-
299567.706089
Discussion of the Aristotelian syllogism over the last sixty years has arguably centered on the question whether syllogisms are inferences or implications. But the significance of this debate at times has been taken to concern whether the syllogistic is a logic or a theory, and how it ought to be represented by modern systems.
-
321056.706242
The Turing test for machine thought has an interrogator communicate (by typing) with a human and a machine both of which try to convince the interrogator that they are human. The interrogator then guesses which is human. …
-
538347.706252
I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor’s account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory.
-
596189.706263
In classical first-order logic (FOL), let T be a theory with an unspecified (arbitrary) constant c, where the symbol c does not occur in any of the axioms of T. Let psi(x) be a formula in the language of T that does not contain the symbol c. In a well-known result due to Shoenfield (the “theorem on constants”), it is proven that if psi(c) is provable in T, then so is psi(x), where x is the only free variable in psi(x). In the proof of this result, Shoenfield starts with the hypothesis that P is a valid proof of psi(c) in T, and then replaces each occurrence of c in P by a variable to obtain a valid proof of psi(x) in T, the argument being that no axiom of T is violated by this replacement. In this paper, we demonstrate that the theorem on constants leads to a meta-inconsistency in FOL (i.e., a logical inconsistency in the metatheory of T in which Shoenfield’s proof is executed), the root cause of which is the existence of arbitrary constants. In previous papers, the author has proposed a finitistic paraconsistent logic (NAFL) in which it is provable that arbitrary constants do not exist. The nonclassical reasons for this nonexistence are briefly examined and shown to be relevant to the above example.
-
870608.706277
How can the Biblical God be the Lord and King who, being typically unseen and even self-veiled at times, authoritatively leads people for divine purposes? This article’s main thesis is that the answer is in divine moral leading via human moral experience of God (of a kind to be clarified). The Hebrew Bible speaks of God as ‘king,’ including for a time prior to the Jewish human monarchy. Ancient Judaism, as Martin Buber has observed, acknowledged direct and indirect forms of divine rule and thus of theocracy. This article explores the importance of divine rule as divine direct leading, particularly in moral matters, without reliance on indirect theocracy supervised by humans. It thus considers a role for God as Über-King superior to any human king, maintaining a direct moral theocracy without a need for indirect theocracy. The divine goal, in this perspective, is a universal commonwealth in righteousness, while allowing for variation in political structure. The article identifies the importance in the Hebrew Bible of letting God be God as an Über-King who, although self-veiled at times, leads willing people directly and thereby rules over them uncoercively. It also clarifies a purpose for divine self-veiling neglected by Buber and many others, and it offers a morally sensitive test for unveiled authenticity in divine moral leading.
-
928366.706288
This is an attempt to axiomatise the natural laws. Note especially axiom 4, which is expressed in third order predicate logic, and which permits a solution to the problem of causation in nature without stating that “everything has a cause”. The undefined term “difference” constitutes the basic element and each difference is postulated to have an exact position and to have a discrete cause. The set of causes belonging to a natural set of dimensions is defined as a law. This means that a natural law is determined by the discrete causes tied to a natural set of dimensions. A law is defined as “defined” in a point if a difference there has a cause. Given that there is a point for which the law is not defined it is shown that a difference is caused that connects two points in two separate sets of dimensions.
-
928388.706294
Ontology and theology cannot be combined if ontology excludes non physical causes. This paper examines some possibilities for ontology to be combined with theology in so far as non physical causes are permitted. The paper builds on metaphysical findings that shows that separate ontological domains can interact causally indirectly via interfaces. As interfaces are not universes a first universe is allowed to be caused by an interface without violating the principle of causal closure of any universe. Formal theology can therefore be based on the assumption that the (first) universe is caused by God if God is defined as the first cause. Given this, formal theology and science can have the same ontological base.
-
1448122.7063
This paper develops the idea that valid arguments are equivalent to true conditionals by combining Kripke’s theory of truth with the evidential account of conditionals offered by Crupi and Iacona. As will be shown, in a first-order language that contains a naïve truth predicate and a suitable conditional, one can define a validity predicate in accordance with the thesis that the inference from a conjunction of premises to a conclusion is valid when the corresponding conditional is true. The validity predicate so defined significantly increases our expressive resources and provides a coherent formal treatment of paradoxical arguments.
-
1448146.706306
Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral proof rules has a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself.
-
1491175.706312
In the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) Existence: Settings where no countably additive probability exists though finitely additive probabilities do. (2) Complete Additivity: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the Comparative principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the Comparative principle [C ].
-
1535719.706323
Some people want to be able to compare the sizes of infinite sets while preserving the "Euclidean" proper subset principle that holds for finite sets:
- If A is a proper subset of B, then A < B. We also want to make sure that our comparison agrees with how we compare finite sets:
- If A and B are finite, then A ≤ B if and only if A has no more elements than B. …
-
1572815.70633
This essay examines the philosophical significance of Ω-logic in Zermelo- Fraenkel set theory with choice (ZFC). The categorical duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The hyperintensional profile of Ω-logical validity can then be countenanced within a coalgebraic logic. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal and hyperintensional profiles of Ω-logical validity correspond to those of second-order logical consequence, Ω-logical validity is genuinely logical. Second, the foregoing provides a hyperintensional account of the interpretation of mathematical vocabulary.
-
1621459.706336
Fragmentalism allows incompatible facts to constitute reality in an absolute manner, provided that they fail to obtain together. In recent years, the view has been extensively discussed, with a focus on its formalisation in model-theoretic terms. This paper focuses on three formalisations: Lipman’s approach, the subvaluationist interpretation, and a novel view that has been so far overlooked. The aim of the paper is to explore the application of these formalisations to the alethic modal case. This logical exploration will allow us to study (i) cases of metaphysical incompatibility between modal facts and (ii) cases of modal dialetheias. In turn, this will enrich our understanding of the role of impossibility in the fragmentalist framework.
-
2265038.706342
Citation: Ellerman, D. A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics.
-
2303155.706348
In this paper, we discuss J. Michael Dunn’s foundational work on the semantics for First Degree Entailment logic (FDE), also known as Belnap–Dunn logic (or Sanjaya–Belnap–Smiley–Dunn Four-valued Logic, as suggested by Dunn himself). More specifically, by building on the framework due to Dunn, we sketch a broad picture towards a systematic understanding of contra-classicality. Our focus will be on a simple propositional language with negation, conjunction, and disjunction, and we will systematically explore variants of FDE, K3, and LP by tweaking the falsity condition for negation.
-
2303179.706368
In this paper, we apply a Herzberger-style semantics to deal with the question: is the de Finetti conditional a conditional? The question is pressing, in view of the inferential behavior of the de Finetti conditional: it allows for inferences that seem quite unexpected for a conditional. The semantics we advance here for the de Finetti conditional is simply the classical semantics for material conditional, with a further dimension whose understanding depends on the kind of application one has in mind. We discuss such possible applications and how they cover ground already advanced in the literature.
-
2303201.706376
Connexive logic is a topic in non-classical logic that is receiving a lot of attention these days; however, that is not to say that it is a recent topic by any means. Some have claimed that it has its historical roots in antiquity, others have disputed this. Much less controversial is the important role of two connexive logical systems in the much more recent history of the subject. The first is the system CC1 that is due to Richard Angell [1] and Storrs McCall [15] and marks, in many ways, the modern inception of connexive logic as a unified topic in logical research. While that topic never quite disappeared after the seminal work by Angell and McCall, it was only after Heinrich Wansing started making his contributions some forty years later that it truly started to blossom. For that reason alone, the first connexive logic Wansing introduced in [33], which is called C, has an indisputably important place in the history of connexivity. Also, it offered one of the most elegant semantics as well as proof systems for connexive logics to date (see also [22, p.178]).
-
2303224.706383
In his “The simple argument for subclassical logic,” Jc Beall advances an argument that led him to take FDE as the one true logic (the latter point is explicitly made clear in his “FDE as the One True Logic”). The aim of this article is to point out that if we follow Beall’s line of reasoning for endorsing FDE, there are at least two additional reasons to consider that FDE is too strong for Beall’s purposes. In fact, we claim that Beall should consider another weaker subclassical logic as the logic adequate for his project. To this end, we first briefly present Beall’s argument for FDE. Then, we discuss two specific topics that seem to motivate us to weaken FDE. We then introduce a subsystem that will enjoy all the benefits of Beall’s suggestion.
-
2303254.706389
A BSTRACT. S ören Halld´en’s logic of nonsense is one of the most well-known many-valued logics available in the literature. In this paper, we discuss Peter Woodruff’s as yet rather unexplored attempt to advance a version of such a logic built on the top of a constructive logical basis. We start by recalling the basics of Woodruff’s system and by bringing to light some of its notable features. We then go on to elaborate on some of the difficulties attached to it; on our way to offer a possible solution to such difficulties, we discuss the relation between Woodruff’s system and two-dimensional semantics for many-valued logics, as developed by Hans Herzberger. Keywords: Peter Woodruff, logic of nonsense, constructive logic, Hans Herzberger, two-dimensional semantics.
-
2314340.706401
It is often assumed that concepts from the formal sciences, such as mathematics and logic, have to be treated differently from concepts from non-formal sciences. This is especially relevant in cases of concept defectiveness, as in the empirical sciences defectiveness is an essential component of lager disruptive or transformative processes such as concept change or concept fragmentation.
-
2317758.706411
This paper explores how, given a proof, we can systematically transform it into a proof that contains no irrelevancies and which is as strong as possible. I define a weaker and stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. Using classical core logic to study classical validities and core logic to study intuitionistic validities, I show that every core proof or classical core proof can be transformed into a perfect proof. In a sequel paper, I show how proofs in core logic can also be transformed into gaunt proofs and I observe that this property fails for classical core logic.
-
2317883.706424
This paper is the second part of a series exploring how, given a proof, we can inductively transform it into a proof that contains no irrelevancies and is as strong as possible. In the prequel paper, I defined a weaker and a stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. There, I showed how proofs in core logic and classical core logic can be transformed into perfect proofs. In this paper I study gaunt proofs. I show how proofs in core logic can be inductively transformed into gaunt core proofs, but that this property fails for the natural deduction system of classical core logic.
-
2317908.706433
Set-theoretic potentialism is the view that the universe of sets is potentially infinite: it can always, necessarily, be expanded to a more inclusive universe of sets. One version of this view is that the set-theoretic universe can always be expanded to a forcing extension in particular. This view has primarily been studied from a technical point of view, however; in this chapter I explore what philosophical conceptions of set theory might motivate forcing potentialism. I begin by raising an explanatory challenge for any form of width potentialism based on the iterative conception of set, and then sharpen this challenge to argue that any broadly iterative conception of set is inconsistent with the claim that the possible width extension of a universe are exactly its forcing extensions. Finally, I suggest one possible way meeting the explanatory challenge by disentangling the iterative conception of set-formation from the combinatorial conception of sethood. This makes room for what I call the iterative logical conception of set. I sketch a toy model of a potentialist system which is both height- and width-potentialist, and where the width extension include all (but not only) the forcing extensions.
-
2317931.706442
Being relevant to some topic can be informally understood as making a difference or having something to contribute. Given a sequent ∆ ⇒ Γ, we can say, somewhat schematically, that a component of the sequent is relevant to the sequent when it contributes to the validity of the sequent. Different ways of making precise the idea of contributing to the validity and different understandings of the components of a sequent lead to a hierarchy of explications of relevance. I identify four key explications, called gaunt validity, perfect validity, relevant validity, and perfectibility. Each is shown to enjoy an interesting variable sharing property. Furthermore, if we begin with a standard sequent calculus for classical logic and introduce some simple constraints on the rules, the result is a fragment of classical logic that proves exactly the gauntly valid sequents.
-
2322650.706451
Relevant logics, in the tradition coming out of the work of Anderson and Belnap (1975), are concerned with implication. Relevant logics constitute a large family with great variety, even restricting attention to the comparatively well known logics. Perhaps unsurprisingly, different philosophical motivations have been given for relevant logics, targeted at different subfamilies of the broader group. In this article I will survey the different philosophical views motivating relevant logics, indicating how they secure relevance, what logics are most clearly supported by those views, and the presentation of the logic most naturally supported by the view.
-
2372074.706458
I shall discuss here the topics of existence and nonexistence, of what it is for an individual to be actual and what it is for an individual not to be actual. What I shall have to say about these matters offers little toward our primordial need to discover the Meaning of Existence, but I hope to say some things that will satisfy the more modest ambition of those of us who wish to know the meaning of ‘existence’. I shall also say some things that bear on issues in the grandest traditions of Philosophy.
-
2545192.706467
My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language.
-
2703308.706473
I investigate whether Wittgenstein’s “weakly exclusive” Tractarian semantics (as reconstructed by Rogers and Wehmeier 2012) is compositional. In both Tarskian and Wittgensteinian semantics, one has the choice of either working exclusively with total variable assignments or allowing partial assignments; the choice has no bearing on the compositionality of Tarskian semantics, but turns out to make a difference in the Wittgensteinian case. Some philosophical ramifications of this observation are discussed.
-
2714174.706482
Finite set theory FST consists of all of the axioms of ZF without the axiom of infinity and has the model HF of hereditarily finite sets as its canonical model. In this paper, we shall consider all combinatorially possible systems corresponding to subsets of the axioms of finite set theory and develop a general technique called axiom closure of graphs; for each of the 2 = 64 combinatorially possible systems we shall either show that it cannot hold in a transitive submodel of the hereditarily finite sets or provide a concrete model in which it holds (cf. Table 1).
-
2718364.706489
Identity is a peculiar notion. On the one hand, surely everything is just the thing that it is and nothing else. But on the other hand, we are tempted to think of it as a relation. After all, it appears as a relational predicate in formal languages (e.g. ‘? = ?’) and in natural languages (e.g. ‘Eric is identical to George’). However, the idea of identity as a relation that might hold between two or more things is absurd for, after all, the whole idea of identity is that it concerns just one thing. At best, we can think of it as a relation that everything bears only to itself.