https://orcid.org/0000-0003-3968-3550
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Abstract
This paper sketches the antilogism of Christine Ladd-Franklin and historical advancement about antilogism, mainly constructs an axiomatic system Atl based on first-order logic with equality and the wholly-exclusion and not-wholly-exclusion relations abstracted from the algebra of Ladd-Franklin, with soundness and completeness of Atl proved, providing a simple and convenient tool on syllogistic reasoning. Atl depicts the empty class and the whole class differently from normal set theories, e.g. ZFC, revealing another perspective on sets and set theories. Two series of Dotterer and the theorem proved by Russinoff are re-characterized in antilogistic formulas corresponding to the square of opposition and twenty-four moods of syllogism. Then the restricted equivalence between two forms of antilogism and minimal inconsistency is built up, implying the internal connection between antilogisms and minimal inconsistency and leaving two related conjectures not proved. Besides, this paper provides a new explanation for the reason why contrariety, subcontrariety, and subalternation and the nine special moods require the non-empty assumption.
Keywords:
Acknowledgement
I would like to thank my supervisor, Professor Xinwen Liu, for his guidance on the topic and his suggestions on modifying this paper.
Notes
1 Below, this paper will use the symbol to represent the not-wholly-exclusion relation instead of the symbol
that Ladd-Franklin used, in order to distinguish the relation from the disjunction. Besides, in Ladd-Franklin's using, these two copulas represent not only the relations between classes but also some relations among propositions, but in this paper, we use the symbols
and
only for the relations between classes. The inconsistency among propositions used in section 5 will be expressed by classical symbols of propositional logic or first-order logic.
2 Here we see the traditional version of the theorem presented by Russinoff. In Section 5.3, we will present and prove another extensional version which is closer to Ladd-Franklin's.
1a For more details on how antilogisms support and imply syllogistic reasoning, see chapter 5 of my master thesis 许方洲 Citation2023 in Chinese.
3 For Ladd-Franklin, the word ‘antilogism’ merely means the inconsistent triad of propositions. While, this paper uses the word ‘antilogism’ to designate some groups of formulas of particular forms as defined below, to prevent the ambiguity between the original ‘antilogism’ and minimally inconsistent groups of atomic propositions.
4 Actually, if T is inconsistent, then is inconsistent, where
is an arbitrary closed term, and t is an arbitrary atomic class symbol in T.