We write $\mathcal {S}_n(A)$ for the set of permutations of a set A with n non-fixed points and $\mathrm {{seq}}^{1-1}_n(A)$ for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than $1$. With the Axiom of Choice, $|\mathcal {S}_n(A)|$ and $|\mathrm {{seq}}^{1-1}_n(A)|$ are equal for all infinite sets A. Among our results, we show, in ZF, that $|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$ for any infinite set A if ${\mathrm {AC}}_{\leq n}$ is (...) assumed and this assumption cannot be removed. In the other direction, we show that $|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A and the subscript $n+1$ cannot be reduced to n. Moreover, we also show that “ $|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A” is not provable in ZF. (shrink)

How are permutation arguments for the inscrutability of reference to be formulated in the context of a Davidsonian truth-theoretic semantics? Davidson takes these arguments to establish that there are no grounds for favouring a reference scheme that assigns London to “Londres”, rather than one that assigns Sydney to that name. We shall see, however, that it is far from clear whether permutation arguments work when set out in the context of the kind of truth-theoretic semantics which Davidson favours. (...) The principle required to make the argument work allows us to resurrect Foster problems against the Davidsonian position. The Foster problems and the permutation inscrutability problems stand or fall together: they are one puzzle, not two. (shrink)

The purpose of this article is to give a general overview of permutations in physics, particularly the symmetry of theories under permutations. Particular attention is paid to classical mechanics, classical statistical mechanics and quantum mechanics. There are two recurring themes: (i) the metaphysical dispute between haecceitism and anti-haecceitism, and the extent to which this dispute may be settled empirically; and relatedly, (ii) the way in which elementary systems are individuated in a theory's formalism, either primitively or in terms of the (...) properties and relations those systems are represented as bearing. Section 1 introduces permutations and provides a brief outline of the symmetric and braid groups. Section 2 discusses permutations in the general setting provided by model theory, in particular providing some definitions and elementary results regarding the permutability and indiscernibility of objects. Section 3 lays some philosophical groundwork for later sections, in particular articulated the distinction between haecceitism and anti-haecceitism and the distinction between transcendental and qualitative individuation. Section 4 addresses classical mechanics and introduces the procedure of quotienting, under which permutable states are identified. Section 5 addresses classical statistical mechanics, and outlines a number of equivalent ways to implement permutation invariance. I also briefly outline how particles may be qualitatively individuated in this framework. Section 6 addresses quantum mechanics. This contains an outline of: the representation theory of the symmetric groups; the topological approach to quantum statistics, in which the braid groups become relevant; and a brief proposal for qualitatively individuating quantum particles, and its implications for entanglement. Section 7 concludes with a discussion of equilibrium ensembles in the classical and quantum theories under permutation invariance. A (much) shorter version of this paper was published as a chapter in E. Knox & A. Wilson (eds), the Routledge Companion to Philosophy of Physics (Routledge, 2021), pp. 578-594. (shrink)

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993 ) concerning the modelling conditions for the axioms of assertion A → (( A → B ) → B ) (there called c 6) and permutation ( A → ( B → C )) → ( B → ( A → C )) (there called c 7). We show that the modelling conditions for assertion and (...) class='Hi'>permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001 ), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections. (shrink)

This essay reviews the contributions of Charles S. McCoy in three areas: religion and higher education, theology and ethics. I analyze McCoy’s primary ideas as a blending of influences from covenantal theology, Plato, Michael Polanyi and H. Richard Niebuhr.

It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function which is peculiar to NF turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente.

Let M be a model of ZFAC (ZFC modified to allow a set of atoms), and let N be an inner model with the same set of atoms and the same pure sets (sets with no atoms in their transitive closure) as M. We show that N is a permutation submodel of M if and only if N satisfies the principle SVC (Small Violations of Choice), a weak form of the axiom of choice which says that in some sense, (...) all violations of choice are localized in a set. A special case is considered in which there exists an SVC witness which satisfies a certain homogeneity condition. (shrink)

The article presents an analysis of a mystical practice of letter permutation conceived as part of the practice of “kavannah” in prayer. This practice was articulated by a 13th century anonymous ecstatic kabbalist writing in Catalonia. The anonymous author draws on earlier sources in the kabbalah and Ashkenazi spirituality. The article explores the wider connection between ecstasy and ritual, particularly prayer in the earlier stages of Judaism and its development in medieval theology and kabbalah. The anonymous author describes a (...) unique permutation technique capable of inducing ecstatic experiences as part of the liturgical ritual. (shrink)

The paper exposes the relevance of permuting conversions (in natural-deduction systems) to the role of such systems in the theory of meaning known as proof-theoretic semantics, by relating permuting conversion to harmony, hitherto related to normalisation only. This is achieved by showing the connection of permuting conversion to the general notion of canonicity, once applied to arbitrary derivations from open assumption. In the course of exposing the relationship of permuting conversions to harmony, a general definition of the former is proposed, (...) generalising the specific cases of disjunction and existential quantifiers considered in the literature. (shrink)

Is there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial (...) automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more. (shrink)

We write and for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality, where n is a natural number greater than 1. With the Axiom of Choice, and are equal for all infinite cardinals. We show, in ZF, that if is assumed, then for any infinite cardinal. Moreover, the assumption cannot be removed for and the superscript cannot be replaced by n. We (...) also show under that for any infinite cardinal, implies is Dedekind‐infinite. (shrink)

I offer a variant of Putnam’s “permutation argument,” originally an argument against metaphysical realism. This variant is called the “natural permutation argument.” I explain how the natural permutation argument generates a form of referential inscrutability which is not resolvable by consideration of “natural properties” in the sense of Lewis’s response to Putnam. However, unlike the classical permutation argument (which is applicable to nearly all interpretations of all first-order theories), the natural permutation argument only applies to (...) interpretations which have some special symmetries. I give an analysis of the interpretations to which the natural permutation argument does apply, and I explain how, when it fails to apply, the referential inscrutability generated by permutation arguments is resolvable by a Lewisian strategy. In order to demonstrate how these problems of referential inscrutability play out in an a priori setting relevant to philosophy, I discuss the applicability of the natural permutation argument in set-theoretic reasoning. I use the well-known Kunen inconsistency theorem to show that, in Zermelo–Fraenkel set theory, the Axiom of Choice is sufficient to resolve referential inscrutability. I then explain how, as a result of a recent theorem of Daghighi–Golshani–Hamkins–Jeřábek, in certain non-well-founded set theories the natural permutation argument does yield an intractable inscrutability of reference. (shrink)

Permutation invariance is often presented as the correct criterion for logicality. The basic idea is that one can demarcate the realm of logic by isolating specific entities—logical notions or constants—and that permutation invariance would provide a philosophically motivated and technically sophisticated criterion for what counts as a logical notion. The thesis of permutation invariance as a criterion for logicality has received considerable attention in the literature in recent decades, and much of the debate is developed against the (...) background of ideas put forth by Tarski in a 1966 lecture (Tarski 1966/1986). But as noted by Tarski himself in the lecture, the permutation invariance criterion yields a class of putative ‘logical constants’ that are essentially only sensitive to the number of elements in classes of individuals. Thus, to hold the permutation invariance thesis essentially amounts to limiting the scope of logic to quantificational phenomena, which is controversial at best and possibly simply wrong. In this paper, I argue that permutation invariance is a misguided approach to the nature of logic because it is not an adequate formal explanans for the informal notion of the generality of logic. In particular, I discuss some cases of undergeneration of the criterion, i.e. the fact that it excludes from the realm of logic operators that we have good reason to regard as logical, especially some modal operators. (shrink)

In this article I expound an understanding of the quantum mechanics of so-called “indistinguishable” systems in which permutation invariance is taken as a symmetry of a special kind, namely the result of representational redundancy. This understand- ing has heterodox consequences for the understanding of the states of constituent systems in an assembly and for the notion of entanglement. It corrects widespread misconceptions about the inter-theoretic relations between quantum mechanics and both classical particle mechanics and quantum field theory. The most (...) striking of the heterodox consequences are: that fermionic states ought not always to be considered entangled; it is possible for two fermions or two bosons to be discerned using purely monadic quantities; and that fermions may always be so discerned. (shrink)

In this paper we study some questions concerning Łukasiewicz implication algebras. In particular, we show that every subquasivariety of Łukasiewicz implication algebras is, in fact, a variety. We also derive some characterizations of congruence permutable algebras. The starting point for these results is a representation of finite Łukasiewicz implication algebras as upwardly-closed subsets in direct products of MV-chains.

There has been considerable recent philosophical debate over the implications of many particle quantum mechanics for the metaphysics of individuality (cf. Huggett [1997]). In this paper I look at things from a rather different perspective: by investigating the significance of permutation symmetry. I consider how various philosophical positions link up to the physical postulate of the indistinguishability of permuted states-permutation invariance-and how this postulate is used to explain quantum statistics. I offer an explanation of the statistics that relies (...) on the neglected parallel between permutations and covariant spatial transformation. And I explore the parallel, showing that a further kind of symmetry explains why permutations are invariant when spatial symmetries are not. (shrink)

We give a derivation of exclusion principles for the elementary particles of the standard model, using simple mathematical principles arising from a set theory of identical particles. We apply the theory of permutation group actions, stating some theorems which are proven elsewhere, and interpreting the results as a heuristic derivation of Pauli's Exclusion Principle (PEP) which dictates the formation of elements in the periodic table and the stability of matter, and also a derivation of quark confinement. We arrive at (...) these properties by using a symmetry property of collections of the particles themselves as compared for example to the symmetry property of their wave function under interchange of two particles. (shrink)

In Section 10 of Grundgesetze, Volume I, Frege advances a mathematical argument (known as the permutation argument), by means of which he intends to show that an arbitrary value-range may be identified with the True, and any other one with the False, without contradicting any stipulations previously introduced (we shall call this claim the identifiability thesis, following Schroeder-Heister (1987)). As far as we are aware, there is no consensus in the literature as to (i) the proper interpretation of the (...) permutation argument and the identifiability thesis, (ii) the validity of the permutation argument, and (iii) the truth of the identifiability thesis. In this paper, we undertake a detailed technical study of the two main lines of interpretation, and gather some evidence for favoring one interpretation over the other. (shrink)

Davidson has claimed that to conclude that reference is inscrutable, one must assume that "If some theory of truth... is satisfactory in the light of all relevant evidence... then any theory that is generated from the first theory by a permutation will also be satisfactory in the light of all relevant evidence." However, given that theories of truth are not directly read off the world, but rather serve as parts of larger theories of behavior, this assumption is far from (...) self-evident. A proper understanding of the role truth theories play in theories of interpretation makes the inscrutability of reference much less wide-spread than Davidson suggests, and, as a result, the radical interpretation methodology is much less likely to saddle its defenders with counterintuitive cases of indeterminacy than is commonly supposed. (shrink)

I argue that fidelity to the context principle requires us to construe reference as a theoretical relation. This point helps us understand the bearing of Putnam's permutation argument on the idea of a systematic theory of meaning. Notwithstanding objections that have been made against Putnam's deployment of that argument, it shows the reference relation to be indeterminate. But since the indeterminacy of reference arises from a metalinguistic perspective, our ability, as object‐language speakers, to talk about the ordinary features of (...) our lives is unaffected. (shrink)

In this paper, I address the problem wherein the same English word permits one of its complement positions to be satisfied by phrases of different categories. A well-known example of such an English word is the copula to be, whose complements include adjective phrases, noun phrases, prepositional phrases and adverbial phrases. I provide a way to treat such words, in particular verbs, as single lexical items through a conservative extension of the usual treatment of word classification as a pair comprising (...) a part of speech category and a complement list. I then show how a further conservative extension of complement lists permits a satisfactory formalization of doubly complemented English verbs which are synonymous under a permutation of their complements. These verbs include, but are not limited to, so-called double object constructions. (shrink)

We show that it is consistent with ZFC + ¬CH that there is a maxima a most disjoint permutation family A ⊆ Symsuch that A is a proper subset of an eventually different family E ⊆ ℕℕ and |A| < |E|. We also ask severa questions in this area.

We extend Shelah's first many model result to show that an unstable theory has 2 κ many non-permutation group isomorphic models of size κ, where κ is an uncountable regular cardinal.

We consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers: the least cardinal number of maximal cofinitary permutation groups; the least cardinal number of maximal almost disjoint permutation families; the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with that ; in fact we show that in the Miller model.

Let λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group B λ (Ω), or simply B λ , is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of B λ if B λ G is the full symmetric group on Ω. In [7], Macpherson and Neumann claimed to (...) have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set Ω is a supplement of B λ if and only if there exists $\Delta \subset \Omega$ with $|\Delta| such that the setwise stabiliser G {Δ} acts as the full symmetric group on $\Omega\setminus\Delta$ . However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory. (shrink)

It has been known for some time that there is a primitive recursive permutation of the nonnegative integers whose inverse is recursive but not primitive recursive. For example one has this result apparently for the first time in Kuznecov [1] and implicitly in Kent [2] or J. Robinson [3], who shows that every singularly recursive function ƒ is representable aswhere A, B, C are primitive recursive and B is a permutation.

The concept of transformative decision rules provides auseful tool for analyzing what is often referred to as the`framing', or `problem specification', or `editing' phase ofdecision making. In the present study we analyze a fundamentalaspect of transformative decision rules, viz. permutability. A setof transformative decision rules is, roughly put, permutable justin case it does not matter in which order the rules are applied.It is argued that in order to be normatively reasonable, sets oftransformative decision rules have to satisfy a number ofstructural (...) conditions that together imply permutability. Thisformal result gives support to a non-sequential theory of framing,i.e., a theory which prescribes no uniform order in which differentsteps in the framing process have to be performed. (shrink)

In an earlier paper published in a neutrosophic math journal (IJNS), we discussed a new approach to technology, which may be called as ‘opti-realism’ or ‘pess-optimism’ as alternative to utopianism based on technocracy, which may lead the world into global technototalitarianism. In this article, we submit a new approach to Nature and technology, which is more modest and humble, rather than a techno-utopianism version of reality that most futurists argue for. Our proposed approach resembles more to Myer-Briggs 16 types of (...) personality, including IJNS, IFNS etc. In our scheme, there are 8 characters of approach toward technology which can lead to many variations or we call it ‘Permutation.” Of course, if the readers ask one variation that we prefer, we would answer: Small Tech, High Touch. (shrink)

In this paper we study some questions concerning Łukasiewicz implication algebras. In particular, we show that every subquasivariety of Łukasiewicz implication algebras is, in fact, a variety. We also derive some characterizations of congruence permutable algebras. The starting point for these results is a representation of finite Łukasiewicz implication algebras as upwardly-closed subsets in direct products of MV-chains.

A permutation group G on a set A is ${\kappa }$ -homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. G is ${\kappa }$ -transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$.Giving a partial answer to a question (...) of P. M. Neumann [6] we show that there is an ${\omega }$ -homogeneous but not ${\omega }$ -transitive permutation group on a cardinal ${\lambda }$ provided (i) ${\lambda } {\omega }$ we give a method to construct large ${\kappa }$ -homogeneous, but not ${\kappa }$ -transitive permutation groups. Using this method we show that there exist ${\kappa }^+$ -homogeneous, but not ${\kappa }^+$ -transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$. (shrink)

The idea that the world is made of particles — little discrete, interacting objects that compose the material bodies of everyday experience — is a durable one. Following the advent of quantum theory, the idea was revised but not abandoned. It remains manifest in the explanatory language of physics, chemistry, and molecular biology. Aside from its durability, there is good reason for the scientific realist to embrace the particle interpretation: such a view can account for the prominent epistemic fact that (...) only limited knowledge of a portion of the material universe is needed in order to make reliable predictions about that portion. Thus, particle interpretations can support an abductive argument from the epistemic facts in favor of a realist reading of physical theory. However, any particle interpretation with this property is untenable. The empirical adequacy of modern particle theories requires adoption of a postulate known as permutation invariance — the claim that interchanging the role of two particles of the same kind in a dynamical state description results in a description of the identical state. It is the central claim of this essay that PI is incompatible with any particle interpretation strong enough to account for the epistemic facts. This incompatibility extends across all physical theories. To frame and motivate the inconsistency argument, I begin by fixing the relevant notion of particle. To single out those accounts of greatest appeal to the realist, I develop the logically weakest particle ontology that entails the epistemic fact that the world is piecewise predictable, an ontology I call ‘minimal atomism’. The entire series of scientific conceptions of the particle, from Newton’s mechanically interacting corpuscles to the ‘centers of force’ in classical field theories, all comport with MA. As long as PI is left out, even quantum mechanics can be viewed this way. To assess the impact of PI on this picture, I present a framework for rigorously connecting interpretations to physical theories. In particular, I represent MA as a set of formal conditions on the models of physical theories, the mathematical structures taken to represent states of the world. I also formulate PI — originally introduced as a postulate of non-relativistic quantum mechanics — in theory independent terms. With all of these pieces in hand, I am then able to present a proof of the inconsistency of PI and MA. In the second part of the essay, I survey responses to the inconsistency result open to the scientific realist. The two most plausible approaches involve abandoning particles in one way or another. The first alternative interpretation considered takes the property bearing objects of the world to be regions of space rather than particles. In this view, the properties once attributed to particles in quantum states are attributed instead to one or more regions of space. PI no longer obtains in this case, at least not as a statement about the permutation symmetry of property bearers. Rather, the new interpretation naturally imposes an analogous constraint on quantum states. The second major approach to evading the inconsistency result is to dispense with objects altogether. This is the recommendation of so-called ‘Ontic Structural Realism’. The central OSR thesis is that structure rather than entities are the basic ontological components of the world. OSR is intended to embrace the ‘miracle’ argument in favor of scientific realism while avoiding the pessimistic meta-induction. I demonstrate that one principal motivation for OSR based on the under-determination of interpretations in QM is actually dissolved by the incompatibility result. At the same time, I suggest reasons to think that OSR fares no better with respect to the pessimistic meta-induction than traditional realism does. Thus, while PI and MA may be incompatible, object ontologies remain the best option for the realist. (shrink)

There is an extensive philosophical literature on the interrelated issues of identity, individuality, and distinguishability. Out of this discussion has arisen a concept called “permutation invariance” that is asserted to apply to quantum systems. I argue that in fact there is no such invariance, and that the best way to understand the permutation of labels in the symmetrized states is as an exchange of haecceities, rather than as an exchange of essences equivalent to permutation invariance. I argue (...) that the strongest notion of haecceity (i.e., "classical haecceity") does not apply at the quantum level, but that in order to properly account for the need for symmetrization in quantum systems, a weaker kind of haecceity must be involved, which I call quantum haecceity. (shrink)

We show that if N and M are transitive models of ZFA such that N M, N and M have the same kernel and same set of atoms, and M AC, then N is a Fraenkel-Mostowski-Specker (FMS) submodel of M if and only if M is a generic extension of N by some almost homogeneous notion of forcing. We also develop a slightly modified notion of FMS submodels to characterize the case where M is a generic extension of N not (...) necessarily by an almost homogeneous notion of forcing. (shrink)