History and Philosophy of Logic 41 (1):1-15 (2019)

Abstract
It has long been recognized that negation in Aristotle’s term logic differs syntactically from negation in classical logic: modern external negation attaches to propositions fully formed, whereas Aristotelian internal negation forms propositions from sentential constituents. Still, modern external negation is used to render Aristotelian internal negation, as may be seen in formalizations of Aristotle’s semantic principles of non-contradiction and of excluded middle. These principles govern the distribution of truth values among pairs of contradictory propositions, and Aristotelian contradictories always consist of an affirmation and a denial. So how should we formalize a false denial? In the literature, we find that a false denial is formalized by means of two negation signs attached to a one-place predicate. However, it can be shown that this rendering leads to an incorrect picture of Aristotle’s principles. In this paper, I propose a solution to this technical problem by devising a formal notation especially for Aristotelian propositions in which internal negation is differentiated from external negation. I will also analyze both principles, each of which has two logically equivalent forms, a positive and a negative one. The fact that Aristotle’s principles are distinct and complementary is reflected in my new formalizations.
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DOI 10.1080/01445340.2019.1683793
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References found in this work BETA

A Natural History of Negation.Jon Barwise & Laurence R. Horn - 1991 - Journal of Symbolic Logic 56 (3):1103.
Aristotle and Łukasiewicz on Existential Import.Stephen Read - 2015 - Journal of the American Philosophical Association 1 (3):535--544.
On the Principle of Contradiction in Aristotle.Jan Lukasiewicz & Vernon Wedin - 1971 - Review of Metaphysics 24 (3):485 - 509.
Aristotle on Non-Contradiction.Paula Gottlieb - 2008 - Stanford Encyclopedia of Philosophy.

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