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

1. General Patterns of Opposition Squares and 2n-gonsKa-fat CHOWThe 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 (1st 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 (2nd Form) – GPSO2 • Given 2 non-trivial distinct propositions s and t such that (a) s  t; (b) they constitute a unilateral entailment: s ut, 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 ux is not unhappy

10. Applications of GPSO2 (ii) • Scope Dominance (studied by Altman, Ben-Avi, Peterzil, Winter): Most boys love no girl uNo 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 ut, 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 (1st Form) – GP2nO1 • Given n (n  3) non-trivial propositions p1, p2 … pn that constitute an n-chotomy (i.e. p1, p2 … pn 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 (2nd Form) – GP2nO2 • Given (n – 1) (n  3) non-trivial distinct propositions s, t1, … tn–2 such that (a) any two of t1, … tn–2 satisfy the subcontrary relation; (b) s  t1 …  tn–2; (c) they constitute (n – 2) co-antecedent unilateral entailments: s ut1 and … s u tn–2, then we have an additional unilateral entailment: s u (s  ~t1 … ~tn–2) and we can construct the following 2nO:

16. GP2nO1  GP2nO2 • Given a 2nO constructed from GP2nO1, then (a) any two of (p1 p3…  pn), … (p1 …  pn–2 pn) satisfy the subcontrary relation • (b) p1  (p1 p3…  pn)  … (p1 …  pn–2 pn); • (c) there are (n – 2) co-antecedent unilateral entailments: p1u(p1 p3…  pn) and … p1u(p1 …  pn–2 pn) • This 2nO also contains an additional unilateral entailment p1u (p1 p2…  pn–1) whose antecedent is p1 and whose consequent has the correct form: p1 p2…  pn–1≡p1 ~(p1 p3…  pn)  … ~(p1 …  pn–2 pn)

17. GP2nO2  GP2nO1 • Given a 2nO constructed from GP2nO2, then s, ~t1 … ~tn–2, (~s  t1  …  tn–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 p1  p4 up1  p2  p4 is missing • The relation between p1  p4and p2  p4 is not among one of the Opposition Relations • We need to generalize the definitions of Opposition Relations

22. Basic Set Relations (BSR) andGeneralized Opposition Relations (GOR) • 15 BSRs • GOR: {<proper subalternation, proper superalternation>, <pre-falsity, post-falsity>, <pre-truth, post-truth>, <anti-subalternation, anti-superalternation>, proper contrariety, proper contradiction, loose relationship, proper subcontrariety}

23. 2n-gon of Opposition (2nO) • Given p1, p2, p3, p4 that constitute a 4-chotomy, we can construct a 24-O based on the GORs

24. Some Statistics of 24O • Can we formulate the GP2nO?