General Patterns of General Patterns of Opposition Squares Opposition Squares

and 2n-gonsand 2n-gons

Ka-fat CHOWKa-fat CHOWThe Hong Kong Polytechnic UniversityThe Hong Kong Polytechnic University

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

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):

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 u t, we can construct the following SO:

GPSO1 GPSO2

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

GPSO2 GPSO1

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

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

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

Applications of GPSO2 (i)

Semiotic Square: given a pair of contrary concepts, eg. happy and unhappy, x is happy u x is not unhappy

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

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.

Hexagon of Opposition (6O): Generalizing GPSO1

6 propositions: p, q, r, (p q), (r q), (p r)

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)

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):

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 u t1 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:

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)

GP2nO2 GP2nO1

Given a 2nO constructed from GP2nO2, then s, ~t1 … ~tn–2, (~s t1 … tn–2) constitute an n-chotomy.

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

Perfection of an Imperfect 2nO (i)

Perfection of an Imperfect 2nO (ii)

2nO is not comprehensive enough

The relation p1 p4 u p1 p2 p4 is missing The relation between p1 p4 and p2 p4 is not among

one of the Opposition Relations We need to generalize the definitions of Opposition

Relations

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}

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

Some Statistics of 24O

Can we formulate the GP2nO?

GOR Number of Instances

<proper subalternation, proper superalternation> 36

<pre-falsity, post-falsity> 14

<pre-truth, post-truth> 14

<anti-subalternation, anti-superalternation> 1

proper contrariety 18

proper contradiction 7

loose relationship 12

proper subcontrariety 18