We introduce the notions of syntactic synonymy and referential syn- onymy due to Moschovakis. Those notions are capable of accounting for fine- grained aspects of the meaning of linguistic expressions, by formalizing the Fregean distinction between sense and denotation. We integrate Moschovakis’s theory with the theory of concepts developed in the foundational ontology DOLCE, in order to enable a formal treatment of equivalence between concepts.

In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely mathematical definition of (...) observational equivalence are unpromising. Despite this critique, I suggest that Werndl’s proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)

This paper presents a simple pair of first-order theories that are not definitionally (nor Morita) equivalent, yet are mutually conservatively translatable and mutually 'surjectively' translatable. We use these results to clarify the overall geography of standards of equivalence and to show that the structural commitments that theories make behave in a more subtle manner than has been recognized.

One disturbing but telling demonstration of how definitions of knowledge can go wrong is due to Hawthorne.[1] Although properly intended as an attack on Dretske’s theory of conclusive reasons, it also works against Nozick’s conditional theory. Hawthorne provides a simple example: suppose, he declaims, that salmon induces hallucinations if eaten in great amounts. Suppose that, to be more precise, were you to eat it in amounts greater than 14 pounds at one go (never mind how that is humanly possible), it (...) could cause you to believe that you’ve only had a few tidbits of the savoury fish. Then, on Dretske’s account, if P is the proposition I ate less than 1 pound of salmon, I could know P. On the other hand, let Q be the proposition I ate less than 14 pounds of salmon. Q is not knowable, because were Q to be false, I might still experience the hallucination that I’ve eaten less than a pound of salmon, and therefore I do not have conclusive reasons for Q. (shrink)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants Kripke-Feferman truth. The first, KF+CONS, features a consistent but partial truth predicate. The second, KF+COMP, an inconsistent but complete truth predicate. It is well-known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability under natural assumptions coincides with (...) class='Hi'>definitional equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF+CONS and KF+COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (shrink)

Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was presented in (...) (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (shrink)

Hume's two definitions of causation have caused an extraordinary amount of controversy. The starting point for the controversy is the fact, well known to most philosophy undergraduates, that the two definitions aren't even extensionally equivalent, let alone semantically equivalent. So how can they both be definitions? One response to this problem has been to argue that Hume intends only the first as a genuine definition—an interpretation that delivers a straightforward regularity interpretation of Hume on causation. By many commentators' lights, however, (...) this is a bug rather than a feature: such an account of the two definitions leaves necessary connection out of Hume's story about the meaning of "cause" .. (shrink)

This paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that.

Although widely used across psychology, economics, and philosophy, the concept ofeffort is rarely ever defined. This article argues that the time is ripe to look for anexplicit general definition of effort, makes some proposals about how to arrive at thisdefinition, and suggests that a force-based approach is the most promising. Section 1presents an interdisciplinary overview of some chief research axes on effort, and arguesthat few, if any, general definitions have been proposed so far. Section 2 argues thatsuch a definition is (...) now needed and proposes a basic methodology to arrive at it, whosefirst step is to make explicit the various tacit assumptions about effort made acrosssciences and ordinary thinking. Section 3 unearths 4 different conceptions of effortfrom research on effort so far:primitive-feelings accounts,comparator-based accounts,resource-based accountsandforce-based accounts. It is then argued that the first 2kinds of accounts, although interesting in their own right, are not strictly speaking abouteffort. Section 4 considers the 2 most promising general approaches to efforts: re-source-based and force-based accounts. It argues that these accounts are not only compatible but actually extensionally equivalent. This notwithstanding, it explains why force-based accounts should be regarded as more fundamental than resource-basedaccounts -/- . (shrink)

Many differences exist between the logical relations “equality” and “equivalence”. In this monograph I point out differences that concern definition, linguistics, computational gates and tables, denotation, application, negation of terms, negation of the relation, relations to other relations, the laws of symmetry, transitivity and reflexivity, the laws of commutation and permutation, the law of tautology, the law of distribution, the law of association, propositional meaning, and “genesis”. I also point out a form of “symmetry breaking”: the negation of some (...) equality may yield some non-equality, while the negation of that non-equality might not yield the same equality. (For full access: use the website link of the title page. -- Copyright License: CC BY-NC-ND. -- DOI: 10.17026/dans-zh3-h2b2 --). (shrink)

Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...) proves, without any use of arithmetical induction, that Frege's definition is extensionally correct and so answers the circularity objections. (shrink)

In his Treatise of Human Nature, Hume offers two definitions of ‘cause’. The first is framed in terms of the precedence and contiguity of objects. The second also mentions precedence and contiguity of objects but speaks also of the mind’s tendency on the appearance of the first object to form the idea of the second. Scholars disagree as to which constitutes Hume’s definition of cause properly speaking. Some hold that the ‘constant conjunction of objects’ version is Hume’s real definition, while (...) others opt for the second, psychological version. I argue that Hume’s method of analysis holds that we explicate the meaning of words by tracing ideas back to their antecedent impressions. Since there are two different sets of antecedent impressions, one consisting of preceding and continuous objects and the other of such objects plus psychological determination, we must attribute to Hume the existence of two non-equivalent definitions of the term. (shrink)

Modern Hume scholarship is still divided into two major camps when it comes to the issue of causation. There are those scholars who interpret Hume as a causal anti-realist, and there are those who interpret him as a causal realist. In my paper, I argue that there is an overwhelming amount of evidence – especially textual evidence – that should lead us to read Hume as being a causal anti-realist. That is to say, one who believes that cause and effect (...) relations are felt and known in the mind of the perceiver alone; not in objects themselves. I then attempt to settle, at least partially, the ‘two definitions of cause dispute’ that was brought about by Hume’s giving us not one, but two non-equivalent and non-substitutable definitions of the cause and effect relation. (shrink)

Ordinary and transfinite recursion and induction and ZF set theory are used to construct from a fully interpreted object language and from an extra formula a new language. It is fully interpreted under a suitably defined interpretation. This interpretation is equivalent to the interpretation by meanings of sentences if the object language is so interpreted. The added formula provides a truth predicate for the constructed language. The so obtained theory of truth satisfies the norms presented in Hannes Leitgeb's paper 'What (...) Theories of Truth Should be Like (but Cannot be)'. (shrink)

For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with (...) Level Theory, as developed in Part 1. (shrink)

On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural (...) extension of BLT is definitionally equivalent with ZF. (shrink)

I explore the logic of ground. I first develop a logic of weak ground. This logic strengthens the logic of weak ground presented by Fine in his ‘Guide to Ground.’ This logic, I argue, generates many plausible principles which Fine’s system leaves out. I then derive from this a logic of strict ground. I argue that there is a strong abductive case for adopting this logic. It’s elegant, parsimonious and explanatorily powerful. Yet, so I suggest, adopting it has important consequences. (...) First, it means we should think of ground as a type of identity. Second, it means we should reject much of Fine’s logic of strict ground. I also show how the logic I develop connects to other systems in the literature. It is definitionally equivalent both to Angell’s logic of analytic containment and to Correia’s system G. (shrink)

It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.

The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then elementary particle, (...) neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore re-presentable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries. (shrink)

In this paper, I outline a novel approach to the semantics of natural language pronouns. On this account, which I call 'demonstrativism', pronouns are semantically equivalent to demonstratives. I begin by presenting some contrasts that provide support for demonstrativism. Then I try to explain these contrasts by developing a particular demonstrativist proposal. I build on the "hidden argument" theory of demonstratives. On this theory, demonstratives are semantically similar to definite descriptions, with one important difference: demonstratives take two arguments, rather than (...) one. Using these ideas, I propose that pronouns also take two (covert) arguments, and that the second argument needs to be sufficiently salient to members of the conversation in order for the use of a pronoun to be felicitous. As for the first argument, I maintain that its content is constrained by the process of noun-phrase deletion. Taken together, I argue that these constraints provide us with a satisfying account of the uses to which pronouns are put. (shrink)

This paper addresses a certain objection to the quantificational theory of definite descriptions. According to this objection, the quantificational account cannot provide correct interpretations of definite descriptions embedded in the non-doxastic attitude ascriptions and therefore ought to be rejected. In brief, the objection says that the quantificational theory is committed to the view that a sentence of the form “The F is G” is equivalent to the claim that there is a unique F and it is G, while the ascription (...) such as, e.g., “S wants the F to be G” is not equivalent to the statement that S wants there to be a F and for it to be G. I argue that this objection is invalid as it rests on a false assumption concerning the substitutivity of the relative clauses in the non-doxastic attitude ascriptions. (shrink)

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these (...) accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic. (shrink)

First, ‘Is Justified True Belief Knowledge’ is imprecise but Gettier is explicit that ‘know’ is analysed as the definiendum is ‘S knows that P’. Second, Gettier does not misrepresent as Plato’s definition as the expressions used are ‘Plato considers’ and ‘seems to accept’. Third, Gettier is not mistaken to apply Plato’s definition to propositions since propositional knowledge is a species of Plato’s definition. Fourth, for Plato true belief temporally precedes an account. ‘Jones owns a Ford’ is never a true opinion, (...) hence no account for it can be given. The counterexample is reconstructed with temporality built into it. Fifth, Gettier does not fail to establish the equivalence of ‘believe’, ‘accepts’ and ‘sure’ in the three versions as this is implicitly established in the shifts made in the paper. Sixth, ‘entails’ logically is used only when the entailing proposition is true, but in the counterexamples a false proposition is taken to imply a true one. ‘Entail’ is to be taken in the ordinary sense of implies. Seventh, in Case I, the implication is preserved with the proper representation: → [Gy ↔ ], which implies { → [Gy ↔ ]}’. Eighth, the counterexample is reworked to avoid the objection that justification for p and justification for q may not be sufficient justification for ‘p & q’. (shrink)

In 'The Reference Book' (2012), Hawthorne and Manley observe the following contrast between (1) and (2): -/- (1) In every race John won. (2) In every race, the colt won. -/- The name 'John' in (1) must intuitively refer to the same single individual for each race. However, the description 'the colt' in (2) has a co-varying reading, i.e. a reading where for each race it refers to a different colt. This observation is a prima facie problem for proponents of (...) so-called The-Predicativism which is the view that the name in (1) is really a covert definite description, viz. 'the John'. If the The-Predicativism is correct, (1) and (2) are therefore syntactically equivalent, but this makes it mysterious why only (2) would have a co-varying reading. In a recent paper, Fara (2015) argues that there is a simple and elegant way for proponents of The-Predicativism to explain this contrast. This explanation relies on discerning some subtle syntactic differences between (1) and (2) which in turn are based on assumptions about nominal restrictions a la Stanley and Szabó (2000). In this short paper, I demonstrate that Fara's proposed explanation has a variety of serious shortcomings and hence that the contrast between (1) and (2) remains a significant problem for Predicativism. (shrink)

A strongly deterministic theory of physics is one that permits exactly one possible history of the universe. In the words of Penrose (1989), "it is not just a matter of the future being determined by the past; the entire history of the universe is fixed, according to some precise mathematical scheme, for all time.” Such an extraordinary feature may appear unattainable in a world like ours. In this paper, I show that it can be achieved in a simple way and (...) discuss its implications for metaphysics and philosophy of science, including natural properties, free will, explanation, and modality. First, I propose a precise definition of strong determinism. Next, I discuss its philosophical ramifications and a toy example. Finally, I provide a realistic example of a strongly deterministic (and simple) physical theory---the Everettian Wentaculus. A surprising consequence is that whether or not our world is strongly deterministic may be empirically underdetermined. (shrink)

In his influential article ‘Essence and Modality’, Fine proposes a definition of necessity in terms of the primitive essentialist notion ‘true in virtue of the nature of’. Fine’s proposal is suggestive, but it admits of different interpretations, leaving it unsettled what the precise formulation of an Essentialist definition of necessity should be. In this paper, four different versions of the definition are discussed: a singular, a plural reading, and an existential variant of Fine’s original suggestion and an alternative version proposed (...) by Correia which is not based on Fine’s primitive essentialist notion. The first main point of the paper is that the singular reading is untenable. The second that given plausible background assumptions, the remaining three definitions are extensionally equivalent. The third is that, this equivalence notwithstanding, Essentialists should adopt Correia’s version of the definition, since both the existential variant, which has de facto been adopted as the standard version of the definition in the literature, and the plural reading suffer from problems connected to Fine’s primitive essentialist notion. (shrink)

Supererogatory acts are those that lie “beyond the call of duty.” There are two standard ways to define this idea more precisely. Although the definitions are often seen as equivalent, I argue that they can diverge when options are infinite, or when there are cycles of better options; moreover, each definition is acceptable in only one case. I consider two ways out of this dilemma.

Gödel’s slingshot-argument proceeds from a referential theory of definite descriptions and from the principle of compositionality for reference. It outlines a metasemantic proof of Frege’s thesis that all true sentences refer to the same object—as well as all false ones. Whereas Frege drew from this the conclusion that sentences refer to truth-values, Gödel rejected a referential theory of definite descriptions. By formalising Gödel’s argument, it is possible to reconstruct all premises that are needed for the derivation of Frege’s thesis. For (...) this purpose, a reference-theoretical semantics for a language of first-order predicate logic with identity and referentially treated definite descriptions will be defined. Some of the premises of Gödel’s argument will be proven by such a reference-theoretical semantics, whereas others can only be postulated. For example, the principle that logically equivalent sentences refer to the same object cannot be proven but must be assumed in order to derive Frege’s thesis. However, different true (or false) sentences can refer to different states of affairs if the latter principle is rejected and the other two premises are maintained. This is shown using an identity criterion for states of affairs according to which two states of affairs are identical if and only if they involve the same objects and have the same necessary and sufficient condition for obtaining. (shrink)

Alonzo Church famously provided three principal competing criteria for “strict synonymy,” i.e., sameness of semantic content. These are his Alternatives (0), (1), and (2)—numbered in order of increasing course-grainedness of content. On Alternative (2), expressions are deemed strictly synonymous iff they are logically equivalent. This criterion seems hopeless as an account of the objects of propositional attitude. On Alternative (1), expressions are deemed synonymous iff they are λ-convertible. Alternative (1) also evidently conflicts with discourse about the attitudes. On Alternative (0), (...) expressions are deemed strictly synonymous iff they are “synonymously isomorphic” in Church’s sense. On Alternative (0), semantic content is so fine-grained that not even “Romeo loves Juliet” and “Juliet is loved by Romeo” qualify as strictly synonymous. A fourth alternative, here called “Alternative (3),” deems expressions strictly synonymous iff they are co-intensional, i.e., they have the same semantic extension with respect to the same possible worlds. This criterion’s notion of content is even more course grained than that of Alternative (2). Several objections to Alternative (3) are considered. (This article is contained within the author’s more definitive essay, “Singular Concepts”.). (shrink)

Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,f(7)]. Let B denote the system of equations: {x_j!=x_k: i,k∈{1,...,9}}∪{x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. The system of equations {x_1!=x_1, x_1 \cdot x_1=x_2, x_2!=x_3, x_3!=x_4, x_4!=x_5, x_5!=x_6, x_6!=x_7, x_7!=x_8, x_8!=x_9} has exactly two solutions in positive integers x_1,...,x_9, namely (1,...,1) and (f(1),...,f(9)). No known system S⊆B with a finite (...) number of solutions in positive integers x_1,...,x_9 has a solution (x_1,...,x_9)∈(N\{0})^9 satisfying max(x_1,...,x_9)>f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. Let Λ denote the statement: if the system of equations {x_2!=x_3, x_3!=x_4, x_5!=x_6, x_8!=x_9, x_1 \cdot x_1=x_2, x_3 \cdot x_5=x_6, x_4 \cdot x_8=x_9, x_5 \cdot x_7=x_8} has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. Algorithms always terminate. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known). Assuming that the infiniteness of a set X⊆N is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. *** (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) No known algorithm with no input returns the logical value of the statement card(X)=ω. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies Condition (1) for X=P(n^2+1). The set X={n∈N: the interval [-1,n] contains more than 29.5+\frac{11!}{3n+1}⋅sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. 501893∈X. Condition (1) holds with n=501893. card(X∩[0,501893])=159827. X∩[501894,∞)= {n∈N: the interval [-1,n] contains at least 30 primes of the form k!+1}. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption. (shrink)

What is cognition? Equivalently, what is cognition good for? Or, what is it that would not be but for human cognition? But for human cognition, there would not be science. Based on this kinship between individual cognition and collective science, here we put forward Isbell conjugacy---the adjointness between objective geometry and subjective algebra---as a scientific method for developing cognitive science. We begin with the correspondence between categorical perception and category theory. Next, we show how the Gestalt maxim is subsumed by (...) the mathematical construct of colimit, a generalization of summation. The universal mapping property definitions of mathematical constructs, by virtue of being the best with respect to the universe of discourse, can be learned using reinforcement learning algorithms, which raises the possibility of abstracting the architecture of mathematics by artificial intelligence. Subsequently, we present naturality (to be contrasted with miracles), understood as 'Becoming consistent with Being', which governs the transformations of both things and their theories, as the zeroth law of change. Furthermore, the contrast---physical [mechanism] vs. biological [organism]---is smoothed via natural transformation, wherein transformations are respectful of the cohesion of the objects transformed. In closing, upon recognizing the scientific value of learning difficult-to-master differential calculus by physicists, of learning a strange four-letter language by biologists, and of learning the grammar of our respective mother tongues, we make a case for learning the theory of naturality / category theory for developing cognitive science. (shrink)

Quine argues that if sentences that are set theoretically equivalent are interchangeable salva veritate, then all transparent operators are truth-functional. Criticisms of this argument fail to take into account the conditional character of the conclusion. Quine also argues that, for any person P with minimal logical acuity, if ‘belief’ has a sense in which it is a transparent operator, then, in that sense of the word, P believes everything if P believes anything. The suggestion is made that he intends that (...) result to show us that ‘believes’ has no transparent sense. Criticisms of this argument are either based on unwarranted assertions or on definitions of key terms that depart from Quine’s usage of those terms. (shrink)

The concept of quantum information is introduced as both normed superposition of two orthogonal sub-spaces of the separable complex Hilbert space and in-variance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen. The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing (...) a state of a quantum system) as its value as the bound variable. A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is re-presentable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above. Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and in-variance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper. Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries of the Standard model. (shrink)

An important moral category—dishonest speech—has been overlooked in theoretical ethics despite its importance in legal, political, and everyday social exchanges. Discussion in this area has instead been fixated on a binary debate over the contrast between lying and ‘merely misleading’. Some see lying as a distinctive wrong; others see it as morally equivalent to deliberately omitting relevant truths, falsely insinuating, or any other species of attempted verbal deception. Parties to this debate have missed the relevance to their disagreement of the (...) notion of communicative dishonesty. Communicative dishonesty need not take the form of a lie, yet its wrongness does not reduce to the wrongness of seeking to deceive. This paper therefore proposes a major shift of attention away from the lying/misleading debate and towards the topic of communicative dishonesty. Dishonesty is not a simple notion to define, however. It presupposes a difficult distinction between what is and is not expressed in a given utterance. This differs from the more familiar distinction between what is and is not said, the distinction at the heart of the lying/misleading debate. This paper uses an idea central to speech act theory to characterize dishonesty in terms of the utterer’s communicative intentions, and applies the resulting definition to a variety of contexts. (shrink)

Conspiracy theories should be neither believed nor investigated - that is the conventional wisdom. I argue that it is sometimes permissible both to investigate and to believe. Hence this is a dispute in the ethics of belief. I defend epistemic ‘oughts’ that apply in the first instance to belief-forming strategies that are partly under our control. I argue that the policy of systematically doubting or disbelieving conspiracy theories would be both a political disaster and the epistemic equivalent of self-mutilation, since (...) it leads to the conclusion that history is bunk and the nightly news unbelievable. In fact (of course) the policy is not employed systematically but is only wheeled on to do down theories that the speaker happens to dislike. I develop a deductive argument from hard-to-deny premises that if you are not a ‘conspiracy theorist’ in my anodyne sense of the word then you are an ‘idiot’ in the Greek sense of the word, that is, someone so politically purblind as to have no opinions about either history or public affairs. The conventional wisdom can only be saved (if at all) if ‘conspiracy theory’ is given a slanted definition. I discuss some slanted definitions apparently presupposed by proponents of the conventional wisdom (including, amongst others, Tony Blair) and conclude that even with these definitions the conventional wisdom comes out as deeply unwise. I finish up with a little harmless fun at the expense of David Aaronvitch whose abilities as a rhetorician and a popular historian are not perhaps matched by a corresponding capacity for logical thought. (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. (...) Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

The concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of (...) elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information. (shrink)

The conditional interpretation of general categorical statements like ‘All men are animals’ as universally quantified material conditionals ‘For all x, if x is F, then x is G’ suggests that the logical structure of law statements is conditional rather than categorical. Disregarding the problem that the universally quantified material conditional is trivially true whenever there are no xs that are F, there are some reasons to be sceptical of Frege’s equivalence between categorical and conditional expressions. -/- Now many philosophers (...) will claim that the material conditional interpretation of laws statements, dispositions ascriptions, or any causal claim is generally accepted as wrong and outdated. Still, there seem to be some basic logical assumptions that are shared by most of the participants in the debate on causal matters which at least stems from the traditional truth functional interpretation of conditionals. This is indicated by the vocabulary in the philosophical debate on causation, where one often speaks of ‘counterfactuals’, ‘possible worlds’ and ‘necessity’ without being explicit on whether or to what extent one accepts the logical-technical definition of these notions. To guarantee a non-Humean and non-extensional approach to causal relations, it is therefore important to be aware of the logical and metaphysical implications of the technical vocabulary. -/- In this paper we want to show why extensional logic cannot deal with causal relations. Via a logical analysis of law-like statements ‘All Fs are Gs’ we hope to throw some new light on interrelated notions like causation, laws, induction, hypotheticality and modality. If successful, our analysis should be of relevance for a deeper understanding of any type of causal relations, whether we understand them to be laws, dispositions, singulars or categoricals. (shrink)

The word pro-tasis is etymologically a near equivalent of pre-mise, pro-position, and ante-cedent—all having positional, relational connotations now totally absent in contemporary use of proposition. Taking protasis for premise, Aristotle’s statement (24a16) -/- A protasis is a sentence affirming or denying something of something…. -/- is not a definition of premise—intensionally: the relational feature is absent. Likewise, it is not a general definition of proposition—extensionally: it is too narrow. This paper explores recent literature on these issues.

The monograph is an English, expanded and revised version of the book Cheshko, V. T., Ivanitskaya, L.V., & Glazko, V.I. (2018). Anthropocene. Philosophy of Biotechnology. Moscow, Course. The manuscript was completed by me on November 15, 2019. It is a study devoted to the development of the concept of a stable evolutionary human strategy as a unique phenomenon of global evolution. The name “An Evolutionary Metaphysics (Cheshko, 2012; Glazko et al., 2016). With equal rights, this study could be entitled “Biotechnology (...) as a result and factor of the evolutionary processˮ. The choice in favor of used “The Evolutionary Metaphysics of Human Enhancement Technologiesˮ was made in accordance with the basic principle of modern post-academician and human-sized science, a classic example of which is biotechnology. The “Metaphysics of Evolution” and “Evolutionary Metaphysics” concepts are used in several ways in modern philosophical discourse. In any case, the values contain a logical or associative reference to the teleological nature of the evolutionary process (Hull, 1967, 1989; Apel, 1995; Faye, 2016; Dupre, 2017; Rose, 2018, etc). In our study, the “evolutionary metaphysics” serves to denote the thesis of the rationalization and technologization of global evolution and anthropogenesis, in particular. At the same time, the postulate of an open future remains relevant in relation to the results of the evolutionary process. The theory of evolution of complex, including the humans system and algorithm for its constructing are а synthesis of evolutionary epistemology, philosophical anthropology and concrete scientific empirical basis in modern science. ln other words, natural philosophy is regaining the status bar element theoretical science in the era of technology-driven evolution. The co-evolutionary concept of 3-modal stable evolutionary strategy of Homo sapiens is developed. The concept based оn the principle of evolutionary complementarity of anthropogenesis: value of evolutionary risk and evolutionary path of human evolution are defined bу descriptive (evolutionary efficiency) and creative-teleological (evolutionary correctness) parameters simultaneously, that cannot bе instrumental reduced to others ones. Resulting volume of both parameters define the vectors of blological, social, cultural and techno-rationalistic human evolution Ьу two gear mechanism genetic and cultural co-evolution and techno-humanitarian balance. The resultant each of them сап estimated Ьу the ratio of socio-psychological predispositions of humanization / dehumanization in mentality. Explanatory model and methodology of evaluation of creatively teleological evolutionary risk component of NBIC technological complex is proposed. Integral part of the model is evolutionary semantics (time-varying semantic code, the compliance of the blological, socio-cultural and techno-rationalist adaptive modules of human stable evolutionary strategy). It is seem necessary to make three clarifications. First, logical construct, “evolutionary metaphysics” contains an internal contradiction, because it unites two alternative explanatory models. “Metaphysics”, as a subject, implies deducibility of the process from the initial general abstract principle, and, consequently, the outcome of the development of the object is uniquely determined by the initial conditions. Predicate, “evolutionary”, means stochastic mechanism of realizing the same principle by memorizing and replicating random choices in all variants of the post-Darwin paradigm. In philosophy, random choice corresponds to the category of “free will” of a reasonable agent. In evolutionary theory, the same phenomenon is reflected in the concept of “covariant replication”. Authors will attempt to synthesize both of these models in a single transdisciplinary theoretical framework. Secondly, the interpretation of the term “evolutionary (adaptive) strategyˮ is different from the classical definition. The difference is that the adaptive strategy in this context is equivalent to the survival, i.e. it includes the adaptation to the environment and the transformation (construction) of the medium in accordance with the objectives of survival. To emphasize this difference authors used verbal construction “adaptiveˮ (rather than “evolutionaryˮ) strategy as more adequate. In all other cases, the two terms may be regarded as synonymous. Thirdly, the initial two essays of this series were published in one book in 2012. Their main goal was the development of the logically consistent methodological concept of stable adaptive (evolutionary) strategy of hominines and the argumentation of its heuristic possibilities as a transdisciplinary scientific paradigm of modern anthropology. The task was to demonstrate the possibilities of the SESH concept in describing and explaining the evolutionary prospects for the interaction of social organization and technology (techno-humanitarian balance) and the associated biological and cultural mechanisms of the genesis of religion (gene-cultural co-evolution). In other words, it was related to the sphere of cultural and philosophical anthropology, i.e. to the axiological component of any theoretical constructions describing the behavior of self-organizing systems with human participation. In contrast, the present work is an attempt to introduce this concept into the sphere of biological anthropology and, consequently, its main goal is to demonstrate the possibility of verification of its main provisions by means of procedures developed by natural science, i.e. refers to the descriptive component of the same theoretical constructions. The result of this in the future should be methods for assessing, calculating and predicting the risk of loss of biological and cultural identity of a person, associated with a permanent and continuously deepening process of development of science and technology. (shrink)

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. -/- It is well known from universal algebra that, for every signature Sigma, there exist algebras over Sigma which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of Sigma-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of Sigma-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor U, from the category of Sigma-multialgebras to Set, does not have a left adjoint. -/- In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that U does not have a left adjoint. (shrink)

Fisher criticised the Neyman-Pearson approach to hypothesis testing by arguing that it relies on the assumption of “repeated sampling from the same population.” The present article considers the responses to this criticism provided by Pearson and Neyman. Pearson interpreted alpha levels in relation to imaginary replications of the original test. This interpretation is appropriate when test users are sure that their replications will be equivalent to one another. However, by definition, scientific researchers do not possess sufficient knowledge about the relevant (...) and irrelevant aspects of their tests and populations to be sure that their replications will be equivalent to one another. Pearson also interpreted the alpha level as a personal rule that guides researchers’ behavior during hypothesis testing. However, this interpretation fails to acknowledge that the same researcher may use different alpha levels in different testing situations. Addressing this problem, Neyman proposed that the average alpha level adopted by a particular researcher can be viewed as an indicator of that researcher’s typical Type I error rate. Researchers’ average alpha levels may be informative from a metascientific perspective. However, they are not useful from a scientific perspective. Scientists are more concerned with the error rates of specific tests of specific hypotheses, rather than the error rates of their colleagues. It is concluded that neither Neyman nor Pearson adequately rebutted Fisher’s “repeated sampling” criticism. Fisher’s significance testing approach is briefly considered as an alternative to the Neyman-Pearson approach. (shrink)

JOHN CORCORAN AND WILIAM FRANK. Surprises in logic. Bulletin of Symbolic Logic. 19 253. Some people, not just beginning students, are at first surprised to learn that the proposition “If zero is odd, then zero is not odd” is not self-contradictory. Some people are surprised to find out that there are logically equivalent false universal propositions that have no counterexamples in common, i. e., that no counterexample for one is a counterexample for the other. Some people would be surprised to (...) find out that in normal first-order logic existential import is quite common: some universals “Everything that is S is P” —actually quite a few—imply their corresponding existentials “Something that is S is P”. Anyway, perhaps contrary to its title, this paper is not a cataloging of surprises in logic but rather about the mistakes that did or might have or might still lead people to think that there are no surprises in logic. The paper cataloging of surprises in logic is on our “to-do” list. -/- ► JOHN CORCORAN AND WILIAM FRANK, Surprises in logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] There are many surprises in logic. Peirce gave us a few. Russell gave Frege one. Löwenheim gave Zermelo one. Gödel gave some to Hilbert. Tarski gave us several. When we get a surprise, we are often delighted, puzzled, or skeptical. Sometimes we feel or say “Nice!”, “Wow, I didn’t know that!”, “Is that so?”, or the like. Every surprise belongs to someone. There are no disembodied surprises. Saying there are surprises in logic means that logicians experience surprises doing logic—not that among logical propositions some are intrinsically or objectively “surprising”. The expression “That isn’t surprising” often denigrates logical results. Logicians often aim for surprises. In fact, [1] argues that logic’s potential for surprises helps motivate its study and, indeed, helps justify logic’s existence as a discipline. Besides big surprises that change logicians’ perspectives, the logician’s daily life brings little surprises, e.g. that Gödel’s induction axiom alone implies Robinson’s axiom. Sometimes wild guesses succeed. Sometimes promising ideas fail. Perhaps one of the least surprising things about logic is that it is full of surprises. Against the above is Wittgenstein’s surprising conclusion : “Hence there can never be surprises in logic”. This paper unearths basic mistakes in [2] that might help to explain how Wittgenstein arrived at his false conclusion and why he never caught it. The mistakes include: unawareness that surprise is personal, confusing logicians having certainty with propositions having logical necessity, confusing definitions with criteria, and thinking that facts demonstrate truths. People demonstrate truths using their deductive know-how and their knowledge of facts: facts per se are epistemically inert. [1] JOHN CORCORAN, Hidden consequence and hidden independence. This Bulletin, vol.16, p. 443. [2] LUDWIG WITTGENSTEIN, Tractatus Logico-Philosophicus, Kegan Paul, London, 1921. -/-. (shrink)

How does the critical Kant view ontology? There is no shared scholarly answer to this question. Norbert Hinske sees in the Critique of Pure Reason a “farewell to ontology,” albeit one that took Kant long to bid (Hinske 2009). Karl Ameriks has found evidence in Kant’s metaphysics lectures from the critical period that he “was unwilling to break away fully from traditional ontology” (Ameriks 1992: 272). Gualtiero Lorini argues that a decisive break with the tradition of ontology is essential to (...) Kant’s critical reform of metaphysics, as is reflected in his shift from “ontology” to “transcendental philosophy,” two notions that Lorini takes to be related by mere “analogy” (Lorini 2015). I agree with Lorini that a thorough reform of ontology is a pivotal part of Kant’s critical plan for metaphysics and that ontology somehow “survives within the critical philosophy” (Lorini 2015: 76). To make this case, however, I deem it important to identify “ontology” and “transcendental philosophy” in the sense of extensional equivalence. While we can detect this identification in Kant’s writings, only from his metaphysics lectures can we get a full sense of its historical and philosophical significance. In this chapter I focus on how it represents a definitive turn from as well as notable continuity with traditional treatments of ontology, particularly the Wolffian one. (shrink)

There are two ways to characterize symmetric relations. One is intensional: necessarily, _Rxy_ iff _Ryx_. In some discussions of relations, however, what is important is whether or not a relation gives rise to the same completion of a given type (fact, state of affairs, or proposition) for each of its possible applications to some fixed relata. Kit Fine calls relations that do ‘strictly symmetric’. Is there is a difference between the notions of necessary and strict symmetry that would prevent them (...) from being used interchangeably in such discussions? I show that there is. While the notions coincide assuming an intensional account of relations and their completions, according to which relations/completions are identical if they are necessarily coinstantiated/equivalent, they come apart assuming a hyperintensional account, which individuates relations and completions more finely on the basis of relations’ real definitions. I establish this by identifying two definable relations, each of which is necessarily symmetric but nonetheless results in distinct facts when it applies to the same objects in opposite orders. In each case, I argue that these facts are distinct because they have different grounds. (shrink)

To solve the ship of Theseus puzzle, you need several keys. First of all, we need to fix the application condition of the physical object—it is a vehicle or an object memory; then the identification criterion—the ship was named The Ship of Theseus when he boarded it; and finally the re-application criterion—a ship (or an object memory) when it is the same ship (or object memory) as before. The degree of change from the previous and reference states must be given. (...) The identity of physical objects over time can be expressed by the uniformity of the time slices of the persisting objects. Uniformity exists from the point of view that these instances are temporal cross-sections of the same object; they may differ in other characteristics; there are no general a priori principles independent of practice. The definition of identity is decided by the community of language users. The two states of an object are the same as long as the rate of change of state remains below the threshold value relative to the reference. The two states of an object are similar as long as the change is below a threshold value with respect to each other. Similarity can be described by a tolerance relationship. The mathematical relation describing uniformity is formally an equivalence relation, just as the relation describing identity is an equivalence relation. A physical object is identical to an equivalence class when we speak of the object's identity over time and identical or not identical to an element of the equivalence class when we speak of the object having changed or ceased to exist. (shrink)

The starting point of this paper is an argument to the conclusion that the definition of metaphysical possibility in terms of correct conceivability, conceivability informed by knowledge of relevant essences, found in Rosen (2006) is equivalent to a version of the essentialist definition of metaphysical necessity. This argument appears to show that correct conceivability is a notion of conceivability by name only and is therefore of no interest to epistemologists of modality. In this paper, I present the equivalence argument, (...) explain the idealizing assumptions involved in it and sketch a version of the conceivability approach which weakens these assumptions in order to show that the notion of correct conceivability can still play a specific limited role in the epistemology of modality. (shrink)

Brock and Glasgow recently introduced a new definition of paradox and argue that this conception of paradox itself leads to paradox, the so-called Paradox Paradox. I show that they beg the questions during the course of their argument, but, more importantly, do so in a philosophically interesting way: it reveals a counterexample to the equivalence between being a logical truth and having a probability of one. This has consequences regarding norms of rationality, undermining the grounds for the Conjunction Fallacy.

The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the context (...) of a subobject classifier, there should be a neutral label for the generic subobject, or if the label "true" appears instead, it carries no additional properties or assumptions. It has been shown that a natural isomorphism occurring in one of the definitions of topos need not be an isomorphism of the corresponding algebraic structures on the sets, so that a co-Heyting algebraic structure can be defined on the set of characteristic morphisms, which moreover will also be natural. The analyzes suggest that the notion of co-topos may be considered correct. However, it should be strongly emphasized that although the possibility of defining the notion of co-topos has been shown, its further full consequences should be very carefully examined, which can severely limit its logical applications. (shrink)