In this paper we define the forcing relation and prove its basic properties in the context of the theory ZFCA, i.e., ZFC minus the Foundation axiom and plus the Anti-Foundation axiom.

We introduce a new simple way of defining the forcing method that works well in the usual setting under FA, the Foundation Axiom, and moreover works even under Aczel's AFA, the Anti-Foundation Axiom. This new way allows us to have an intuition about what happens in defining the forcing relation. The main tool is H. Friedman's method of defining the extensional membership relation ∈ by means of the intensional membership relation ε .Analogously to the usual (...) forcing and the usual generic extension for FA-models, we can justify the existence of generic filters and can obtain the Forcing Theorem and the Minimal Model Theorem with some modifications. These results are on the line of works to investigate whether model theory for AFA-set theory can be developed in a similar way to that for FA-set theory.Aczel pointed out that the quotient of transition systems by the largest bisimulation and transition relations have the essentially same theory as the set theory with AFA. Therefore, we could hope that, by using our new method, some open problems about transition systems turn out to be consistent or independent. (shrink)

The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength.

In Zermelo-Fraenkel set theory without the Axiom of Foundation we study the schema version of the principle of dependent choices in connection with Aczel's antifoundation axiom , Boffa's anti-foundation axiom, and axiom of collection.

In Aczel [1], the existence of largest fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms.

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X1 introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X1 in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. (...) Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. (shrink)

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti-well-founded sets, i. e., non-well-founded sets whose all maximal ∈-paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets (...) with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈-sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbáček concerning nonstandard Boffa universes of sets is considerably extended. (shrink)

The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the (...) new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic. (shrink)

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models (...) in which foundation fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

Aczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the "circular logic" of [3]. This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical "extension" to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible (...) sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU - , that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between "non-wellfounded sets over the admissible set A" and the fragment L A of the modal language L ∞. (shrink)

Recent interest in non-well-founded set theories has been concentrated on Aczel's anti-foundation axiom AFA. I compare this axiom with some others considered by Aczel, and argue that another axiom, FAFA, is superior in that it gives the richest possible universe of sets consistent with respecting the spirit of extensionality. I illustrate how using FAFA instead of AFA might result in an improvement to Barwise and Etchemendy's treatment of the liar paradox.

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom (...) AFA is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)

The paper aims to provide precise proof theoretic characterizations of Myhill–Friedman-style “weak” constructive extensional set theories and Aczel–Rathjen analogous constructive set theories both enriched by Mostowski-style collapsing axioms and/or related anti-foundation axioms. The main results include full intuitionistic conservations over the corresponding purely arithmetical formalisms that are well known in the reverse mathematics – which strengthens analogous results obtained by the author in the 80s. The present research was inspired by the more recent Sato-style “weak weak” classical extensional (...) set theories whose proof theoretic strengths are shown to strongly exceed the ones of the intuitionistic counterparts in the presence of the collapsing axioms. (shrink)

This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain access to the abstract posits of platonist accounts of arithmetic, analysis, geometry, (...) etc., including numbers, functions, sets, points, lines, spaces, etc., whether it be such objects themselves or the possibilities of such, as postulated by modal and modal structural accounts?2 What real difference does it make, whether it be the existence of such things or the mathematical possibility of such things? Even fictionalist views seem to confront analogous problems. After all, we rely on mathematics for myriad scientific applications; so such mathematics had better at least be coherent, even if not true. But coherence requires at least formal consistency; so we seem implicitly to be committed to things like formal derivations, viz. the absence of any such having contradictory consequences, framed within the appropriate system of axioms and rules. But derivations are themselves akin to mathematical objects, as derivations are strings of strings of symbols. Moreover, derivations provided by axioms and rules form an infinite class, such that, at any future time, infinitely many will not have been written down. Moreover, such strings, as Gödel famously showed, can be coded as natural numbers. Thus, even fictionalist accounts (such as that of Field in his Science without Numbers [2016]) seem to confront problems very much like those plaguing platonist accounts. (shrink)

We are now in a position to examine the claim that Pavlovian physiology and Marxist-Leninist philosophy form two complementary systems.There is certainly a similarity between the Leninist theory of reflection and Pavlov's theory of higher nervous activity. Both present so-called psychic phenomena as a reaction of the organism to the stimuli of the outer world and both insist that this reflection is not a passive reception of impressions but is an active response on the part of the organism.Again both systems (...) are monist; they are united in excluding the possibility of having recourse to a non-material substance as the basis for psychic phenomena. But for Pavlov this exclusion is a scientific axiom while for Marxism-Leninism it is founded on philosophical materialism. However, the most important difference between Pavlov's theories and Marxism-Leninism on this point is that Pavlov's approach to psychic is fundamentally mechanistic and reductionist whereas that of Marxism-Leninism is dialectical and consequently anti-reductionist and anti-mechanist. Soviet psychology is, in consequence, founded partly on a mechanist system which is not materialist in the full sense of the word, and partly on a materialist system which is definitely not mechanist. From this point of view there is a definite discrepancy between the two traditions on which Soviet psychology is founded and which goes a long way towards explaining many of the inconsistencies in Soviet psychological theory. (shrink)

Weber’s concept of “vocation” in science implies “anti-monumentalism”: research can always be continued, and the results obtained can be used in various ways. The scientist cannot be completely aware of the final impact of their work, so they are faced with a paradox of consequences. This paradox is based on value polytheism, a concept put forward by Weber. There are two ideas central to polytheism: first, one must recognize the internal logic of value spheres and, second, one must consider (...) their fundamental incommensurability. But how does this idea emerge in Weber’s theory? Interpretations of value polytheism as a “fact” of a cultural situation and as the logical foundation of science do not allow one to answer the question of its origin. The conceptual bridge is found in Weber’s sociology of religion. Tenbruck’s, Schluchter’s, and Hennis’s models are examined to identify variations of value polytheism. However, their macro-orientation does not demonstrate the internal structure and functioning of polytheism. The present paper explicates the logical-methodological foundations of Weber’s scientific programme to clarify these points. Primarily, it investigates the problem of the consequences of an action carried out in a “vocation” mode and the boundaries of “adequate” causal explanations as presented in Weber’s works. It makes it possible to consider Weber’s value polytheism and concepts associated with it not as value metaphysics or unreasonable axioms,but as a methodologically based conceptual apparatus. (shrink)

We present two results which shed some more light on the deep connection between ZFA and the standard ZF set theory: First of all we refine a result of Forti and Honsell in order to prove that the universe of ZFA can also be obtained as the least fixed point of a continuous operator and not only as the greatest fixed point of the powerset operator. Next we show that it is possible to define a new absolute Gödel operation in (...) addition to the standard ones in order to obtain the “constructible” model of ZFA as the least fixed point of the continuous operator of Gödel closure with respect to the standard and the new Gödel operations. (shrink)

We show that there are universes of sets which contain descending ϵ-sequences of length α for every ordinal α, though they do not contain any ϵ-cycle. It is also shown that there is no set universe containing a descending ϵ-sequence of length On. MSC: 03E30; 03E65.

This note argues against Barwise and Etchemendy's claim that their semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, ones irrelevant to the task at hand" (The Liar, p. 35). Switching from ZF to AFA neither adds nor precludes any isomorphism types of sets. So it makes no difference to ordinary mathematics. I argue against the author's claim that a certain kind of 'naturalness' nevertheless makes (...) AFA preferable to ZF for their purposes. I cast their semantics in a natural, isomorphism invariant form with self-reference as a fixed point property for propositional operators. Independent of the particulars of any set theory, this form is somewhat simpler than theirs and easier to adapt to other theories of self-reference. (shrink)

“I am me”, but what does this mean? For centuries humans identified themselves as conscious beings with free will, beings that are important in the cosmos they live in. However, modern science has been trying to reduce us into unimportant pawns in a cold universe and diminish our sense of consciousness into a mere illusion generated by lifeless matter. Our identity in the cosmos is nothing more than a deception and all the scientific evidence seem to support this idea. Or (...) is it not? The goal of this paper is to discard current underlying dogmatism (axioms taken for granted as "self-evident") of modern mind research and to show that consciousness seems to be the ultimate frontier that will cause a major change in the way exact sciences think. If we want to re-discover our identity as luminous beings in the cosmos, we must first try to pinpoint our prejudices and discard them. Materialism is an obsolete philosophical dogma and modern scientists should try to also use other premises as the foundation of their theories to approach the mysteries of the self. Exact sciences need to examine the world with a more open mind, accepting potentially different interpretations of existing experimental data in the fields of brain research, which are currently not considered simply on the basis of a strong anti-spiritual dogmatism. Such interpretations can be compatible with the notion of an immaterial spirit proposed by religion for thousands of years. Mind seems that is not the by-product of matter, but the opposite: its master. No current materialistic theory can explain how matter may give rise to what we call “self” and only a drastic paradigm shift towards more idealistic theories will help us avoid rejecting our own nature. (shrink)

We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

John Stuart Mill considered his A System of Logic , first published in 1843, the methodological foundation and intellectual groundwork of his later works in ethical, social, and political theory. Yet no book has attempted in the past to engage with the most important aspects of Mill's Logic . This volume brings together leading scholars to elucidate the key themes of this influential work, looking at such topics as his philosophy of language and mathematics, his view on logic, induction (...) and deduction, free will, argumentation, ethology and psychology, as well as his account of normativity, kinds of pleasure, philosophical and political method and the "Art of Life.". (shrink)

This essay examines sections relevant to libertarianism in Will Kymlicka’s Contemporary Political Philosophy: An Introduction (2nd ed.), making and explaining the following criticisms. Kymlicka’s “preface” misconstrues political philosophy’s progress, purpose, and its relation to libertarianism. In his “introduction”, his “project” mistakes libertarianism as “right-wing”, justice as compromise among “existing theories”, and equality as the “ultimate value.” His “a note on method” in effect takes as axioms, beyond philosophical examination, various alleged desiderata and the necessary moral role of the state. Moreover, (...) his “ultimate test” being “our considered convictions” amounts to a conservative and illogical justificationism at odds with radical and coherent critical rationalism. Kymlicka’s chapter on “libertarianism” mistakes it as, inherently and unavoidably, free-marketist, anti-consequentialist, deontological, Nozickian, requiring “a foundational moral premiss, without objective content, unmaximisable, indistinguishable from license, equality-based, anti-anarchist, “self-defeating”, indefensibly “unfair”, impractically “philosophical”, and without influence. A different version of libertarianism easily withstands all Kymlicka’s antipathetic criticisms. (shrink)

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one (...) can get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealism. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of (...) energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

Jakob Friedrich Fries is one of the most important representatives of the Critical Philosophy, someone who built immediately on the original Kantian philosophy. -/- Fries was born in 1773 in Barby (on the Elbe). In 1805 he was extraordinary professor for philosophy in Jena and in the same year was ordinary professor for philosophy in Heidelberg. Returning to Jena in 1816, one year later he was compulsorily retired because of his participation at the nationalistic and republican Wartburg Festival. In 1924 (...) he obtained a professorship for physics and mathematics, and in 1838 he was given back a professorship for philosophy. He died in 1843 in Jena. -/- The book summarizes the research results of the DFG-Project "Jakob Friedrich Fries' Influence on the Sciences of the 19th Century". The research project was carried out by Dr. Kay Herrmann (Institute of Philosophy, Jena University) and Prof. Dr. Wolfram Hogrebe (Institute of Philosophy, Bonn University). Such a study has special importance. There is available a large amount of literature about the "speculative contemporaries" of Fries, like Fichte, Schelling, and Hegel. In contrast to the "speculative philosophy", there has been published only a few studies about the Friesian natural philosophy. Fries was, in his natural-philosophical studies, looking for a link between philosophy and modern sciences, wheras his "speculative philosophical" contemporaries felt obligated to stick primarily to a descriptive, phenomenal view of nature. So far the question "How was mathematical natural philosophy regarded by scientists and mathematicians of the 19th century?" has hardly been investigated. Archival studies showed that this gap in Fries-research can be filled. The Friesian correspondence turned out to be a rich gold mine. -/- The present publication is more than a research report. The monographic first part is intended to introduce the foundations of the Friesian theory of cognition, the Friesian methodology, and the Friesian natural philosophy. This should facilitate entry into Friesian philosophy. -/- The Friesian theory is analyzed from two points of view: •How did Fries suceed in continuing and improving the Kantian approach? Is Fries able to remove the weak points of Kantian philosophy? -/- •What is the current significance of the Friesian approach? There are some interesting similarities between the Friesian approach and modern philosophical theories (such like Chomsky's theory of "universal grammer"). The lasting core of the Friesian theory of cognition is: To use empirical studies for working on philosophical problems. -/- Chapters 3 and 4 are scientific-historically oriented. These chapters analyze the Friesian position in scientific and mathematical debates (debates about the a priori foundations of physics, the problem of the identification of physics as an independent discipline, the problem of the boundary between chemistry and physics, the problem of mathematization of the sciences, the theory of the imponderabilia, the systematics and structure of sciences and mathematics, problems of infinity, the differential calculus, the theory of parallel lines) and the relation between Fries and the scientists of the 19th century. The book contains the latest findings gained by evaluation of the Friesian unpublished work (for example the correspondence with W. Weber, C. F. Gauß, E. F. Apelt, O. Schlömilch, Ch. Reichel, B. A. v. Lindenau, L. Gmelin, E. G. Fischer, A. N. Scherer, J. S. C. Schweigger) -/- One result of the research project is that some important scientists took a favourable view of the Friesian theory, but the influence of the Friesian philosophy on the sciences of the 19th century was very limited. The causes are very complex: An anti-natural-philosophical spirit of age, the limits of the Kantian inspired philosophy and some unfavourable aspects in the biography of Fries. -/- For the first time the voluminous Fries-Reichel-correspondence was evaluated. The Fries-Reichel-correspondence contains the Friesian approach to prove the 11th Euclidean axiom, and the whole transcript of the Friesian attempt at proof is given. // Der erste Teil des Buches will in die Grundprobleme Fries’scher Erkenntnis­theorie, Methodenlehre und Naturphilosophie einführen, wobei das Hauptaugenmerk auf die Fortführung der kantischen Ansätze durch Fries sowie auf die aktuelle Interpretation der Fries’schen Lehre gerichtet ist. Der wissenschaftshistorisch ausgerichtete zweite Teil analysiert Fries’ Stellung zu naturwissenschaftlichen und mathematischen Diskussionsrichtungen (Probleme der Identifizierung der Physik als eigenständige Disziplin, der Grenzziehung zwischen Physik und Chemie, der Mathematisierung der Naturwissenschaften, der Imponderabilientheorie, der Systematik von Naturwissenschaft und Mathematik, des Unendlichen, der Parallelentheorie usw.) sowie sein Verhältnis zu Naturwissenschaftlern und Mathematikern seiner Zeit. Das Buch enthält neue Erkenntnisse, die aus der Auswertung zahlreicher Nachlassmaterialien gewonnen wurden. Erstmalig wird unter dem Blickwinkel „Fries als Naturwissenschaftler und Mathematiker“ auch der sehr umfangreiche Reichel-Briefwechsel ausgewertet. Dem Reichel-Briefwechsel entstammt auch Fries’ Versuch eines Beweises des Parallelenaxioms, der in diesem Buch erstmalig in transkribierter Form vollständig vorliegt. -/- . (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

This essay examines sections relevant to libertarianism in Will Kymlicka’s Contemporary Political Philosophy: An Introduction (2nd ed.), making and explaining the following criticisms. Kymlicka’s “preface” misconstrues political philosophy’s progress, purpose, and its relation to libertarianism. In his “introduction”, his “project” mistakes libertarianism as “right-wing”, justice as compromise among “existing theories”, and equality as the “ultimate value.” His “a note on method” in effect takes as axioms, beyond philosophical examination, various alleged desiderata and the necessary moral role of the state. Moreover, (...) his “ultimate test” being “our considered convictions” amounts to a conservative and illogical justificationism at odds with radical and coherent critical rationalism. Kymlicka’s chapter on “libertarianism” mistakes it as, inherently and unavoidably, free-marketist, anti-consequentialist, deontological, Nozickian, requiring “a foundational moral premiss, without objective content, unmaximisable, indistinguishable from license, equality-based, anti-anarchist, “self-defeating”, indefensibly “unfair”, impractically “philosophical”, and without influence. A different version of libertarianism easily withstands all Kymlicka’s antipathetic criticisms. (shrink)

Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. Of (...) course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution. (shrink)

Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. Of (...) course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution. (shrink)

This chapter provides a conceptual analysis of Tommy J. Curry’s anti-ethical stance. In what sense is anti-ethics opposed to, or against, ethics? I argue that, despite appearances, anti-ethics is a kind of ethical theory. Close analysis reveals that the term “ethics” in the term “anti-ethics” does not refer to ethics per se, but to an idealist approach to ethics that frames white people as virtuous. I argue that Leonard Harris’ insurrectionist ethics provides a naturalistically-informed deontological framework (...) that is useful for analyzing the normative dimensions of anti-ethics. In this way, insurrectionist ethics enables us to analyze the positive content of anti-ethics, an ethics founded on a form of ethical hope that abandons the premise of white virtue. (shrink)

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...) the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

We use NF to designate the system known as Quine's New Foundations, and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic”. This system ML is just the systemPproposed by Wang in [4], and essentially includes NF as a part.The pripcipal results of the present paper are:A. In NF the axiom of infinity is equivalent to (...) the definability of an ordered pair having the same type as its constituents.B. Considering the formulas of NF as having translations in the formalism of ML in the sense defined in [4], we find that a formula of NF is provable in NF if and only if its translation is provable in ML. From the fact that the axiom of infinity is provable in ML, one might be tempted to conclude that the axiom of infinity is provable in NF. It is explained why this is not a valid inference. Consequently, it would appear that there is a sense in which NF + AF is stronger than ML. (shrink)

The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on (...) the foundations of mathematics. (shrink)

This paper seeks to question the position of ZF as the dominant system of set theory, and in particular to examine whether there is any philosophical justification for the axiom of foundation. After some historical observations regarding Poincare and Russell, and the notions of circularity and hierarchy, the iterative conception of set is argued to be a semi-constructvist hybrid without philosophical coherence. ZF cannot be justified as necessary to avoid paradoxes, as axiomatizing a coherent notion of set, nor (...) on pragmatic grounds, and should be replaced by a system allowing non-well-founded sets. (shrink)

Perhaps there has been no greater opportunity than in this “VOLUME FIFTEEN of our Death And Anti-Death set of anthologies” to write about how might think about life and how to avoid death. There are two reasons to discuss “life”, the first being enhancing our understanding of who we are and why we may be here in the Universe. The second is more practical: how humans meet the physical challenges brought about by the way they have interacted with their (...) environment. Many persons discussing “life” beg the question about what “life” is. Surely, when one discusses how to overcome its opposite, death, they are not referring to another “living” thing such as a plant. There seems to be a commonality, though, and it is this commonality is one needing elaboration. It ostensibly seems to be the boundary condition separating what is completely passive (inert) from what attempts to maintain its integrity, as well as fulfilling other conditions we think “life” has. In our present discussion, there will be a reminder that it by no means has been unequivocally established what life really is by placing quotes around the word, namely, “life”. Consider it a tag representing a bundle of philosophical ideas that will be unpacked in this paper. (shrink)

This paper seeks to question the position of ZF as the dominant system of set theory, and in particular to examine whether there is any philosophical justification for the axiom of foundation. After some historical observations regarding Poincare and Russell, and the notions of circularity and hierarchy, the iterative conception of set is argued to be a semi-constructvist hybrid without philosophical coherence. ZF cannot be justified as necessary to avoid paradoxes, as axiomatizing a coherent notion of set, nor (...) on pragmatic grounds, and should be replaced by a system allowing non-well-founded sets. (shrink)

We discuss a `negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the `non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ+ is the extension of Λ with a set of non-ξ rules, then Λ+ is (...) strongly sound and complete with respect to the class of frames determined by the axioms and the rules. (shrink)

The still unsettled decision problem for the restricted purely universal formulae 0-formulae) of the first order set-theoretic language based over =, ∈ is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the 0-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property (...) over the hereditarily finite sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable . Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the set-satisfiability problem for the full class of Δ0-formulae is shown to be semidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the Δ0-formulae. (shrink)

In 'No Experience Necessary: Foundationalism and the Retreat from Culture in Environmental Ethics'. Ben A. Minteer forgivably misconstrues my critique of moral pluralism. Contrary to Minteer’s representation: I do not accuse moral pluralists of ‘moral promiscuity’: nor do I posit a ‘master principle’ to govern all human action respecting the environment: and although I offer conceptual foundations for environmental ethics, I do not claim that they rest on certain, a priori, and non-empirical intuitions. Rather, the conceptual foundations I offer for (...) environmental ethics are largely scientific. Contrary to Minteer’s representation: I do consider a multiplicity of contexts in which ethical actions are situated: and I do respectfully attend to and creatively engage a variety of cultural points of view, both western and nonwestern. in constructing environmental ethics. Anti-foundationalists, such as Minteer and Bryan G. Norton, ironically pose an insidious threat to democratic discussion and debate of environmental values, because they themselves posit, but do not frankly acknowledge, foundational beliefs. (shrink)

We show that the Axiom of Foundation, as well as the Antifoundation Axiom AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language , and a sentence F of the form with quantifier-free in the same language, are there models of T in which F is true? Furthermore we show that the Extensionality Axiom is quite irrelevant in that respect.