Infinitary Logic

The article approaches cut elimination from a new angle. On the basis of an arbitrary inference relation among logically atomic formulae, an inference relation on a language possessing logical operators is defined by means of inductive clauses similar to the operator-introducing rules of a cut-free intuitionistic sequent calculus. The logical terminology of the richer language is not uniquely specified, but assumed to satisfy certain conditions of a general nature, allowing for, but not requiring, the existence of infinite conjunctions and disjunctions. (...) We investigate to what extent structural properties of the given atomic relation are preserved through the extension to the full language. While closure under the Cut rule narrowly construed is not in general thus preserved, two properties jointly amounting to closure under the ordinary structural rules, including Cut, are. -/- Attention is then narrowed down to the special case of a standard first-order language, where a similar result is obtained also for closure under substitution of terms for individual parameters. Taken together, the three preservation results imply the familiar cut-elimination theorem for pure first-order intuitionistic sequent calculus. -/- In the interest of conceptual economy, all deducibility relations are specified purely inductively, rather than in terms of the existence of formal proofs of any kind. (shrink)

It is common for contemporary metaphysical realists to adopt Quine's criterion of ontological commitment while at the same time repudiating his ontological pragmatism. 2 Drawing heavily from the work of others—especially Joseph Melia and Stephen Yablo—I will argue that the resulting approach to meta-ontology is unstable. In particular, if we are metaphysical realists, we need not accept ontological commitment to whatever is quantified over by our best first-order theories.

Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via "Gödel numbering") and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning "languages" some of whose formulas would be naturally identified as infinite sets . A "language" of this kind is called (...) an infinitary language : in this article I discuss those infinitary languages which can be obtained in a straightforward manner from first-order languages by allowing conjunctions, disjunctions and, possibly, quantifier sequences, to be of infinite length. In the course of the discussion it will be seen that, while the expressive power of such languages far exceeds that of their finitary (first-order) counterparts, very few of them possess the "attractive" features (e.g., compactness and completeness) of the latter. Accordingly, the infinitary languages that do in fact possess these features merit special attention. (shrink)

The discussion of supervenience is replete with the use of in?nitary logical operations. For instance, one may often ?nd a supervenient property that corresponds to an in?nite collection of supervenience-base properties, and then ask about the in?nite disjunction of all those base properties. This is crucial to a well-known argument of Kim (1984) that supervenience comes nearer to reduction than many non-reductive physicalists suppose. It also appears in recent discussions such as Jackson (1998).

Let K 0 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ is disjoint from a club, and let K 1 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ contains a club. We prove that if $\lambda = \lambda^{ is regular, then no sentence of L λ+κ separates K 0 λ and K 1 λ . On the other hand, we prove that if $\lambda = \mu^+,\mu = \mu^{ , and a forcing axiom (...) holds (and ℵ L 1 = ℵ 1 if μ = ℵ 0 ), then there is a sentence of L λλ which separates K 0 λ and K 1 λ. (shrink)

Let L ω ∞ω be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. It is shown that if p(n) ≪ n -1 satisfies certain simple conditions on its growth rate, then for every σ∈ L ω ∞ω , the probability that σ holds for the (...) random graph on n vertices converges. In fact, if $p(n) = n^{-\alpha}, \alpha > 1$ , then the probability is either smaller than 2 -n d for some $d > 0$ , or it is asymptotic to cn -d for some $c > 0, d \geq 0$ . Results on the difficulty of computing the asymptotic probability are given. (shrink)

Proceeding of the work in trivalent logic developped under the direction of the professor Hugo Padilla Chacón at the 90's at the National Autonome University of México. Program in RLisp.

A logic with normal modal operators and countable infinite conjunctions and disjunctions is introduced. A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories. It is also shown that the logic has the interpolation property.