I come not to clarify Aristotle’s defence of the principle of non-contradiction, but to put it in its proper context. I argue that remarks in Metaphysics IV.3 together with the argument of IV.4, 1006a11-31 show that Aristotle practises Plato’s method of dialectic in his defence of PNC. I mean this in the strong sense that he uses the very methodology described in the middle books of the Republic and, I claim, illustrated in such dialogues as Parmenides, Sophist and Theaetetus.

This paper focuses on Aristotle’s discussion of PNC in Metaphysics Gamma and argues that the argument operates at three different levels: ontological, doxastic, and semantic through the invocation of three philosophical personae: the first one can only state what is otherwise unprovable, the second one can only confirm that we should trust PNC, the third one denies PNC and must be silenced. Aristotle cannot prove what is beyond proof. This situation results in a fundamental ambiguity in the figure of the (...) philosopher. The Metaphysics is written from the standpoint of an investigative thinker who admits her puzzlement before a question that will forever remain open and imagines another philosopher who has achieved a god-like insight into the first principles of all things. The path from the first figure to the second one, however, remains an enigmatic leap. (shrink)

The purpose of this dissertation is to provide a defence of Aristotle's principle of contradiction against the critique made on it by Jan Lukasiewicz in an article he wrote in 1910 which was translated and published in the March 1971 number of The Review of Metaphysics. Lukasiewicz maintains in general that the law of contradiction has no logical worth. Specifically, he charges Aristotle with having several laws of contradiction instead of one as Aristotle claims; with attempting to prove the law (...) despite his claim that this is impossible and finally with failing in the very attempt to prove the law, or at least one of its formulations. In 20 chapters, each dealing with a respective section of Lukasiewicz's article, I attempt to show that all of Lukasiewicz's allegations are unfounded. My methodology is simply to follow Lukasiewicz through his 20 sections and compare what he says with what Aristotle actually wrote. If successful, this dissertation will show, in basic agreement with Professor Joseph Owens, that the best way to read Aristotle is on his own terms and not, as in the case of Lukasiewicz, via the latest developments in symbolic logic. I reiterate, with Aristotle, that the law of contradiction is the basic principle of being qua being and has logical worth precisely because of that fact. (shrink)

This paper is dedicated to the memory of Mike Dunn. His untimely death is a loss not only to logic, computer science, and philosophy, but to all of us who knew and loved him. The paper gives an argument for closure under γ in standard systems of relevance logic (first proved by Meyer and Dunn 1969). For definiteness, I chose the example of R. The proof also applies to E and to the quantified systems RQ and EQ. The argument uses (...) semantic tableaux (with one exceptional rule not satisfying the subformula property). It avoids the previous arguments’ use of cutting down inconsistent sets of formulas to consistent sets. Like all tableau arguments, it extends partial valuations to total valuations. (shrink)

In this contribution, I explore the treatment that Plato devotes to Protagoras’ relativism in the first section of the Theaetetus (151 E 1–186 E 12) where, among other things, the definition that knowledge is perception is put under scrutiny. What I aim to do is to understand the subtlety of Plato’s argument about Protagorean relativism and, at the same time, to assess its philosophical significance by revealing the inextric¬ability of ontological and epistemological aspects on which it is built (for this (...) latter aspect, I refer to contemporary discussions of relativism, mainly to Margolis’ robust relativism). I then turn to Aristotle’s treatment of Protagoras’ relativism in Metaphysics Γ, sections 5 and 6, in order to show that Plato and Aristotle surprisingly share the same view as regards the philosophical content of Protagoras’ relativism (in doing so, I take position against the standard opinion among scholars that Plato and Aristotle understand Protagoras’ relativism in different, even incompatible, ways). What I ultimately aim to demonstrate is that Protagoras’ relativism, as understood by both Plato and Aristotle, is a coherent, even attractive, philosophical position. (shrink)

Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We classify the closed permutation subgroups containing the group $\operatorname {Aut}(\Gamma)^{\ast}$ , where $\operatorname {Aut}(\Gamma)^{\ast}$ is the group of all isomorphisms and anti-isomorphisms of $\Gamma$ preserving the two sides. Our results rely on a combinatorial theorem of Nešetřil and Rödl and a strong (...) finite submodel property for $\Gamma$. (shrink)

El objetivo de este trabajo es proponer un acercamiento metafísico al problema del método filosófico como proceso de reflexión y, con ello, enunciar que toda investigación filosófica se sustenta en la noción de relación como concepto metafísico absoluto. La tesis que sostengo señala, primero, que la relación es la expresión ontológica fundamental, y segundo, la noción de porción es el concepto mínimo de sentido en toda multiplicidad ontológica y todo criterio gnoseológico. Al asumirlas como premisas deduzco de ellas la proporción (...) (analogía) que hace posible establecer un vínculo congruente entre un mundo ontológico α y un criterio gnoseológico β apoyado en una lógica de magmas, indispensable para acudir a que llamo lógica gamma. (shrink)

Some of Aristotle’s statements about the indemonstrability of the Principle of Non-Contradiction (PNC) in Metaphysics Γ 4 merit more attention. The consensus seems to be that Aristotle provides two arguments against the demonstrability of the PNC, with one located in Γ 3 and the other found in the first paragraph of Γ 4. In this article, I argue that Aristotle also relies upon a third argument for the same conclusion: the argument from truth. Although Aristotle does not explicitly state this (...) argument, it is the best argument that he could use to defend some of his statements in the second paragraph of Γ 4. Since the argument relies on only a few of Aristotle’s core views about truth, I propose that it is faithful to his considered position throughout his corpus, and it may be the strongest argument he could offer for the indemonstrability of the PNC. (shrink)

Given an ideal $I$ , let $\mathbb{P}_{I}$ denote the forcing with $I$ -positive sets. We consider models of forcing axioms $MA(\Gamma)$ which also have a normal ideal $I$ with completeness $\omega_{2}$ such that $\mathbb{P}_{I}\in \Gamma$ . Using a bit more than a superhuge cardinal, we produce a model of PFA (proper forcing axiom) which has many ideals on $\omega_{2}$ whose associated forcings are proper; a similar phenomenon is also observed in the standard model of $MA^{+\omega_{1}}(\sigma\mbox{-closed})$ obtained from a supercompact cardinal. (...) Our model of PFA also exhibits weaker versions of ideal properties, which were shown by Foreman and Magidor to be inconsistent with PFA. Along the way, we also show (1) the diagonal reflection principle for internally club sets ( $\mathit{DRP}(IC_{\omega_{1}})$ ) introduced by the author in earlier work is equivalent to a natural weakening of “there is an ideal $I$ such that $\mathbb{P}_{I}$ is proper”; and (2) for many natural classes $\Gamma$ of posets, $MA^{+\omega_{1}}(\Gamma)$ is equivalent to an apparently stronger version which we call $MA^{+\operatorname{Diag}}(\Gamma)$. (shrink)