Karol Wapniarski

We look into how logical diagrams used throughout the history of logic might have influenced the debate on the use of empty terms in logic. To this end, we survey Byzantine diagrams for syllogistic figures, Arabic linear diagrams of Al-Barakt, European linear diagrams of Leibniz and Lambert, Euler circles, Venn, and Venn-Peirce diagrams. The diagrams are analysed with respect to their ability or inability to account for and represent (non-)empty terms. Additionally, we sketch parallels between the evolution of the diagrams and the historical perspectives on empty terms, arguing that an influence, albeit not necessarily conscious, might have been at work. Attention is also drawn to how Venn and post-Venn diagrams explicitly addressed the issue of representing non-emptiness – this is in order to show the influence that the development of modern formal logic has exerted on visually representing syllogistic.

From the seventeenth century, an ideal system of knowledge had to be built on clear axioms and developed "more geometrico", modelled on Euclid's Elements. Yet only decades earlier, the Elements themselves were debated regarding their ability to provide unquestionable knowledge. In the sixteenth century, Aristotelian doctrine with syllogistic logic remained the model of science. As doubts arose about mathematical certainty, attempts were made to formalise the Elements, reducing them to Aristotelian syllogisms. This paper presents such attempts together with a syllogistic formalisation of Proposition I.5 and argues for the crucial importance of "syllogising Euclid" for history of mathematics and logic alike.

The purpose of the paper is to answer the question of what additional existential premises are needed in order to render Aristotelian syllogisms provable in First-Order Logic and to give an overview of how the issue of ontological commitments in syllogistic was handled throughout the history. In contemporary discussions concerning the history of logic, there is a widespread assumption that the Aristotelian syllogistic, as it is the case with the modern formal logic, did not allow for the use of empty terms. The issue is however much more complex, and ontological commitments of syllogistic were discussed extensively throughout the history. In the paper, we relate these discussions, covering four main areas: the logic of Aristotle, Arabic logic, medieval European logic, and later discussions up till the emergence of modern formal logic, as well as provide our own view of the issue.

The history of the search for a minimal set of inference rules for syllogistic reasoning starts with Aristotle, with one of the metatheorems proved in Prior Analytics stating that all syllogistic deductions are ultimately reducible to two universal moods in the first figure: Barbara and Celarent. In the present work, we will consider what is the greatest reduction one can achieve when proving all the moods indirectly. The question we shall answer is whether, when proved indirectly, syllogisms behave the same way and, consequently, whether the Aristotelian reduction to Barbara and Celarent holds for indirect proofs as well. To achieve this, we shall consider four different scenarios of applying single-premise inference rules to syllogism. The results for each scenario will then be analyzed and displayed as tables. Considering the tables, we shall claim that in certain cases, the Aristotelian reduction does not hold and further propose our division of all the syllogistic moods, which will be different from the usual figure arrangement. Finally, we will analyze the results quantitatively and show that they can provide insight into the nature of the division to regular and the so-called subalternated moods.