1. General Patterns of Opposition Squares and 2n-gons Ka-fat CHOW The Hong Kong Polytechnic University

2. General Remarks Definitions of Opposition Relations: Subalternate: Unilateral entailment Contrary: Mutually exclusive but not collectively exhaustive Subcontrary: Collectively exhaustive but not mutually exclusive Contradictory: Both mutually exclusive and collectively exhaustive Do not consider inner / outer negations, duality Adopt a graph-theoretic rather than geometrical view on the logical figures which will be represented as 2- dimensional labeled multidigraphs

3. General Pattern of Squares of Opposition (1 st Form) – GPSO1 Given 3 non-trivial propositions p, q and r that constitute a trichotomy (i.e. p, q, r are pairwise mutually exclusive and collectively exhaustive), we can construct the following square of opposition (SO):

4. General Pattern of Squares of Opposition (2 nd Form) – GPSO2 Given 2 non-trivial distinct propositions s and t such that (a) s t; (b) they constitute a unilateral entailment: s u t, we can construct the following SO:

5. GPSO1 GPSO2 Given a SO constructed from GPSO1, then we have a unilateral entailment: p u (p q) such that p (p q).

6. GPSO2 GPSO1 Given a SO constructed from GPSO2, then s, ~t and (~s t) constitute a trichotomy.

7. Applications of GPSO1 (i) Let 50 < n < 100. Then [0, 100 – n), [100 – n, n] and (n, 100] is a tripartition of [0, 100] NB: Less than (100 – n)% of S is P More than n% is not P; At most n% of S is P At least (100 – n)% of S is not P

8. Applications of GPSO1 (ii) In the pre-1789 French Estates General, clergyman, nobleman, commoner constitute a trichotomy NB: clergyman nobleman = privileged class; commoner nobleman = secular class

9. Applications of GPSO2 (i) Semiotic Square: given a pair of contrary concepts, eg. happy and unhappy, x is happy u x is not unhappy

10. Applications of GPSO2 (ii) Scope Dominance (studied by Altman, Ben-Avi, Peterzil, Winter): Most boys love no girl u No girl is loved by most boys

11. Asymmetry of GPSO1 While each of p and r appears as independent propositions in the two upper corners, q only appears as parts of two disjunctions in the lower corners.

12. Hexagon of Opposition (6O): Generalizing GPSO1 6 propositions: p, q, r, (p q), (r q), (p r)

13. Hexagon of Opposition: Generalizing GPSO2 Apart from the original unilateral entailment, s u t, there is an additional unilateral entailment, s u (s ~t) 6 propositions: s, t, (s ~t), ~s, ~t, (~s t)

14. General Pattern of 2n-gons of Opposition (1 st Form) – GP2nO1 Given n (n 3) non-trivial propositions p 1, p 2 … p n that constitute an n-chotomy (i.e. p 1, p 2 … p n are collectively exhaustive and pairwise mutually exclusive), we can construct the following 2n-gon of opposition (2nO):

15. General Pattern of 2n-gons of Opposition (2 nd Form) – GP2nO2 Given (n – 1) (n 3) non-trivial distinct propositions s, t 1, … t n–2 such that (a) any two of t 1, … t n–2 satisfy the subcontrary relation; (b) s t 1 … t n–2 ; (c) they constitute (n – 2) co-antecedent unilateral entailments: s u t 1 and … s u t n–2, then we have an additional unilateral entailment: s u (s ~t 1 … ~t n–2 ) and we can construct the following 2nO:

16. GP2nO1 GP2nO2 Given a 2nO constructed from GP2nO1, then (a) any two of (p 1 p 3 … p n ), … (p 1 … p n–2 p n ) satisfy the subcontrary relation (b) p 1 (p 1 p 3 … p n ) … (p 1 … p n–2 p n ); (c) there are (n – 2) co-antecedent unilateral entailments: p 1 u (p 1 p 3 … p n ) and … p 1 u (p 1 … p n–2 p n ) This 2nO also contains an additional unilateral entailment p 1 u (p 1 p 2 … p n–1 ) whose antecedent is p 1 and whose consequent has the correct form: p 1 p 2 … p n–1 p 1 ~(p 1 p 3 … p n ) … ~(p 1 … p n–2 p n )

17. GP2nO2 GP2nO1 Given a 2nO constructed from GP2nO2, then s, ~t 1 … ~t n–2, (~s t 1 … t n–2 ) constitute an n-chotomy.

18. The Notion of Perfection A 2nO is perfect if the disjunction of all upper-row propositions the disjunction of all lower-row propositions T; otherwise it is imperfect A 2mO (m < n and m 2) which is a proper subpart of a perfect 2nO is imperfect Any SO (i.e. 4O) must be imperfect An imperfect 2mO may be perfected at different fine- grainedness by combining or splitting concepts

19. Perfection of an Imperfect 2nO (i)

20. Perfection of an Imperfect 2nO (ii)

21. 2nO is not comprehensive enough The relation p 1 p 4 u p 1 p 2 p 4 is missing The relation between p 1 p 4 and p 2 p 4 is not among one of the Opposition Relations We need to generalize the definitions of Opposition Relations

22. Basic Set Relations (BSR) and Generalized Opposition Relations (GOR) 15 BSRs GOR: {,,,, proper contrariety, proper contradiction, loose relationship, proper subcontrariety}

23. 2 n -gon of Opposition (2 n O) Given p 1, p 2, p 3, p 4 that constitute a 4-chotomy, we can construct a 2 4 - O based on the GORs

24. Some Statistics of 2 4 O Can we formulate the GP2 n O? GORNumber of Instances 36 14 14 1 proper contrariety18 proper contradiction7 loose relationship12 proper subcontrariety18