In this paper, the independence of irrelevant alternatives and the Pareto principle are simultaneously weakened in the Arrovian framework of social choice. Moreover, we also relax transitivity of social preferences. We show that impossibility remains under weaker versions of Arrow’s original conditions. Our results complement the recent work by Coban and Sanver (Soc Choice Welf 43(4):953–961, 2014).

We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of (...) inductive methods from alternative perceptions of 'irrelevance' is an indication that this notion is imperfectly understood at present. (shrink)

In this paper, I demonstrate that Kant's commitment to an asymmetry between the control conditions for praise and blame is explained by his endorsement of the principle Ought Implies Can (OIC). I argue that Kant accepts only a relatively weak version of OIC and that he is hence committed only to a relatively weak requirement of alternate possibilities for moral blame. This suggests that whether we are transcendentally free is irrelevant to questions about moral permissibility and moral (...) blameworthiness. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:HUME, THE CAUSAL PRINCIPLE, AN'D KEMP SMITH When we say of a proposition that it is possible, we sometimes mean no more than that it is logically possible, that is, consistent with itself. A proposition can be possible in stronger senses than this, but not in any weaker one. For a sense of "p is possible" that did not entail "p is self-consistent, "would have to be a (...) sense of "p is possible" which was consistent with "p is self-inconsistent" And it is obvious that there can be no such sense as that. One of the stronger senses in which a proposition can be possible is this: consistent (with itself and in addition) with every observation-statement. (I mean by an observationstatement, a proposition which, if it were true, could in principle be discovered by experience to be true.) If a proposition? is contingent, (that is, neither necessarily true??t necessarily false), then its negation (its contradictory) not-p is contingent too; and since contingent, not necessarily false; and since not necessarily false, consistent with itself. So, given any contingent propositions, its falsity is among those propositions which are possible in the first and weak sense mentioned above. If a proposition? is not only contingent but unverifiable,(that is, not deducible from any observation- statement), then not-p is consistent both with itself and with every observation -statement. So, given any unverifiable contingent proposition, its falsity is among those propositions which are possible in the second and stronger sense mentioned above. Some stock examples. "Whatever is a raven is black" is contingent, and its falsity therefore possible in the first sense; so too is its contradictory "There are such things as non-black ravens" contingent, and its falsity possible in the first sense. But the latter proposition is not unverifiable, since there are observation-statements from which it is deducible. ("There are such things as green ravens", for example; or "A green raven was in captivity at Taronga Park Zoo in January 31st 1975".) The former proposition, on the other hand, is not only contingent but unverifiable; since there are no observation -statements from which it is deducible. Hence, "whatever is a raven is black" is a proposition the falsity of which is possible both in the first and in the second sense noticed above. A third and still stronger sense in which a proposition can be possible is the following:? is possible if and only if (p is consistent with itself,every observation-statement is consistent with p, and in addition)? is neither more nor less probable in relation to the conjunction of e and n, where e is any observation-statement and? any necessary truth, than it is in relation to? alone. The last condition may be expressed, symbolically, as "P(p/n.e) » P(p/n) for all necessarily true? and all observation-statements e"; or in Carnap' s phrase as, "every observation- statement is initially irrelevant to p." Some examples will quickly make the idea of initial irrelevance, and its opposite, more familiar; and hence will make the third sense of "p is possible" more familiar. Let? be any necessary truth and e any observation-statement. Now let? be, for- example,some necessarily false proposition; then, since P(p/n) « o = P(p/n.e), e is initially irrelevant to p. Or let? be necessarily true; then, since P(p/n) ¦ 1 = P(p/n.e), e is again initially irrelevant to p. Again, let? be "Whatever is a raven is black": then some observation- statements e are initially relevant to p. For if e is "This raven is black," for example, P(p/n.e)>P(p/n) ; or at any rate, that is what we think if we are not inductive sceptic?. Again, if? is "The Holy Ghost proceeds from the Father and the Son", it is at least very plausible to say that every observational e is initially irrelevant to p; since there appears to be no observational e such that P(p/n.e)>P(p/n), or P(p/n.e). (shrink)

As the demand for healthcare rises, so does the need for priority setting in healthcare. In this paper, I consider a prominent priority-setting principle: proportional shortfall. My purpose is to argue that proportional shortfall, as a principle, should not be adopted. My key criticism is that proportional shortfall fails to consider past health.Proportional shortfall is justified as it supposedly balances concern for prospective health while still accounting for lifetime health, even though past health is deemed irrelevant. Accounting for (...) this lifetime perspective means that the principle may indirectly consider past health by accounting for how far an individual is from achieving a complete, healthy life. I argue that proportional shortfall does not account for this lifetime perspective as it fails to incorporate the fair innings argument as originally claimed, undermining its purported justification.I go on to demonstrate that the case for ignoring past health is weak, and argue that past health is at least sometimes relevant for priority-setting decisions. Specifically, when an individual’s past health has a direct impact on current or future health, and when one individual has enjoyed significantly more healthy life years than another.Finally, I demonstrate that by ignoring past illnesses, even those entirely unrelated to their current illness, proportional shortfall can lead to instances of double jeopardy, a highly problematic implication. These arguments give us reason to reject proportional shortfall. (shrink)

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the (...) existence of a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $. (shrink)

We study the extension 123) of the theory S21 by instances of the dual weak pigeonhole principle for p-time functions, dWPHPx2x. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S21+dWPHP. We construct a propositional proof system WF , which captures the Π1b-consequences of S21+dWPHP. We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. 256). We (...) prove that dWPHP is equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction in a conservative extension of S21+dWPHP. (shrink)

A modern assessment of the classical Boltzmann-Schuetz argument for large-scale entropy fluctuations as the origin of our observable cosmological domain is given.The emphasis is put on the central implication of this picture which flatly contradicts the weak anthropic principle as an epistemological statement about the universe. Therefore, to associate this picture with the anthropic principle as it is usually done is unwarranted. In particular, Feynman's criticism of theanthropic principle based on the entropy-fluctuation picture is a product (...) of this semantic confusion. (shrink)

It is well known that S12 cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective weak pigeonhole principle in S12 for provably weaker function classes.

The design argument for God’s existence was critically assessed when in the growth of modern science the cognitive value of teleological categories was called into question. In recent discussions dealing with anthropic principles there has appeared a new version of the design argument, in which cosmic design is described without the use of teleological terms. The weak anthropic principle (WAP), a most critical version of all these principles, describes the fine-tuning of physical parameters necessary to the genesis of (...) carbon-based life. Consequently, a new version of the philosophical design argument can be developed on the basis of the weak anthropic principle.(edited). (shrink)

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of. Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which (...) states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model. (shrink)

The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well-orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo-Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals is provable without the axiom of choice.

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos (...) as defined by van den Berg. (shrink)

We bring together some facts about the weak pigeonhole principle from bounded arithmetic, complexity theory, cryptography and abstract model theory. We characterize the models of arithmetic in which WPHP fails as those which are determined by an initial segment and prove a conditional separation result in bounded arithmetic, that PV + lies strictly between PV and S21 in strength, assuming that the cryptosystem RSA is secure.

We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of S21 exists in which NP is not in the second level of the linear hierarchy; and that a model of S21 exists in which the polynomial hierarchy collapses to the linear hierarchy. Our methods are model-theoretic. We use the assumption about factoring to get a model in which the (...) class='Hi'>weak pigeonhole principle fails in a certain way, and then work with this failure to obtain our results. As a corollary of one of the proofs, we also show that in S21 the failure of WPHP (for Σb1 definable relations) implies that the strict version of PH does not collapse to a finite level. (shrink)

Context: Is non-dualist epistemology, based on the unity of descriptions and objects, logically consistent? Problem: What is the status of the infinite regresses that the non-dualist Josef Mitterer, in his book The Beyond of Philosophy, censures in dualist thought? Their academic discussion is still in its infancy. Method: An attempt to reconstruct and differentiate Mitterer’s infinite regress accusations against dualism (originating from the 1970s) with today’s means and distinctions. Results: A weak and a strong linguistic principle are presented (...) (non-dualism being subsumed under the strong linguistic principle), which are defined as such depending on whether the infinite regresses of dualism are interpreted as benign or as vicious. Implications: Further penetrating investigation of the infinite regresses highlighted by Mitterer is of crucial importance because these regresses have not yet been discussed and classified in relevant current publications on infinite regress arguments by authors such as Rescher or Gratton. Such proper classification is indispensable, however, if the value of the non-dualist alternative to, and its critique of, dualist thought is to be assessed adequately. (shrink)

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ+-saturated.1 The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣*−, and show that if it (...) holds on a stationary subset S of λ, then no normal filter on S is λ+-saturated. Under the above assumptions, ♣*− is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣*− holds modulo an ideal obtained through the weak reflection. (shrink)

We consider a weak version of Schindler’s remarkable cardinals that may fail to be ${{\rm{\Sigma }}_2}$-reflecting. We show that the ${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and that the existence of a non-${{\rm{\Sigma }}_2}$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an ω-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + “Ord is (...) not ${{\rm{\Delta }}_2}$-Mahlo” and ${\text{gVP}}$ + “there is no proper class of remarkable cardinals” are both equiconsistent with the existence of a proper class of ω-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler. (shrink)

The weakly compact reflection principle\\) states that \ is a weakly compact cardinal and every weakly compact subset of \ has a weakly compact proper initial segment. The weakly compact reflection principle at \ implies that \ is an \-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \ is \\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at \ then there is (...) a forcing extension preserving this in which \ is the least \-weakly compact cardinal. Along the way we generalize the well-known result which states that if \ is a regular cardinal then in any forcing extension by \-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \ is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length \ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal. (shrink)

Incompatibilism about freedom and causal determinism is commonly supported by appeal to versions of the well known Consequence argument. Critics of theConsequence argument have presented counterexamples to the Consequence argument’s central inference principle. The thesis of this article is that proponents of the Consequence argument can easily bypass even the best of these counterexamples.

We introduce the weak partial square principles View the MathML source and View the MathML source, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that □λ,κ does not imply View the MathML source.

One common objection against the Principle of Sufficient Reason is that it leads to a highly counterintuitive position, namely, necessitarianism. In this paper, drawing on Avicenna’s modal theory, I differentiate between two versions of necessitarianism: strong necessitarianism and weak necessitarianism. I argue that the modal intuition driving this objection pertains to strong necessitarianism, while the Principle of Sufficient Reason, at most, leads to weak necessitarianism.

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

We show that the so-called weak Markov's principle which states that every pseudo-positive real number is positive is underivable in [MATHEMATICAL SCRIPT CAPITAL T]ω ≔ E-HAω + AC. Since [MATHEMATICAL SCRIPT CAPITAL T]ω allows one to formalize Bishop's constructive mathematics, this makes it unlikely that WMP can be proved within the framework of Bishop-style mathematics . The underivability even holds if the ine.ective schema of full comprehension for negated formulas is added, which allows one to derive the law (...) of excluded middle for such formulas. (shrink)

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set A, of an infinite subset of A or of its complement. In this paper, we study the infinite pigeonhole pr...

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural numbers.

In Minimal Morality, Michael Moehler justifies what he calls the weak principle of universalization as a principle of pure instrumental morality. This article addresses the application of this principle and problems associated with it. Specifically, the article focuses on the principle’s ability to protect the interests of the most vulnerable members of society: agents without primary moral standing, specifically non-human animals; and the weakest members of society, either as a result of their diminished relative bargaining (...) power in certain cases of conflict, or because of their present standing in society, made worse off by historical injustices. To this end, the article considers, respectively, the problems of moral standing, indivisible goods and historical injustice. (shrink)

This article contends that recent attempts to construct Frankfurt-style cases (FSCs) are irrelevant to the debate over free will. The principle of alternate possibilities (PAP) states that moral responsibility requires indeterminism, or multiple possible futures. Frankfurt's original case purported to demonstrate PAP false by showing an agent can be blameworthy despite not having the ability to choose otherwise; however he admits the agent can come to that choice freely or by force, and thus has alternate possibilities. Neo-FSCs attempt to (...) show that alternate possibilities are irrelevant to explaining an agent's moral responsibility, but a successful Neo-FSC would be consistent with the truth of PAP, and thus is silent on the big metaphysical issues at the center of the free will debate. (shrink)

In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The (...) importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn—Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleene's ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense. (shrink)

Precautionary Principles are often said to be appropriate for decision-making in contexts of uncertainty such as climate policy. Contexts of uncertainty are contrasted to contexts of risk depending on whether we have probabilities or not. Against this view, I argue that the risk-uncertainty distinction is practically irrelevant. I start by noting that the history of the distinction between risk and uncertainty is more varied than is sometimes assumed. In order to examine the distinction, I unpack the idea of having probabilities, (...) in particular by distinguishing three interpretations of probability: objective, epistemic, and subjective probability. I then claim that if we are concerned with whether we have probabilities at all—regardless of how low their epistemic credentials are—then we almost always have probabilities for policy-making. The reason is that subjective and epistemic probability are the relevant interpretations of probability and we almost always have subjective and epistemic probabilities. In contrast, if we are only concerned with probabilities that have sufficiently high epistemic credentials, then we obviously do not always have probabilities. Climate policy, for example, would then be a case of decision-making under uncertainty. But, so I argue, we should not dismiss probabilities with low epistemic credentials. Rather, when they are the best available probabilities our decision principles should make use of them. And, since they are almost always available, the risk-uncertainty distinction remains irrelevant. (shrink)

Mereological principles are sometimes used to support general claims about the structure and arrangement of objects in the world. I focus initially on one such mereological principle, the weak supplementation principle (WSP). It is not obvious that (WSP) is prescribed by ordinary thinking about parthood. Further, (WSP) is not needed for a fairly strong formal characterization of the part–whole relation. For these reasons, some arguments relying on (WSP) might be countered by simply denying (WSP). I argue more (...) generally that there is no reason to assume that one core mereology functions as a common basis for all plausible metaphysical theories. (shrink)

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal (...) one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal. (shrink)

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal princi...

Is there a plausible argument for external world skepticism? Robert Nozick’s well-known discussion focuses upon arguments which utilize the Sensitivity Requirement and the Closure Principle. Nozick claims, correctly, that no such argument succeeds. But he gets almost all the details wrong. The Sensitivity Requirement and the Closure Principle are compatible; the Sensitivity Requirement is incorrect; and even if true, the Closure Principle is structurally incapable of generating a plausible and valid global skeptical argument. It is therefore a (...) mistake to take the Closure Principle as central in discussions of skepticism. The paper concludes by examining the prospects for a plausible skeptical argument. (shrink)

This article investigates the relationship among the weak Pareto principle, the strong Pareto principle, and positive responsiveness in the context of voting. First, it is shown that under a mild domain condition, if an anonymous and neutral collective choice rule (CCR) is complete and transitive, then the weak Pareto principle and the strong Pareto principle are equivalent. Next, it is shown that under another mild domain condition, if a neutral CCR is transitive, then the (...) strong Pareto principle and positive responsiveness are equivalent. By applying these results, we obtain a new characterization of the method of majority decision. (shrink)

The Weak Principle of the Identity of Indiscernibles (weak PII), states that numerically distinct items must be discernible by a symmetrical and irreflexive relation. Recently, some authors have proposed that weak PII holds in non relativistic quantum mechanics, contradicting a long tradition claiming PII to be simply false in that theory. The question that arises then is: are relations allowed in the scope of PII? In this paper, we propose that quantum mechanics does not help us (...) in deciding matters concerning that problem, since that is a metaphysical problem rather than a quantum mechanical one. We argue further that weak PII is unmotivated on metaphysical grounds. We examine three metaphysical theses (bundle theory, counting, empiricism) that may provide reasons for one to sustain PII, and we conclude that weak PII gets no independent motivation from them. (shrink)

ABSTRACTLinguistic evidence supports the claim that certain, weak rejections are less specific than assertions. On the basis of this evidence, it has been argued that rejected sentences cannot be premisses and conclusions in inferences. We give examples of inferences with weakly rejected sentences as premisses and conclusions. We then propose a logic of weak rejection which accounts for the relevant phenomena and is motivated by principles of coherence in dialogue. We give a semantics for which this logic is (...) sound and complete, show that it axiomatizes the modal logic KD45 and prove that it still derives classical logic on its asserted fragment. Finally, we defend previous logics of strong rejection as being about the linguistically preferred interpretations of weak rejections. (shrink)

E.J. Lowe is one of the few philosophers who defend both the existence of spatially coincident entities and the Principle of Weak Extensionality that no two objects which have proper parts have exactly the same proper parts at the same time. Lowe maintains that when spatially coincident things like the statue and the lump of bronze are in a constitution relation, the constituted entity (the statue) has parts that the constituting entity (the lump) doesn’t, hence the compatibility with (...) Weak Extensionality. My contention is that his argument for why the statue has parts the lump of bronze lacks can also be used to show that the lump of bronze has parts the statue doesn’t. This will mean that there is no basis for saying the statue and the lump are in a constitution relation. I argue for accepting a modified account of constitution and abandoning the Principle of Weak Extensionality. (shrink)