechanism to ensure compliance and control of IFIs with the principles of Islamic law. However, the levels of Shariah governance are different from each other due to the different legal affiliations of each country, the level of development of Islamic finance, and regime differences. International Islamic financial institutions such as Islamic Financial Services Board and The Accounting and Auditing Organization for Islamic Financial Institutions have published standards to define the general lines of Shariah governance and have brought Shariah governance practices to a certain point, at least in principle. After the Shariah governance standards of IFSB and AAOIFI, studies on Shariah governance have increased rapidly. By 2010, while there were only 18 publications on Shariah governance in the two most well-known indexes, this number increased to 42 in 2015 and 159 in 2020. This academic interest shows that Shariah governance issues are followed closely. In this context, there have been some remarkable developments at the regulatory and supervisory level in terms of Shariah governance in Turkey, and the TKBB Advisory Board was established in 2018, bringing one step closer to international Shariah governance practices. (shrink)

The considerations of Part I are extended and the experimental data and hypotheses that led to the establishment of the general theory of relativity are analyzed. It is found that one of the fundamental assumptions is that light is propagated homogeneously; i.e., by using arbitrary systems of coordinates, propagation of light can be represented by a homogeneous quadratic form. This is shown to be an assumption that can be verified by experiment, at least in principle. As a result (...) of adding a number of further assumptions to this, the usual formalism of the general theory of relativity can be established. In the above point of view, the general theory of relativity—like any other theory—cannot be built upad hoc, but is built on distinct physical hypotheses, each of which can be subjected to test by experiment. (shrink)

A number of philosophers argue that the moral value of national identity is sufficient to justify at least a prima facie right of a national community to create its own independent, sovereign state. In the literature, this argument is commonly referred to as the identity argument. In this paper, I consider whether the identity argument successfully proves that a national group is entitled to a state of its own. To do so, I first explain three important steps in (...) the argument and then consider whether they lead to the desired conclusion. My examination reveals that the identity argument relies on the Optimal Protection Principle; however, this principle does not apply to the case of a national community. As a result, the identity argument fails to justify even a prima facie right of a national community to establish its own state. (shrink)

Among the quantificational notions neglected by classical logic are "many," "few," and "nearly all." Despite the apparent vagueness associated with these terms in ordinary discourse, in specific contexts we can and do draw strict inferences from statements in which they occur. In this pioneering work, Altham has attempted to uncover something of the formal logic that justifies such inferences. He begins by showing the mutual interdefinability of the three terms. If negation and any one of them are taken as primitive, (...) the other two are definable in terms of these. Selecting as his basic semantic notion the idea of a manifold or many-membered-set, he shows it is only necessary to fix upon a number n, greater than one, to represent the least number of elements to constitute a specific manifold and then one can construct a natural deductive system which uses the standard predicate logic. He develops at great length a weak system that yields principles of inference which are valid in every non-empty domain and in which the classical predicate logic appears as a limiting case. One point that emerges is that the three plurality operators may all be exhibited as involving a double quantification not characteristic of "some," "none," or "all." Next, two stronger systems are developed more briefly which are valid only in many-membered domains. The strongest can represent the largest set of informal arguments and seems to be the most natural of the three. Nevertheless, since these three systems admittedly do not give an exhaustive account of the use of these quantifiers in informal discourse, Altham explains three further systems which correspond more closely to the natural language use of these terms. Finally an appendix indicates what modifications would be required for an intuitionistic logic like Heyting's or the minimal logic of Kolmogorov-Johansson. The text is clearly written and requires no greater logical sophistication to understand than would be needed to follow Lemmon's Beginning Logic or Mates' Elementary Logic.--A. B. W. (shrink)