Online research in philosophy

We characterize the generalized quantifiers Q which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$ , the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

If $\{{\cal A}(V)\}$ is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and $V_1$ and $V_2$ are spacelike separated spacetime regions, then the system $({\cal A}(V_1),{\cal A}(V_2),\phi)$ is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections $A\in{\cal A}(V_1)$, $B\in{\cal A}(V_2)$ correlated in the normal state $\phi$ there exists a projection $C$ belonging to a von Neumann algebra associated with a spacetime region $V$ contained in the union of the backward light cones of $V_1$ and $V_2$ and disjoint from both $V_1$ and $V_2$, a projection having the properties of a Reichenbachian common cause of the correlation between $A$ and $B$. It is shown that if the net has the local primitive causality property then every local system $({\cal A}(V_1),{\cal A}(V_2),\phi)$ with a locally normal and locally faithful state $\phi$ and open bounded $V_1$ and $V_2$ satisfies the Weak Reichenbach's Common Cause Principle.

We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, φ, built up from propositional variables (p,q,r,...) and falsity $(\perp)$ using conjunction $(\wedge)$ , disjunction (∨) and implication (→). Write $\vdash\phi$ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula φ there exists a formula Apφ (effectively computable from φ), containing only variables not equal to p which occur in φ, and such that for all formulas ψ not involving $p, \vdash \psi \rightarrow A_p\phi$ if and only if $\vdash \psi \rightarrow \phi$ . Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions. An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

Given well ordered countable sets of the form $\Lambda_\phi$ , we consider Borel mappings from $\Lambda_\phi^\omega$ with countable image inside the ordinals. The ordinals and $\Lambda_\phi^\omega$ are respectively equipped with the discrete topology and the product of the discrete topology on $\Lambda_\phi$ . The Steel well-ordering on such mappings is defined by $\phi \leq \psi$ iff there exists a continuous function f such that $\phi(x) \leq \psi \circ f(x)$ holds for any $x \in \Lambda_\phi^\omega$ . It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal α, a mapping T(α) whose rank is precisely α in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by α. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

Łukasiewicz distinguishes three formulations of the principle of contradiction in Aristotle’s works: ontological, logical, and psychological. The first two formulations are equivalent though not synonymous, but neither of them is equivalent to the psychological one, which expresses not a principle but only an empirical law. Furthermore, the principle of contradiction is neither a simple and ultimate law nor is it necessary for conducting an inference, because the syllogism is independent of it. The further explanation of this concept leads Łukasiewicz to formulate the idea of a non-Aristotelian logic, that is, a logic operating without the principle of contradiction. If the principle of contradiction shall be valid, it must be proved. A proof can be supplied only on the basis of a definition of object, as something that cannot have and not have the same property at the same time. However, this definition does not hold for all objects, i.e., for contradictory objects. In virtue of its ontological character the Aristotelian principle of contradiction is then different from that of symbolic logic.

Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following: Let T be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula in T built up from the propositional variables $q,p_{1},...,p_{n}$ , using logical connectives and the predicate Pr, has the same "fixed-points" relative to q (that is, formulas $\psi (p_{1},...,p_{n})$ for which for all $p_{1},...,p_{n}\vdash _{T}\phi (\psi (p_{1},...,p_{n}),p_{1},...,p_{n})\leftrightarrow \psi (p_{1},...,p_{n})$ ) of a formula $\phi ^{\ast}$ of the same kind, obtained from φ in an effective way. Moreover, such $\phi ^{\ast}$ is provably equivalent to the formula obtained from φ substituting with $\phi ^{\ast}$ itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in φ, $\phi ^{\ast}$ is the unique (up to provable equivalence) "fixed-point" of φ. Since this result is proved only assuming Pr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma. All the results are proved more generally in the intuitionistic case.

We show that for any pair $\phi$ and $\psi$ of contradictory formulas of dependence logic there is a formula $\theta$ of the same logic such that $\phi\equiv\theta$ and $\psi\equiv\neg\theta$. This generalizes a result of Burgess.

F. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic HAH: $\forall X\lbrack\forall x(x \in X \vee \neg x \in X) \wedge \forall Y(\forall x(x \in Y \vee \neg x \in Y) \rightarrow \forall x(x \in X \rightarrow x \in Y) \vee \forall x \neg(x \in X \wedge x \in Y)) \rightarrow \exists n\forall x(x \in X \rightarrow x = n)\rbrack$ , and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of HAH in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP. In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that RP is derivable in HAH + CT + MP + ECT0, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau system with labels, signs, and suffixes is defined, extending the basic language $\mathscr{L}_{\mathbf{QB}}$ by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since $\phi \wedge \neg\phi$ does not ``explode'' with arbitrary syntactic consequences.

The goals of this paper are two-fold: I wish to clarify the Aristotelian conception of the law of non-contradiction as a metaphysical rather than a semantic or logical principle, and to defend the truth of the principle in this sense. First I will explain what it in fact means that the law of non-contradiction is a metaphysical principle. The core idea is that the law of non-contradiction is a general principle derived from how things are in the world. For example, there are certain constraints as to what kind of properties an object can have, and especially: some of these properties are mutually exclusive. Given this characterisation, I will advance to examine what kind of challenges the law of non-contradiction faces; the main opponent here is Graham Priest. I will consider these challenges and conclude that they do not threaten the truth of the law of non-contradiction understood as a metaphysical principle.