The proposed heavy dependence on Pavlovian conditioning to account for social behavior confounds phylogenically and ontogenically selected behavior patterns and ignores the extension of the principle of selection by consequences from biological to learning theory. Instead of acknowledging operant relations, Domjan et al. construct vaguely specified mechanisms based upon anticipatory cost-benefit considerations that are not supported by the Pavlovian conditioning literature.

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...) (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

Rather than reading Plato's philosophy of arithmetic ‘charitably’, it is better to try to explain its failure to generate any fruitful ideas. Prominent in the explanation is Plato's focus on predicates assigning cardinalities and on ‘groups’ falling under them. This focus left Plato unable to envisage the possibility, emerging in Dedekind and Frege but which arithmetic in Plato's time would not easily have suggested, of regarding numbers as objects essentially ranged in the structure of a progression.

Leibniz said with a mixture of admiration and inspiration that the Duchess Sophie of Hannover always wanted to know the reason why behind the reason why. And that is just how rationality works: it wants to leave no loose ends to understanding, seeking to enable us to understand things through to the bitter end. In the twelve chapters that make up Satisfying Reason, Rescher develops and defends the following perspective: That rationality is a cardinal virtue in cognitive matters. That (...) this is not something simple and cut-and-dried: in the pursuit of truth through the development of knowledge we face obstacles -- sometimes even insuperable ones. All that we can do is the best we can, realizing that even our very best may still be imperfect. Nevertheless, the venture is far from hopeless. While absolutes are unattainable in the cognitive venture, some solutions are situationally optimal, being comparatively the best that can be managed under the circumstances. That reason itself enables us to come to terms with this state of affairs, urging us to accept the best we can do as good enough. Satisfying Reason is an explanation of the presuppositions and methods of rational enquiry, an original exercise in metaknowledge, developing a systematic body of knowledge about the scope and limits of knowledge itself. (shrink)

Augustinian Just War Theory and the Wars in Afghanistan and Iraq: Confessions, Contentions and the Lust for Power,edited by Craig J. N. de Paulo, Senior Editor, et al. New York: Peter Lang Publishing, 2011. Details: A work concerning Augustine’s influence on Christian just war theory and the rhetoric of just war theorists from two symposia in addition to an Augustinian critique of the wars. Preface by Most Rev. Sean Cardinal O’ Malley, O.F.M. Cap., Archbishop of Boston. Foreword by Roland (...) J. Teske, S.J. Chapter One is a brief history of Augustine’s influence on the theory. Chapter Two includes a transcript of a symposium on the topic that includes the following distinguished contributors: Avery Cardinal Dulles, S.J., John D. Caputo, Most Rev. Edwin Cardinal O’Brien, Ambassador Thomas Melady, Col. Jack Jacobs, Dr. Joseph Hagan. Chapter Three includes a transcript of a colloquium on the topic that includes the following distinguished scholars and contributors: Joseph Margolis, Frederick Van Fleteren, Brian Kane, et al. (Advance Praise by James J. O’Donnell and Arthur Waldron.). (shrink)

A leading Catholic philosopher, he won a wide audience through such books as The Four Cardinal Virtues and About Love.This book is one of few extended studies ...

There are a number of contentious issues in the study of philosophy of science. There is the issue of method, there is the issue of subject-matter, there is the issue of truth and certainty as well as the issue of rationality, and the utility of scientific discoveries. Popper demonstrated a lot of interest in the issue of method, stressing ways and means science as a living enterprise could make progress. His theory of conjecture and refutation, or falsifiability is in pursuance (...) of this. He rejected induction as a method of science, insisting that falsifiability is a cardinal factor in any scientific research programme. In this paper I shall examine what Popper consider critical in the advancement of scientific knowledge. That is the task of this paper. (shrink)

This paper explores the modal interpretation of ?ukasiewicz's n -truth-values, his conditional and the puzzles they generate by exploring his suggestion that by ?necessity? he intends the concept used in traditional philosophy. Scalar adjectives form families with nested extensions over the left and right fields of an ordering relation described by an associated comparative adjective. Associated is a privative negation that reverses the ?rank? of a predicate within the field. If the scalar semantics is interpreted over a totally ordered domain (...) of cardinality n and metric ?, an n-valued Lukasiewicz algebra is definable. Privation is analysed in terms of non-scalar adjectives. Any Boolean algebra of 2 n ?properties? determines an n + 1 valued Lukasiewicz algebra. The Neoplatonic ?hierarchy of Being? is essentially the order presupposed by natural language modal scalars. ?ukasiewicz's ≈ is privative negation, and ? proves to stand for the extensional (antitonic) dual if ? then for scalar adjectives, especially modals. Relations to product logics and frequency interpretations of probability are sketched. (shrink)

I show that the two most devastating objections to Shogenji's formal account of coherence necessarily involve information sets of cardinality . Given this, I surmise that the problem with Shogenji's measure has more to do with his means of generalizing the measure than with the measure itself. I defend this claim by offering an alternative generalization of Shogenji's measure. This alternative retains the intuitive merits of the original measure while avoiding both of the relevant problems that befall it. In the (...) light of all of this, I suggest that there is new hope for Shogenji's analysis: Shogenji's early and influential attempt at measuring coherence, when generalized in a subset-sensitive way, is able to clear its most troubling objections. (shrink)

The work aims to demonstrate that at the heart of Eriugena’s approach to Christian theology there lies a profoundly philosophical interest in the necessity of a cardinal shift in the paradigms of thinking – namely, that from the metaphysical to the dialectical one, which wins him a reputation of the ‘Hegel of the ninth century,’ as scholars in Post-Hegelian Germany called him. The prime concern of Eriugena’s discourse is to prove that the actual adoption of the salvific truth of (...) Christ’s revelation about all humans’ Sonship to God directly depends on the way the truth of God’s Oneness is consistently thought of. It is exactly the dialectic of the universal and particular which allows Eriugena both to tackle the dichotomy between being and non-being and to proceed towards raising a question how the totality of God’s being can be approached so as to let him radically reconsider a predominantly metaphysical view of creation the theological reflection traditionally relies on. According to the dialectical understanding of unity that Eriugena does adhere to, the reality of creation cannot be thought of, and therefore known, otherwise than in the way of being inseparable from the universal Principle of all. This is the Principle abandoned by nothing, unless the mind corrupted by the senses thinks otherwise and, following the metaphysical pattern of dichotomy, improperly sets the creation and its Principle apart. Restoration of the mind to the proper rational motion of recta ratio freed, as Eriugena argues, from the dictates of senses therefore becomes the way of both the epistemological breakthrough to the infinite whole and practical return from the world of finite things to living in the divine reality of creation. The work’s argument is based on the assumption of close affinity between Eriugena’s discourse and that of his Islamic contemporaries, who developed their dialectical ideas within the Mu’tazilah tradition of a philosophically disciplined approach to the truth of God’s Oneness. In particular, al-Nazzam’s engagement with Parmenides’ Periphyseon and his resistance to the danger of a dualistic interpretation of its ontology seem to provoke Eriugena’s innovative approach to Christian theology with a view to suggesting a mode of overcoming dualism as the main obstacle on the way to the Truth revealed. This vision of the meaning of Eriugena’s undertaking allows us not only to better understand the novelty of his approach to Christian theology, but also reconsider some of the key points of his discourse that seem to have become a sort of commonplace in Eriugenian studies: 1. Unlike the prevalent opinion, not the forms of the division of Nature but the modes of interpreting being and non-being are to be understood to constitute the genuine subject-matter of each book of the Periphyseon and, hence, of the five parts of his system. 2. The fourfold division of Nature is to be interpreted not as a basic structure of the system offered by Eriugena, but as a means of introducing dialectic to the body of theology by refuting Augustine’s metaphysical vision of a hierarchical model of the universe and indicating the way of resolution of the cardinally theological contradiction – God does and does not create at the same time. 3. All this gives reason to disagree with a general tendency of associating Eriugena’s work with exploration of the division of God’s Nature and to reinterpret it as an immense anti-division project to be understood as an important turn in the history of Christian thought entirely focused on the truth of God’s Oneness and human life in conformity to it. *** I affirm that this thesis is entirely my own work and has not been submitted for examination in any form elsewhere. (shrink)

Institutional Review Boards and their federal overseers protect human subjects, but this vital work is often dysfunctional despite their conscientious efforts. A cardinal, but unrecognized, explanation is that IRBs are performing a specific function – the management of risk – using a flawed theoretical and practical approach. At the time of the IRB system’s creation, risk management theory emphasized the suppression of risk. Since then, scholars of governance, studying the experience of business and government, have learned that we must (...) distinguish pure from opportunity risks. Pure risks should be suppressed. Some opportunity risks, in contrast, must be accepted if the institution is to meet its goals. Contemporary theory shows how institutions may make these decisions wisely. It also shows how a sound organizational understanding of risk, a proper locus of responsibility, and appropriate institutional oversight all contribute to effective risk management. We can apply this general theory, developed in other contexts, to the problems of the IRB system. Doing so provides a unifying explanation for IRBs’ disparate dysfunctions by spotlighting five related deficiencies in IRB theory and structure. These deficiencies are inability to focus on greater risks, loss of balanced theory, inaccessibility to guidance from senior leadership, unbalanced federal oversight, and inflexibility. These flaws are deeply rooted in the system, and superficial reform cannot resolve them. Congress should overhaul the system to meet contemporary standards of risk management; this would benefit subjects, scientists, and the public that needs the fruits of research. (shrink)

We continue the research of an extension of the divisibility relation to the Stone‐Čech compactification. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in and nonstandard extensions of are answered, providing a few more equivalent conditions for divisibility in. Results on uncountable chains in are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of (...) a chain of type in. Finally, we consider ultrafilters without divisors in and among them find the greatest class. (shrink)

We give modest upper bounds for consistency strengths for two well-studied combinatorial principles. These bounds range at the level of subcompact cardinals, which is significantly below a κ+-supercompact cardinal. All previously known upper bounds on these principles ranged at the level of some degree of supercompactness. We show that by using any of the standard modified Prikry forcings it is possible to turn a measurable subcompact cardinal into ℵω and make the principle □ℵω,<ω fail in the generic extension. (...) We also show that by using Lévy collapse followed by standard iterated club shooting it is possible to turn a subcompact cardinal into ℵ2 and arrange in the generic extension that simultaneous reflection holds at ℵ2, and at the same time, every stationary subset of ℵ3 concentrating on points of cofinality ω has a reflection point of cofinality ω1. (shrink)

We discuss saturating ultrafilters on N, relating them to other types of nonprincipal ultrafilter. (a) There is an (ω,c)-saturating ultrafilter on $\mathbf{N} \operatorname{iff} 2^\lambda \leq \mathfrak{c}$ for every $\lambda and there is no cover of R by fewer than c nowhere dense sets. (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafilter on N is (ω,κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω,κ)-saturating, and (ii) the set of (ω,κ)-saturating ultrafilters is invariant (...) under homeomorphisms of $\beta\mathbf{N\backslash N}$ . (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω,c)-saturating. (shrink)

In this paper, assuming large cardinals, we prove the consistency of the following: Let n ∈ ω and k1, k2 ≤ n. Let f: ω → {k1, k2} be such that for all $n_1 n, cf(\mathscr{B} \cap \omega_m) = \omega_{f(m)}$.

Fix 2 < n < ω. Let CA n denote the class of cylindric algebras of dimension n, and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n, an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n, namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n. Unlike Boolean (...) algebras having a Stone representation theorem, RCA n ⊊ CA n. Using combinatorial game theory, we show that the existence of certain finite relation algebras RA, which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for L n. Using special cases of such finite RA ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on RCA n and we re-prove Hirsch and Hodkinson’s result that the class of completely representable CA n ’s is not first-order definable. We show that if T is an L n countable theory that admits elimination of quantifiers, λ is a cardinal < 2 ℵ 0, and F = 〈 Γ i : i < λ 〉 is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T. (shrink)

We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ) equal to κ++, and use it to prove the following theorem on mutual stationarity at n.Let ω1 denote the first uncountable cardinal of V and set to be the class of ordinals of cofinality ω1.TheoremIf every sequence n m. In particular, there is such a model in (...) which for all sufficiently large m<ω, the class of measurables λ with oM≥ωm is, in V, stationary below m+2. (shrink)

This paper argues that (cardinal) numbers are originally given to us in the context ‘Fs exist n-wise’, and accordingly, numbers are certain manners or modes of existence, by addressing two objections both of which are due to Frege. First, the so-called Caesar objection will be answered by explaining exactly what kind of manner or mode numbers are. And then what we shall call the Functionality of Cardinality objection will be answered by establishing the fact that for any numbers m (...) and n, if there are exactly m Fs and also there are exactly n Fs, then m = n. (shrink)

Let Mn♯ denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn the class-sized model obtained by iterating the topmost measure of Mn class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn, under the assumption that projective games on reals are determined:1. for even n, Σ1Mn=⅁RΠn+11;2. for odd n, Σ1Mn=⅁RΣn+11.This generalizes a theorem of Martin and Steel for L, (...) that is, the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in R is equivalent to the statement that Mn♯ exists for all n∈N, and that determinacy of all projective games of length ω2 with moves in N is equivalent to the statement that Mn♯ exists and satisfies AD for all n∈N. (shrink)

We investigate splitting number and reaping number for the structure (ω) ω of infinite partitions of ω. We prove that ${\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}}$ and ${\mathfrak{s}_{d}\geq\mathfrak{b}}$ . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}$ . To prove the consistency ${\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})}$ we introduce new cardinal invariants ${\mathfrak{r}_{pair}}$ and ${\mathfrak{s}_{pair}}$ . We also study the relation between ${\mathfrak{r}_{pair}, \mathfrak{s}_{pair}}$ and other cardinal invariants. (...) We show that ${\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\ma thfrak{r}}$ and ${\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})}$. (shrink)

Letκbe the cardinality of some saturated model of Peano Arithmetic. There is a set of${2^{{\aleph _0}}}$saturated models of PA, each having cardinalityκ, such that wheneverMandNare two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).

Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $ “E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$ ”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ (...) satisfies the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$ ”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “ $V=\mathrm {HOD}$ ”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$. (shrink)

This paper continues the study of covering properties of models closed under countable sequences. In a previous article we focused on C. Chang's Model . Our purpose is now to deal with the model N = ∪ { L [A]: A countable ⊂ Ord}. We study here relations between covering properties, satisfaction of ZF by N , and cardinality of power sets. Under large cardinal assumptions N is strictly included in Chang's Model C , it may thus be interesting (...) to obtain an analogue for N of the covering result for C . As in [13], we say that N satisfies the covering property if, for any set of ordinals X in the Universe, there exists a set Y in N such that X ⊂ Y and |Y| = |X| ℵ0 . In the first part, we show that if N does not satisfy the covering property, then for any α ω 1 , there exists an inner model with α measurable cardinals. Throughout this paper, we compare three situations: 1. V is the universe constructible from a countable sequence of ultrafilters on different measurable cardinals. 2. N does not satisfy the covering property. 3. There exist ω 1 measurable cardinals or a compact cardinal. Section 2 is devoted to the relation between satisfaction of ZF by N and the covering property. If holds, then N satisfies both ZF and the covering property. On the contrary, in case , N satisfies neither ZF nor the covering property. It is thus natural to obtain that the satisfaction of ZF by N implies the satisfaction of the covering property. Finally we study the cardinality of P ∩ N for a cardinal λ in the different cases. For example we show that if N does not satisfy the covering property, then + is inaccessible in any L [ A ] where A is a countable set of ordinals. This is not the case if holds. Let the expression “ λ + is inaccessible in N ” mean that for any α λ + | P ∩ N | ⩽ λ . If situation occurs, then there exists a measurable cardinal κ such that “ κ + is inaccessible in N ”. The large cardinal assumption is necessary since we show that the last fact implies the violation of the covering property by N. (shrink)