Ph.D. dissertation in K.U. Leuven. Summary of Ch. 1 and 6: Contradictorial gradualism vs. discontinuism

1. Fuzziness and the Sorites Paradox
by Marcelo Vásconez
Chapter 1: Introduction
1b Principles of Bivalence. Sensu stricto: T and F jointly exhaustive and mutually exclusive. Rejected.
Loose sense: "p" is true or false. Accepted.
1c Principles of Excluded Middle: Strong: (3) "pw¬p" true, to some degree however small.
Absolute: (4) "H(pw~p)", (5) "Hpw¬p" both totally false.
Weak or simple: (1) "pw~p" at least 50% true, and at most 50% false. (2) "Lpw¬p" completely true.
FUZZINESS
3bi Fuzzy fact: (6) ~(ΔφxwΔ~φx) = (5) ~Δφxv~Δ~φx ± (8) ~Hφxv~¬φx ˆ (1) ~φxv~~φx
Softly indeterminate.
Fuzziness weakly falsifies the simple PEM & vice versa. However, we can have both.
Absolute PEM: (9) ΔφxwΔ~φx = (11) Hφxw¬φx completely incompatible with (6)
Inverse Co-variance of Opposites (ICO): the more φ x is, the less ~φ it is ± › no total indetermination
3bii Borderline case is contradictory: (13) ~φxvφx, from (1) of 3bi + DN.
Border zone instead of border line. Not all borderline cases are equally distanced from the extremes.
3biii Fuzzy property has many borders. x is <φ than y ± x is >-φ than y ± x is -φ.
There is a soft limit whenever there is a relation of inferiority, because of the ICO, and the:
Aristotelian Rule for Comparatives: possession of φ in a > or < degree implies unqualified possession.
Paraconsistent soft borders: there are φ things on both sides of the border.
3biv Fuzziness generates the sorites, but is not responsible for the absurdity.
SORITES
4b Intuitive Major Premise: The loss of a single hair cannot turn a hairy man bald.
Ψ-φ Correspondence Principle: no tiny alteration in Ψ creates a significant change in φ
Major Premise extends the status of a0 to all other ai. Against transition from φ to not-φ.
What is fuzzy diffuses itself.
4d Soritical Series. Core idea: ai and ai+1 almost completely similar with minimal dissimilarity;
subjectively indiscernible. How to capture this logically?
(SP) φai vφai+1w.~φai v~φai+1 ± (CP) ~(φai v~φai+1) = (Par.P.) ~φai wφai+1
The Fairness Principle: Like cases must be treated alike.
(Pre.P) φaieφai+1. Completely false for the last two members, in bounded series.
It does not represent the relation of contiguous members in a soritical series.
We must render the Major Premise in terms of disjunction + weak negation. Rule: DS.
5 Denials of Major Premise. Discontinuism: (DT): ›ai, ai+1(Hφaiv¬φai+1). Sharp boundary
Criticisms: To deny the intuitive major premise is empirically false.
To deny (Ψ-φ C) goes against common sense truisms, backed up by paraconsistency.
DT is arbitrary, against likeness of adjacent members, and unfair: it discriminates indiscriminable cases.
However, no major premise is completely true, for they all are partially false.
There is a soft boundary: ›ai, ai+1(φaiv~φai+1)
DS is invalid for the weak negation, though valid for the strong negation.
6 The Rejection of the Slippery Slope. If the RI (transitivity of the closeness relation) fails, then:
Maximalism: in order for x to be φ, it is necessary that x be absolutely φ.
Only prototypes. What is good? The optimum.
Alethic Maximalism: a sentence is true only if totally true.
Criticisms: 1) Massive impoverishment of reality. Deficient instances eliminated. Fuzziness abolished.
2) No degrees, and no comparatives.
When the property is unbounded, accept the conclusion of the slippery slope: everything is φ.

2. Summary of Ch. 6: “Contradictorial Gradualism vs. Discontinuism”
2. The Soritical Series. Easy cases: two extremes + every pair so very much alike that:
(CP) Continuation Principle: -(Fai v -Fai+1), or
(SP) Similarity Principle: Fai v Fai+1 w. -Fai v -Fai+1.
N.B. Both principles are formulated with weak negation. Invalid for strong negation, ‘¬’.
3. Nature & Cause of the Transition. Our point of departure: occurrence of a soritical transition.
Q1: What is the nature of the transition? Gradual or abrupt?
Q2: Why does the transition happen? What is its condition of possibility?
4. Is the Transition Possible? If (CP) prohibits a dividing line, how the transition is possible?
If there is a transition ± contradiction: a50 is F and not F, by (SP).
Prima facie incompatibility between the soritical series and the transition.
5. Nihilism. There is no transition.
ASSESSMENT. Nihilism offers no positive clarification, no constructive account of Q1, nor of Q2.
6a. Discontinuism and Abrupt Transition. CP is false. The soritical series is impossible. Then,
(DT) Discontinuity Thesis: ›ai (Fai v ¬Fai+1).
There is a sharp cut-off point. a1 bipartitions the series ± there are no proper borderline cases.
Tertium non datur.
Answer to Q1: sudden transition. Punctual. Death is instantaneous. Change would not be continuous.
ASSESSMENT. Unacceptable dualism for its inadmissible consequences.
Change reduced to a precipitous replacement of two stages.
Transitions are contradictory. Example: walking out of the room.
Reductio ad absurdum not valid for weak negation.
6b. ...and the Cause of Change. Answer to Q2: Passage from ai to ai+1 accounts for the transition.
ASSESSMENT. Not every alteration in the underlying dimension G produces changes in F.
There is lack of proportional correspondence between changes in G and changes in F/not F.
Small quantitative changes in G might produce large changes in the supervening F.
Minimal change in G (losing one hair) does not explain drastic change in F (becoming bald).
Why change? There is no principled ground. Point ai is arbitrary. Enigmatic transition.
7. Contradictorial Gradualism (Lorenzo Peña)
7a. Fuzziness = intermediate zone between the extremes –if any– of the soritical series.
Gradual & Contradictory. Not homogeneous: different proportions of F and not F.
7b. Degrees of Properties. 1) Ancients formulated problem in gradual terms. Little by little.
The sophism affects anything having a measure of extent.
2) If all elements in the series were F to the same extent, F would be unceasing.
F will not stop in a non arbitrary way.
If rigidity were not gradual, there would be no stiffening. If there are no degrees ± no gradual change.
Smooth change made possible only by degrees.
7c. Degrees of Truth. (RT*) Redundancy Truth: That ‘a is F’ is true is equivalent to a is F.
But the right member is gradual. Therefore, the left member also, by replacement of equivalents.
(GRT) Generalized Redundancy Truth: That ‘a is F’ is ... true is equivalent to a is ... F.
Degrees of truth designated & antidesignated to reflect intermediate stages in the transition.
7d. Minimalism vs. Maximalism. Maximalism holds the Maximalization Rule:
(MR) “p” is true | “p” is completely true.
It is far too demanding. We would be deprived of intermediate cases. Analogy with utilitarianism.
Where the series is open on one side, œx-Fx. But this is 1/4 true. Its negation, ›xFx, is 1/4 false.
If (MR) is unpalatable, and intermediate positions arbitrary, we opt for the Acquiescence Rule:
(AR) “p” is more or less true | “p” is true.
For minimalism, “p” is true provided that it is not completely false.
7e. From Degrees to Contradictions. Something not totally F is partially not F.
A fuzzy case is to some extent F & to some extent not F. Applying (AR) ± contradiction.
7f. Gradual Transition. F diminishes in the same amount as not F augments.
a50 is a soft limit. (SP) and (CP) preserved: their truth ranges from 0.5 to 0.99 true.
Therefore, there is no discontinuity.
Answer to Q1: gradual transition occurs through intermediate stages, by inverse covariance of opposites.
Answer to Q2: F changes because of proportional change in the parameter G.