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)

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.

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.

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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.

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.

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)

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.

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)

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)

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)

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)

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)

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 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)

In response to the growing importance of environmental issues, more and more consumers are turning to anti-consumption by reducing, rejecting, or avoiding consumption. Covering the intersection of sustainable consumption and anti-consumption, previous studies relied on socio-cognitive models to explain this decision. In order to extend their findings, we consider the moral and emotional perspectives to examine reducing consumption for environmental reasons in a particular context, i.e. air travel. It is against this backdrop that we propose a conceptual model that includes (...) moral foundations as the main antecedent, followed by anticipated guilt and personal responsibility, while intention to reduce consumption (i.e. air travel) for environmental reasons, positive word of mouth about reducing air travel (WOM) and environmental activism represent the outcomes. The proposed model is tested on a sample of 511 respondents from a UK online consumer panel. Our results confirm the importance of moral foundations, anticipated guilt and personal responsibility and their interplay in the prediction of intention to reduce consumption for environmental reasons. Anticipated guilt influences WOM, while personal responsibility influences activism. In addition, intentions to reduce consumption for environmental reasons have a positive impact on WOM and environmental activism. There are several implications for public policy makers and NGOs that fight against climate change that derive from these findings, as well as research opportunities for academics interested in this topic. (shrink)

The paper begins from a working definition of caste as a contentious form of social belonging and a consideration of casteism as a form of inferiorization. It takes anti-casteism as an ideological critique aimed at unmasking the unethical operations of caste, drawing upon B. R. Ambedkar’s notion of caste as ‘graded inequality’. The politico-legal context of the unfinished trajectory of instituting protection against caste discrimination in Britain provides the backdrop for thinking through the philosophical foundations of anti-casteism. The peculiar religio-discursive (...) aspect of ‘emergent vulnerability’ is noted, which explains the recent introduction of the trope of ‘institutional casteism’ used as a shield by deniers of caste against accusations of casteism. The language of protest historically introduced by anti-racists is thus usurped and inverted in a simulated language of anti-colonialism. It is suggested that the stymieing of the UK legislation on caste is an effect of collective hypocrisies, the refusal to acknowledge caste privilege, and the continuity of an agonistic intellectual inheritance, exemplified in the deep differences between Ambedkar and Gandhi in the Indian nationalist discourse on caste. The paper argues that for a modern anti-casteism to develop, at stake is the possibility of an ethical social solidarity. Following Ambedkar, this expansive solidarity can only be found through our willingness to subject received opinions and traditions to critical scrutiny. Since opposed groups ‘make sense’ of their worlds in ways that might generate collective hypocrisies of denial of caste effects, anti-casteism must be geared to expose the lie that caste as the system of graded inequality is benign and seamlessly self-perpetuating, when it is everywhere enforced through penalties for transgression of local caste norms with the complicity of the privileged castes. The ideal for modern anti-casteism is Maitri formed through praxis, eschewing birth-ascribed caste status and loyalties. (shrink)

Review of Maddy, Penelope "Defending the Axioms".

The central premise of Michael Dummett's global argument for anti-realism is the thesis that a speaker's grasp of the meaning of a declarative, indexical-free sentence must be manifested in her uses of that sentence. This enigmatic thesis has been the subject of a great deal of discussion, and something of a consensus has emerged about its content and justification. The received view is that the manifestation thesis expresses a behaviorist and reductive theory of meaning, essentially in agreement with Quine's view (...) of language, and motivated by worries about the epistemology of communication. In the present paper I begin by arguing that this standard interpretation of the manifestation thesis is neither particularly faithful to Dummett's writings nor philosophically compelling. I then continue by reconstructing, from Dummett's texts, an account of the manifestation thesis, and of its justification, that differ sharply from the received view. On my reading, the thesis is motivated not epistemologically, but conceptually. I argue that connections among our conceptions of meaning, assertion, and justification lead to a conclusion about the metaphysics of meaning: we cannot form a clearly coherent conception of how two speakers can attach different meanings to a sentence without at the same time differing in what they count as justifying assertions made with that sentence. I conclude with some suggestions about how Dummett's argument for global anti-realism should be understood, given my account of the manifestation thesis. (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)