Paolo Maffezioli

In this note I focus on a controversial passage of APr. II 22 where Aristotle allegedly claimed that two terms A and B convert without the latter being predicated of the former. The passage is controversial given Aristotle’s endorsement in APr. II 5 of quite the opposite view, namely that converting terms are predicated of each other. I provide an interpretation of the final clause πλὴν αὐτοῦ τοῦ A at 68a21 which is consistent with the idea that B is predicated of A. Then, I attempt a reconstruction of the argument in order to show that Aristotle was indeed committed to the claim that B is predicated of A, hence the two terms convert precisely in the sense of APr. II 5. Such a reconstruction, unlike the one provided by the consensus view, does not presuppose at all the orthodox dictum (in fact, it is arguably neutral with respect to which semantics for assertoric syllogistic is adopted). Besides being closer to Aristotle’s text, the reconstruction has the advantage that it can be endorsed even by those who do accept the heterodox interpretation of the dictum.

In this note I focus on a controversial passage of APr. II 22 where Aristotle allegedly claimed that two terms A and B convert without the latter being predicated of the former. The passage is controversial given Aristotle’s endorsement in APr. II 5 of quite the opposite view, namely that converting terms are predicated of each other. I provide an interpretation of the final clause πλὴν αὐτοῦ τοῦ A at 68a21 which is consistent with the idea that B is predicated of A. Then, I attempt a reconstruction of the argument in order to show that Aristotle was indeed committed to the claim that B is predicated of A, hence the two terms convert precisely in the sense of APr. II 5. Such a reconstruction, unlike the one provided by the consensus view, does not presuppose at all the orthodox dictum (in fact, it is arguably neutral with respect to which semantics for assertoric syllogistic is adopted). Besides being closer to Aristotle’s text, the reconstruction has the advantage that it can be endorsed even by those who do accept the heterodox interpretation of the dictum.

I focus on the commonly shared view that Hume’s monetary theory is inconsistent. I review several attempts to solve the alleged inconsistency in Hume’s monetary theory, including the consensus interpretation according to which Hume was committed to the neutrality of money only in the long run, while he conceded that money can be non-neutral in the short run. Then, building on a monetary version of the logical fallacy of monotonic counterfactuals in the essay Of the Balance of Trade, I argue for the consistency of Hume’s theory of money by ascribing to Hume a distinction between money as collectively neutral and distributively non-neutral.

In this paper we show how to extend the standard cut-elimination procedure from first-order intuitionistic stable logic to a class of intuitionistic stable theories. Building on previous works by Negri and von Plato, we aptly modify the underlying calculus for first-order intuitionistic logic so as to preserve the admissibility of all the structural rules, including cut, in the presence of a restricted version of the rule of classical reductio ad absurdum and of a special case of universal rules.

The paper provides a cut-free sequent calculus for the theory of tight apartness in a language where both apartness and equality are primitive notions. The result is obtained by aptly modifying the underlying logical calculus for intuitionistic logic and adding rules of inference corresponding to the axioms of apartness and the principles governing the mutual deductive relationships between apartness and equality. While the rules for apartness are found directly from the axioms by applying standard proof-theoretic methods, the others, especially the rule corresponding to the principle of tight apartness, are found by exploiting the logical law of _consequentia mirabilis_. Along the way, we also provide a cut-free sequent calculus for the theory of weak tight apartness, also known as the theory of negated equality, thus answering in the positive to an open problem in the existing literature on the subject.

I provide a mereological analysis of Zeno of Sidon’s objection that in Euclid’s Elements we need to supplement the principle that there are no common segments of straight lines and circumferences. The objection is based on the claim that such a principle is presupposed in the proof that the diameter cuts the circle in half. Against Zeno, Posidonius attempts to prove the bisection of the circle without resorting to Zeno’s principle. I show that Posidonius’ proof is flawed as it fails to account for the case in which one of the two circumferences cut by the diameter is a proper part of the other. When such a case is considered, then either the bisection of the circle is false or it presupposes Zeno’s principle, as claimed by Zeno.

The article investigates what happens when philosophy meets and begins to establish connections with two formal research methods such as game theory and network science. We use citation analysis to identify, among the articles published in Synthese and Philosophy of Science between 1985 and 2021, those that cite the specialistic literature in game theory and network science. Then, we investigate the structure of the two corpora thus identified by bibliographic coupling and divide them into clusters of related papers by automatic community detection. Lastly, we construct by the same bibliometric techniques a reference map of philosophy, on which we overlay our corpora to map the diffusion of game theory and network science in the various sub-areas of recent philosophy. Three main results derive from this study. Philosophers are interested not only in using and investigating game theory as a formal method belonging to applied mathematics and sharing many relevant features with social choice theory, but also in considering its applications in more empirically oriented disciplines such as social psychology, cognitive science, or biology. Philosophers focus on networks in two research contexts and in two different ways: in the debate on causality and scientific explanation, they consider the results of network science; in social epistemology, they employ network science as a formal tool. In the reference map, logic—whose use in philosophy dates back to a much earlier period—is distributed in a more uniform way than recently encountered disciplines such as game theory and network science. We conclude by discussing some methodological limitations of our bibliometric approach, especially with reference to the problem of field delineation.

I focus on the commonly shared view that Hume’s monetary theory is inconsistent. I review several attempts to solve the alleged inconsistency in Hume’s monetary theory, including the consensus interpretation according to which Hume was committed to the neutrality of money only in the long run, while he conceded that money can be non-neutral in the short run. Then, building on a monetary version of the logical fallacy of monotonic counterfactuals in the essay Of the Balance of Trade, I argue for the consistency of Hume’s theory of money by ascribing to Hume a distinction between money as collectively neutral and distributively non-neutral.

Using quantitative methods, we investigate the role of logic in analytic philosophy from 1941 to 2010. In particular, a corpus of five journals publishing analytic philosophy is assessed and evaluated against three main criteria: the presence of logic, its role and level of technical sophistication. The analysis reveals that logic is not present at all in nearly three-quarters of the corpus, the instrumental role of logic prevails over the non-instrumental ones, and the level of technical sophistication increases in time, although it remains relatively low. These results are used to challenge the view, widespread among analytic philosophers and labeled here “prevailing view”, that logic is a widely used and highly sophisticated method to analyze philosophical problems.

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and overcome the failure of interpolation for the implication-free fragment.

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

We investigate in intuitionistic first-order logic various principles of preference relations alternative to the standard ones based on the transitivity and completeness of weak preference. In particular, we suggest two ways in which completeness can be formulated while remaining faithful to the spirit of constructive reasoning, and we prove that the cotransitivity of the strict preference relation is a valid intuitionistic alternative to the transitivity of weak preference. Along the way, we also show that the acyclicity axiom is not finitely axiomatizable in first-order logic.

A combination of epistemic logic and dynamic logic of programs is presented. Although rich enough to formalize some simple game-theoretic scenarios, its axiomatization is problematic as it leads to the paradoxical conclusion that agents are omniscient. A cut-free labelled Gentzen-style proof system is then introduced where knowledge and action, as well as their combinations, are formulated as rules of inference, rather than axioms. This provides a logical framework for reasoning about games in a modular and systematic way, and to give a step-by-step reconstruction of agents omniscience. In particular, its semantic assumptions are made explicit and a possible solution can be found in weakening the properties of the knowledge operator.

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism.