Infinitary Logic

En el presente artículo presentaremos algunos de los aspectos centrales del concepto ricoeuriano de hermenéutica crítica a partir de lo que se desprende de su análisis y discusión con las obras de Althusser y Habermas. En términos generales, mientras que las investigaciones históricas del primero apuntan a poner de manifiesto las estrategias de la reproducción de los sistemas de poder, el segundo propone un proyecto liberador basado en una construcción utópica independiente de la historia. El uso que hacen Althusser y (...) Habermas del psicoanálisis nos acercará tangencialmente al vínculo de Ricoeur con esta disciplina. Concluiremos señalando al concepto de “memoria feliz”, desarrollado en La memoria, la historia, el olvido, como un ejemplo de utopía y muestra de que Ricoeur mantendrá algunos aspectos centrales de esta propuesta hasta el final de su vida. (shrink)

One of the core principles of Paul Ricoeur’s hermeneutics is that interpretation culminates in application, or appropriation. But what exactly is an appropriation, and what makes some appropriations better than others? I try to shed light on these difficult matters by examining Ricoeur’s own appropriation of Alasdair MacIntyre’s notion of the narrative unity of a life, and by contrasting it with Richard Rorty’s appropriation of the same notion. I argue that Ricoeur’s appropriation is more successful than Rorty’s, and that the (...) best explanation of its success is that it respects a distinctive norm that governs the activity of appropriation. I conclude by describing this norm, which I call the principle of ultimate compatibility. (shrink)

Philosophy & Social Criticism, Ahead of Print. Ricoeur interpreted the work of compromise as a creative process to imagine a new world by projecting ourselves into other people. The challenge of compromise is to learn to tell our own story differently within the contours of a broader collective narrative, in compliance with the paradigm of translation. As such, Ricoeur’s political ethics of compromise is at risk of highlighting the element of creation, which refers to the social imagination of a shared (...) vision of a better society, at the cost of recognition of the element of renunciation, which refers to the reciprocal shelving of ideas and desires that the other side considers to be truly intolerable. However, I argue that we can read a delicate connection between forgetting, forgoing and forgiving in Ricoeur’s thought. It is, then, the model of forgiveness, as part of the paradigm of translation, that gives due respect to the importance of renunciation through the emphasis on the unjustifiable, for which renunciation even cannot suffice. The ‘poetics of pardon’ brings creation and renunciation together and, by doing so, it highlights that compromise always remains an object of hope. (shrink)

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.