Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. (...) implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω). (shrink)

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)

The theory that ``consistency implies existence'' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the (...) facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems. (shrink)

We determine the consistency strength of determinacy for projective games of length ω^2. Our main theorem is that $\Pi _{n + 1}^1$-determinacy for games of length ω^2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M_n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $A (...) = R$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω^2 with payoff in $R^R\Pi _1^1$ or with σ-projective payoff. (shrink)

We introduce the large-cardinal notions of ξ-greatly-Mahlo and ξ-reflection cardinals and prove (1) in the constructible universe, L, the first ξ-reflection cardinal, for ξ a successor ordinal, is strictly between the first ξ-greatly-Mahlo and the first Π1ξ-indescribable cardinals, (2) assuming the existence of a ξ-reflection cardinal κ in L, ξ a successor ordinal, there exists a forcing notion in L that preserves cardinals and forces that κ is (ξ+1)-stationary, which implies that the consistency strength of the (...) class='Hi'>existence of a (ξ+1)-stationary cardinal is strictly below a Π1ξ-indescribable cardinal. These results generalize to all successor ordinals ξ the original same result of Mekler–Shelah [A. Mekler and S. Shelah, The consistency strength of every stationary set reflects, Israel J. Math.67(3) (1989) 353–365] about a 2-stationary cardinal, i.e. a cardinal that reflects all its stationary sets. (shrink)

Metaphysicians who are aware of modern physics usually follow Putnam (1967) in arguing that Special Theory of Relativity is incompatible with the view that what exists is only what exists now or presently. Partisans of presentism (the motto ‘only present things exist’) had very difficult times since, and no presentist theory of time seems to have been able to satisfactorily counter the objection raised from Special Relativity. One of the strategies offered to the presentist consists in relativizing existence to (...) inertial frames. This unfashionable strategy has been accused of counterfeiting, since the meaning of the concept of existence would be incompatible with its relativization. Therefore, existence could only be relativistically invariant. In this paper, I shall examine whether such an accusation hits its target, and I will do this by examining whether the different criteria of existence that have been suggested by the Philosophical Tradition from Plato onwards imply that existence cannot be relativized. (shrink)

It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ε {1,…,ω}, ∑3 + n1 absoluteness implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑3 + n1 absoluteness is exactly that of n strong cardinals so that (...) in particular projective absoluteness is equiconsistent with the existence of infinitely many strong cardinals. It is also argued how this theorem is to be construed as the first step in the long range program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle positively a conjecture of Woodin and thereby solve the last Delfino problem. (shrink)

We prove the following Main Theorem: $ZF + AD + V = L(R) \Rightarrow DC$ . As a corollary we have that $\operatorname{Con}(ZF + AD) \Rightarrow \operatorname{Con}(ZF + AD + DC)$ . Combined with the result of Woodin that $\operatorname{Con}(ZF + AD) \Rightarrow \operatorname{Con}(ZF + AD + \neg AC^\omega)$ it follows that DC (as well as AC ω ) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬ DC (...) R , where DC R is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R # (i.e. the sharp of L(R)) exists. (shrink)

Simultaneous hypothesis tests can fail to provide results that meet logical requirements. For example, if A and B are two statements such that A implies B, there exist tests that, based on the same data, reject B but not A. Such outcomes are generally inconvenient to statisticians (who want to communicate the results to practitioners in a simple fashion) and non-statisticians (confused by conflicting pieces of information). Based on this inconvenience, one might want to use tests that satisfy logical (...) requirements. However, Izbicki and Esteves shows that the only tests that are in accordance with three logical requirements (monotonicity, invertibility and consonance) are trivial tests based on point estimation, which generally lack statistical optimality. As a possible solution to this dilemma, this paper adapts the above logical requirements to agnostic tests, in which one can accept, reject or remain agnostic with respect to a given hypothesis. Each of the logical requirements is characterized in terms of a Bayesian decision theoretic perspective. Contrary to the results obtained for regular hypothesis tests, there exist agnostic tests that satisfy all logical requirements and also perform well statistically. In particular, agnostic tests that fulfill all logical requirements are characterized as region estimator-based tests. Examples of such tests are provided. (shrink)

It is argued that the goal of Hilbert's program was to prove the model-theoretical consistency of different axiom systems. This Hilbert proposed to do by proving the deductive consistency of the relevant systems. In the extended independence-friendly logic there is a complete proof method for the contradictory negations of independence-friendly sentences, so the existence of a single proposition that is not disprovable from arithmetic axioms can be shown formally in the extended independence-friendly logic. It can also be (...) proved by means of independence-friendly logic that proof-theoretical consistency of a sentence S implies the existence of a model in which S is not false. Hence the consistency of the axioms of arithmetic in the sense of being not-false in a model can be proved. (shrink)

In their paper from 1981, Milner and Sauer conjectured that for any poset P,≤, if , then P must contain an antichain of cardinality κ. The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin’s Maximum and holds for (...) all singular λ above the first strongly compact cardinal. (shrink)

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...) is ramifiable” implies the existence of a proper class of inaccessible cardinals. (shrink)

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that (...) [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe [Formula: see text] (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. We prove, assuming ZF [Formula: see text] [Formula: see text] [Formula: see text] “[Formula: see text] is an even ordinal”, that there is a proper class transitive inner model [Formula: see text] containing [Formula: see text] and satisfying ZF [Formula: see text] [Formula: see text] [Formula: see text] “there is an elementary embedding [Formula: see text]”; in fact we will have [Formula: see text] ⊆[Formula: see text], where [Formula: see text] witnesses [Formula: see text] in [Formula: see text]. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text]-DC holds in [Formula: see text], then the model [Formula: see text] will also satisfy [Formula: see text]-DC. We show that ZFC [Formula: see text] “[Formula: see text] is even” [Formula: see text] [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists. (shrink)

What, if anything, is the correlation between the specialized or technical ideas of the philosopher and the rest of his existence? His everyday life outside his philosophical role. In the specialized reality and reality constitution, when employing the discourse and discipline of philosophy, the philosopher subscribe to many things in an explicit manner and he employs a number of implicit things and assumptions that are not stated explicitly. These things concern the different branches, areas and domains of the philosophical (...) discourse, for example metaphysics, epistemology, ethics, ontology, cognition, consciousness, mind, perception, thinking, etc. Is there a relation between these technical 'beliefs' of the philosopher and other, not philosophical areas of his existence? Does and can all of his non-philosophical existence reflect his philosophical beliefs, statements and expressions? Is it possible that the everyday existence or life world of the philosopher can resemble, confirm and express his philosophical ideas? If this is not the case, what are the factors that are involved in such discrepancies? Do they imply a lack of integrity, of wholeness of integration on behalf of the philosopher? What are the causes of such an illogical or surprising lack of compatibility or similarity between these sets of two or more facts? What are the things that we can look for in such a lack of compatibility – things such as the following? - inconsistency, difference, disparity, variance, variation, deviation, divergence, -/- disagreement, dissimilarity, dissimilitude, mismatch, lack of similarity, contrariety, contradictoriness, disaccord, discordance, incongruity, -/- lack of congruence, incompatibility, irreconcilability, conflict, opposition? -/- I illustrate the above by means of four examples. 1 Let us assume that the embodied human person consists of both voluntary and involuntary processes and activities. Do the philosophy of an individual reflect or express these activities or is it completely or partially unrelated to them? Do the voluntary processes and activities or the voluntary aspects of perception, cognition, brain processes and other activities resemble or express the philosophy of a thinker? 2 Some thinkers are seriously concerned about the importance of discourse and dialogue and the equality of all those involved. We find their concern about the ideal discourse and dialogue described in their written work and talks, but to what an extend do their own discursive and dialogical actions and behaviour reflect the ideal situation they preach? 3 Ethics and morality can be find in and are emphasized in all sorts of disciplines, of course they are major subjects in providing ways to live off for philosophers and we even find institutions devoted to their teaching and investigation. Do the lives, the lived morality and ethics of those individuals who preach the ideals of ethics or ethical ideals reflect the ideals they preach and that provides them with ways to earn a living? Or, is it the case that morality and ethics are merely faked by those individuals living off their preaching of ethics? And, that they merely employ the values, norms, customs and attitudes of a culture, sub-culture, community, group or another social grouping? In other words are the preachers of ethics really more ethical than the rest of the community and society? 4 When the philosopher expresses his ideas by means of all sorts of philosophical tools he is aware of the fact that he employs them, or at least aware of some of them being intentionally employed by him while he is unaware of others that he employs. Do the tools the philosopher employs to theorize (identify problems, the nature of the questions he asks, the nature of his ways of questioning, the assumptions he makes, the forming and testing of hypotheses, the making of generalizations, etc) or philosophize (perceive things, think, think about things, reason, argumentation etc) resemble what he beliefs, asserts and says about thinking, perception, cognition, understanding, subjects, objects, relationships between subjects and objects and other features and processes of epistemology? In short – does the lived philosophy, ontology, epistemology, ethics etc of the philosopher resemble, represent, confirm, substantiate, back, endorse, support, authenticate and corroborate the ideas, assertions and speculations that are expressed by the statements his philosophy or philosophizing consist of? Does he walk the walk of his talk? Are they similar and identical, the same things and merely expressions of the same things in different mediums? (shrink)

Since last year, major initiatives have been undertaken by the chair of theoretical philosophy at the University of Tübingen in order to enhance the reception of analytic metaphysics in the European landscape. Here we review the 2013 summer workshop, intended to be the first of an annual series, on “Existence, Truth and Fundamentality”, the invited speakers being Graham Priest (Melbourne), Stephan Leuenberger (Glasgow), Dan López de Sa (Barcelona), Francesco Berto (Aberdeen), Friederike Moltmann (Paris – Pantheon Sorbonne) and Jason Turner (...) (Leeds). The workshop’s scope is fairly large, focusing on three different aspects of contemporary research in metaphysics. However, the underlying idea of organizers Thomas Sattig and Alessandro Torza consisted in exploring their interconnections and even new areas which possibly share the same problem spectrum. Also, they thought of bringing together both young and senior protagonists of the contemporary debate. In this sense, the main thread followed by the speakers consisted in enquiring the ideas of fundamentality, dependence and grounding with respect to ontology (Berto, López de Sa, Turner) and the truthmaking debate (Priest, Moltmann, Leuenberger), thus pretty much in line with some latest tendencies in metaphysics, such as, prima inter paria, the works of Kit Fine. When we say that something is more fundamental than something else, we move into a non-conventional (for the standards of 20 th century philosophy) metodological framework, hardly focusing on the question about what there is, and rather tackling the issue of how fundamentally there is what there is, or how the existence of something is required as a ground for the existence of other things. Here is how Jonathan Schaffer explains the methodological transition in his influential “On What Grounds What” (2009): -/- Among the many assumptions Quine and Carnap share is that metaphysical questions are existence questions, such as whether numbers exist. They only disagree on the further issue of whether such questions are meaningful (at least as the metaphysician might pose them). But why think that metaphysical questions are existence question of this sort? Return to Aristotle’s Metaphysics. There are virtually no existence question posed. The whole discussion is about substances (fundamental units of being). At one point Aristotle pauses to ask if numbers exist, and his answer is a brief and dismissive yes. [. . . ] For Aristotle, the question about numbers is whether they are transcendent substances or grounded in concreta. The question is not whether numbers exist, but how. (Schaffer, 2009, p. 347) -/- The conference opened with Graham Priest’s intriguing and peculiar review of Buddhist cornerstone “The Fundamental Verses of the Middle Way” (Mūlamadhyakamakārikā) by Nāgārjuna (ca. 150-ca. 250 AD), seen as a unrecognized forerunner of multi-valued semantics and dialetheism. Priest shows that Nagarjuna’s thought indeed has valuable analytic grip (contrary to the traditional mystical dismiss of his works altogether) and is even in accord to some formal schemas rejecting both the laws of excluded middle and non-contradiction. Dan López de Sa was interested in the topic of grounding, raising concerns about how to explain the kind of “reality” exemplified by what is not fundamental, but rather derivative on something else. He proposes a conceptual and non-primitive analysis of grounding as linked to fundamentality and derivation, and argues that the view can be advocated along with the acceptance that at least something is not fundamental. Subsequently, he illustrates an impressive range of applications linked to contemporary debates in meta-ontology, truthmaking and philosophy of time. Stephan Leuenberger discussed a semantics for “total logic”, a variant of first order logic which is claimed to have several applications with respect to the issue of excluding negative existentials from the scope of theories accounting for a given ’totality’ of facts. For example, materialism is the claim that everything there is can be traced back to physical facts or to truth implied by those facts. What about the claim that “There are no angels”? According to Leuenberger, ordinary logic falls prey of modal arguments (if it is possible that there are angels, by modus tollens, that it is not metaphysically necessary that everything there is is physical). So, there is space for the introduction of a new operator, the totality operator, claiming of what enters its scope that “that’s the whole truth”. Consequently, total logic’s semantics tackles David Chalmers’ well-known conceivability-based argument in favour of zombie’s (subjects producing behaviour but experiencing no qualia) possible existence. Friederike Moltmann presented a massive array of natural language-based examples on the role played by case-constructions, such as “it is the case that Caesar crossed the Rubicon”, “the case of the stolen statue”, “the case in which it might rain”, and so forth. She suggests that we should take surface phenomena of our language very seriously with regard to the determination of ontological categories and truth-makers. Cases are provided by many natural languages with their own existence predicates (“to occur” and “to present itself ”), and these do not apply to other kind of entities, such as objects and events. Trying to derive a substantive thesis on their nature, she claims that they are filtered entities, a structure taking into account some features of the corresponding objects and events and preserving their relational pattern. Jason Turner discussed meta-ontological concerns against the claim that ontological debates are defective. Analysed the structure of a metaphysical theory and identified two different forms of defectiveness that can occur in ontological questions, the defection from without and the defection from within, the author provided an original solution in order to avoid such difficulties. He argued thus in favor of a logical constancy, a translation principle thanks to it would be possible reconceive an ontological defection. Lastly, Francesco Berto’s talk closed the workshop with a sagacious discussion of meinongian quantification. He substains that Meinongians and Quineans experience no conceptual equivocation. Even if the former claims that “There are things that do not exist”, something that the latter contrasts with the interpretation of existential quantifiers as all-including, they still rely upon the same concept of “existence” and they understand each other when they are outside the ontology room, so that it cannot be that they lack competence. Subsequently, Berto deals with the objection that Meinongians commit analytic falsehood in separating the meaning of “to be” from that of “exists”. The Quinean has no independent evidence for analytic equivalence aside from her emphatic stress on the “is” in “there is”. Meinongians, however, can retort to linguistic considerations and claim that existence predicates display fairly different functions. -/- As for what regards our internal division of the “philosophical labor”, Fabio Ceravolo reviewed Moltmann and Berto’s talks; whereas Mattia Cozzi took care of Priest’s and Leuenberger’s, and Mattia Sorgon of those given by López de Sa and Turner. Conclusively, we would like to thank Thomas Sattig and Alessandro Torza for all of their organizational efforts and the kind welcome they reserved to us. We hope we will have further opportunities to collaborate with Tübingen in supporting similarly innovative and valiant european initiatives. (shrink)

The existence of natural laws, whether deterministic or indeterministic, and whether exceptionless or ceteris paribus, seems puzzling because it implies that mindless bits of matter behave in a consistent and co-ordinated way. I explain this puzzle by showing that a number of attempted solutions fail. The puzzle could be resolved if it were assumed that natural laws are a manifestation of God’s activity. This argument from natural law to God’s existence differs from its traditional counterparts in that, (...) whereas the latter seek to explain the fact of natural laws, the former seeks to explain their possibility. The customary objections to the traditional arguments cannot be successfully adapted to counter this new argument, with one exception which has only limited effect. I rebut four claims that the theistic solution to the puzzle about natural laws is paradoxical, though I concede that one of these claims has merit. I consider four objections to the new argument but find three of them more or less unsatisfactory. The fourth, if successful, would undermine our claims to know the truth about the world. (shrink)

Dissertatio proposita circa “argumentum ontologicum” pro existentia Dei, quem K. Goedel construxit, versatur. In prima parte structuram logicam dicti argumenti exponimus, singulos gradus argumenti explicamus, “collapsumque modalitatum”, quo argumentum invalidari invenitur, examinamus. Sequenti parte recentiores quasdam confectiones argumenti pertractamus; et scil. praecipue formam eius, quae super conceptum mathematicum multitudinis seu “complexus elementorum terminatorum” fundatur, et formam “algebraicam”, quarum affinitates quasdam notabiles prae oculos ponimus. Ultima parte disceptationes, quae circa huiusce argumenti validitatem ac momentum respectu modernae theisticae philosophiae agebantur, describimus. Loco (...) conclusionis observamus, Goedelii argumentum exemplum esse notabile “fidei quaerentis intellectum”.The article deals with Gödel’s ontological proof of God’s existence. It consists of three parts. In the first part we present the logical structure of the argument, analyse its individual steps and discuss the implied collapse of modalities, which is fatal for the proof. In the second part we focus on some more recent versions of the argument, especially the set-theoretical version and the algebraic version, and we show several interesting connexions between the algebraic and the set-theoretical version. In the final part of the paper we briefly recount the discussions concerning the validity of the argument and its importance for modern theistic philosophy. We conclude by observing that Gödel’s argument is an interesting modern instance of “faith seeking understanding”. (shrink)

The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving (...) aside its application, are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view. (shrink)

A fast consistency prover is a consistent polytime axiomatized theory that has short proofs of the finite consistency statements of any other polytime axiomatized theory. Krajíček and Pudlák have proved that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that NP = coNP implies the existence of a fast consistency prover. The reverse implication is an open question. (...) In this paper we define the notion of an unlikely fast consistency prover and prove that its existence is equivalent to NP = coNP. Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable in polynomial time. (shrink)

The conventional interpretation of Adam Smith is that he is a prophet of commercialism. The liberal capitalist reading of Smith is consistent with the view that history culminates in commercial society. The first part of the article develops this optimistic interpretation of Smith's view of history. Smith implies that commercial society is the end of history because (1) it supplies the ends of nature that he identifies; (2) it is inevitable; and (3) it is permanent. The second part of (...) the article shows that Smith has some dark moments in his writings where he seems to reject completely such teleological notions. In this more civic humanist mood he confesses that commercial society does not supply the ends of nature, nor is it inevitable, nor is it permanent. Both views exist in Smith and the commentator is forced to choose between passages in Smith's work in order to support a particular interpretation of the former's view of history. (shrink)

Durante la Revolución Francesa, el trabajo político del "lado izquierdo", del cual Robespierre fue uno de los portavoces, consistió en implementar los principios de la Declaración de los Derechos del Hombre y del Ciudadano. Esto implica que el pueblo soberano puede controlar el ejercicio del poder de sus agentes. Su función es garantizar el derecho natural del hombre a la existencia, que es la condición principal de la libertad y la razón de ser de la república. During the French Revolution, (...) the political work of the "left side", of which Robespierre was one of the spokespersons, consisted in implementing the principles of the Declaration of the Rights of Man and of the Citizen. These imply that the sovereign people can control the exercise of the power of their agents. Their function is to guarantee the natural right of man to existence which is the main condition of freedom and the raison d'être of the republic. (shrink)

On the one hand, first-order theories are able to assert the existence of objects. For instance, ZF set theory asserts the existence of objects called the power set, while Peano Arithmetic asserts the existence of zero. On the other hand, a first-order theory may or not be consistent: it is if and only if no contradiction is a theorem. Let us ask, What is the connection between consistency and existence?

This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but (...) also in which Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be. (shrink)

Descartes’s Cogito, “I am thinking, therefore I exist,” is perhaps the most famous assertion in the history of philosophy. Thirteen hundred years earlier, St. Augustine formulated a similar claim, arguing “if I am mistaken, I am.” Did St. Augustine anticipate Descartes? We show that Descartes’s dictum is a novel insight and less vulnerable to criticism than the claim of St. Augustine. Whereas Descartes searched for one true proposition on which he could base scientificknowledge, St. Augustine sought to refute the skeptics (...) who had denied the possibility of knowledge. By a twist of irony, the skeptics and St. Augustine reached contradictory (ethical) conclusions based, however, on similar reasoning. (shrink)

We give a simple and direct proof that super-consistency implies the cut-elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut-elimination theorems in higher-order logic that involve V-complexes.

This chapter contains sections titled: Abstract Introduction Defining Terms: Relativism, Pluralism, Tolerance What Is Logic? One Route to Pluralism: Logic ‐ as ‐ Model The Boundary Between Logical and Non ‐ Logical Terminology Vagueness Relativity to Structure References.

Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. -/- Constance Reid was an insider of the Berkeley-Stanford logic circle. Her San Francisco home was in Ashbury Heights near the homes of logicians such as Dana Scott and John Corcoran. Her sister Julia Robinson was one of the top mathematical logicians of her generation, as was Julia’s husband Raphael Robinson for whom Robinson Arithmetic was named. Julia was a Tarski PhD and, in recognition of (...) a distinguished career, was elected President of the American Mathematics Society. https://en.wikipedia.org/wiki/Julia_Robinson http://www.awm-math.org/noetherbrochure/Robinson82.html. (shrink)

Studies of economic decision-making have revealed the existence of consistent contributors, who always make contributions to the collective good. It is difficult to understand such behavior in terms of mutualistic motives. Furthermore, consistent contributors can elicit apparently altruistic behavior from others. Therefore, although mutualistic motives are likely an important contributor to moral action, there is more to morality than mutualism.

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and (...) T prove the existence of infinitely many non-logical objects (T deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of T toCantor's theorem which is somewhat surprising. (shrink)

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of (...) the bounded proper forcing axiom is exactly the existence of a Σ 1 -reflecting cardinal (which is less than the existence of a Mahlo cardinal). We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 10.3 (2003) 265-266 [Access article in PDF] Definition Is Limited and Values Inescapable Richard Mullen THIS IS A welcome paper that lays bare some of the presumptions of those who seek to determine the status of psychiatric disorder. At different times debate on the subject reflects stigma, prejudice, needs for coherent categorization, and occasionally just antipsychiatric resentment. As Pickering hints, much philosophical argument may be (...) selected to support a position already chosen.The article illustrates that the likeness argument is insufficient to demonstrate that mental disorders are or are not disorders in any other more profound sense. Similarly perhaps, if a debate were conducted about whether or not a bony fracture, hypertension, obesity, or bacterial infection were real illnesses, the likeness argument might be unfruitful. This is perhaps unsurprising; notions of health and illness, mental or physical, are largely abstractions. In psychiatry, so-called symptoms are themselves abstract classifications of incompletely articulated mental phenomena far more diverse than the definitions and criteria to be found in a diagnostic manual.Argument by analogy is always limited. If my car has a flat tire, the car will be dysfunctional or disordered. Its loss of performance will be the symptoms of the flat tire, which is the lesion. It is a problem quickly diagnosed and treated. If convenient, I suppose my car could have been said to be ill, but this would not lead to any particular insights into the nature of cars, mechanics, health, or anything else.A variety of concepts can be contained within terms such as illness,disease, and disorder. They include the clinicopathological approach (the disease is the organic lesion), the disease is deviance approach, disease as distress, and approaches that consider disease—particularly mental disorder—as largely socially constructed. None may resolve the questions that Pickering raises, but recognizing that our terms are not always precise or used consistently is part of formulating the problem more clearly. For the most part clinical psychiatry (and the rest of medicine) has no difficulty in properly identifying most of our patients as disordered or ill. The fact that there are some borderline instances, difficult to classify either way, does not invalidate the classification.Consideration of whether or not the features of a disorder can in themselves determine whether or not a person is disordered is timely for two reasons. As stigma about (some) psychiatric disorder lessens, and opportunities for complaint and compensation expand, an era of self-definition as mentally disordered has arrived. How are mental health practitioners to respond to someone whose mental distress appears not to be clinically significant, but who demands access to the sick role? Also of current concern for clinical psychiatry is widespread pressure to define as disordered those persons whose antisocial behavior brings them into conflict with society. Professional recognition that definitions of mental disorder are inadequate, or rather incomplete, [End Page 265] may be valid. Worryingly, there are others who are willing to define it for us.Both major international systems of classification of mental disorder offer definitions of mental disorder that turn on what is to be meant by clinical significance or clinically recognizable. In the current Diagnostic and Statistical Manual, mental disorder implies "clinically significant impairment or distress," and criteria for specific disorders mostly including a comment about "clinically significant distress or impairment in social, occupational, or other important areas of functioning" (APA 2000, 8). In the International Classification of Disease, disorder is used to imply "existence of a clinically recognizable set of symptoms or behavior associated in most cases with distress and with interference with personal function" (WHO 1992, 5). It seems that no definition of mental disorder is adequate for all purposes. Seeking to improve existing definitions by further defining the terms used quickly becomes an exercise in pedantry, far removed from practical utility.Pickering mentions as an example the "symptom" of delusions. Available definitions are widely recognized as incomplete or otherwise unsatisfactory (David 1999). This leads to debate about what the individual elements within the definition precisely mean. However in practice delusions are identified with remarkably high reliability (WHO 1973... (shrink)

It is a widely held view that persons have privileged knowledge about their own minds, although numerous different views on what this privilege exactly consists of exist. One possible way of interpreting it is to claim that persons can refer to their own mental states in a privileged way. I will argue that this view has to be extended. Our common-sense view about reference to mental states implies that besides privileges of first-person reference to one's own mental states, there (...) also exist privileges of third-person reference to the mental states of others: Other persons can refer to all of the mental states of a person in a way that the person cannot. In a next step, I will explain that persons can take two perspectives towards their own mental states: a first-person perspective and a third-person perspective. I will conclude that the possibilities of first-person reference from a third-person perspective are limited. (shrink)

Non-compensatory aggregation rules are applied in a variety of problems such as voting theory, multi-criteria analysis, composite indicators, web ranking algorithms and so on. A major open problem is the fact that non-compensability implies the analytical cost of loosing all available information about intensity of preference, i.e. if some variables are measured on interval or ratio scales, they have to be treated as measured on an ordinal scale. Here this problem has been tackled in its most general formulation, that (...) is when mixed measurement scales (interval, ratio and ordinal) are used and both stochastic and fuzzy uncertainties are present. Objectives of this article are first to present a comprehensive review of useful solutions already proposed in the literature and second to advance the state of the art mainly in the theoretical guarantee that weights have the meaning of importance coefficients and they can be summarized in a voting matrix. This is a key result for using non-compensatory Condorcet consistent rules. A proof on the probability of existence of ties in the voting matrix is also developed. (shrink)

This paper deals with the deuteros plous, literally ‘the second voyage’, proverbially ‘the next best way’, discussed in Plato’s Phaedo, the key passage being Phd. 99e4-100a3. I argue that (a) the ‘flight into the logoi’ can have two different interpretations, a standard one and a non-standard one. The issue is whether at 99e-100a Socrates means that both the student of erga and the student of logoi consider images (‘the standard interpretation’), or the student of logoi does not consider images but (...) that “consistency should suffice for truth” (‘the non-standard interpretation’); (b) the non-standard one implies the problem of the hypothesis, a problem analogous to the problem of the elenchus; (c) there is a structural analogy between Descartes’ ontological argument for the existence of God in his 5th Meditation and the final proof for the immortality of the soul in the Phaedo. (shrink)

According to Eugenie Scott, methodological materialism---the view that science attempts to explain the world using material processes---does not imply philosophical materialism---the view that all that exists are material processes. Thus one can consistently be both a scientist and a theist. According to Phillip Johnson, however, methodological materialism presupposes philosophical materialism. Consequently, scientists are unable to see the cogency of supernatural explanations, like creationism. I argue that both Scott and Johnson are wrong: scientists are not limited to explaining tbe world using (...) material processes and science does not presuppose materialism. Thus scientists’ rejection of creationism is not irrational. (shrink)

Cet article propose une reconstruction de la ligne d’argumentation de l’exposé de 1805 de la Doctrine de la Science de Fichte. Son objet fondamental consiste à répondre à la question : qu’est-ce que le savoir ? ce qui implique de répondre à cette autre question : qu’est-ce que le « est » ? Fichte répond : ce savoir est l’existence absolue, ce qui est équivalent à l’existence de l’absolu – et l’auteur le montre à travers cinq tentatives de (...) démonstration de cette thèse. Cela aboutit à la compréhension du caractère inconditionné de la raison , résultat de la philosophie transcendantale en général.This paper proposes a reconstruction of the argumentative line of the exposition of Fichte’s Science of Knowledge of 1805. Its fundamental object consists in answering the question : what is knowledge ? what implies to answer another question : what ‘is’ is ? Fichte answers that knowledge is absolute existence, what is equivalent to the existence of absolute – and the author demonstrates his thesis through five attempts. It ends up at the comprehension of the unconditioned character of reason , result of the transcendental philosophy in general. (shrink)

Although the present paper looks upon the formal apparatus of quantum mechanics as a calculus of correlations, it goes beyond a purely operationalist interpretation. Having established the consistency of the correlations with the existence of their correlata, and having justified the distinction between a domain in which outcome-indicating events occur and a domain whose properties only exist if their existence is indicated by such events, it explains the difference between the two domains as essentially the difference between (...) the manifested world and its manifestation. A single, intrinsically undifferentiated Being manifests the macroworld by entering into reflexive spatial relations. This atemporal process implies a new kind of causality and sheds new light on the mysterious nonlocality of quantum mechanics. Unlike other realist interpretations, which proceed from an evolving-states formulation, the present interpretation proceeds from Feynman’s formulation of the theory, and it introduces a new interpretive principle, replacing the collapse postulate and the eigenvalue–eigenstate link of evolving-states formulations. Applied to alternatives involving distinctions between regions of space, this principle implies that the spatiotemporal differentiation of the physical world is incomplete. Applied to alternatives involving distinctions between things, it warrants the claim that, intrinsically, all fundamental particles are identical in the strong sense of numerical identical. They are the aforementioned intrinsically undifferentiated Being, which manifests the macroworld by entering into reflexive spatial relations. (shrink)

We show that MRP + MA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force (...) MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω 2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω 1 ) for all λ ≥ ω 2. (shrink)