2nd International Colloquium on Colous and Numbers Fortaleza https://sites.google.com/site/iccn2015/ March 26, 2015

On a philosophical motivation for mutilating truth tables Marcos Silva [email protected]

Outline: O. Three preliminary remarks I. A image of Logic or how colors challenge this image of logic. II. Colors as a logical problem! III. Systematic mutilation of truth tables (and rules)

IV.Two forms of mutilation?

Three preliminary remarks (a methodological, a philosophical and a logical remark!)

1) Methodological: Wittgenstein (1918 e 1929) and internal approach!

2) Philosophical: What is the nature of negation? Which is its logical behavior? What does it mean to negate p? •Suppression, extraction, abstraction, delimitation, privation, rejection/refusal, addition, determination etc.

•Absurd (inconsistency): (contradictionXcontrarieties)

It

hides

•Exclusion. To negate means to exclude!

what

an

inconsistency

is

a) Inversion of truth conditions (sense). To negate p means to contradict p. (5.513 and 5.1241). Bestehen oder nicht bestehen (bipolarity), artikuliert oder nicht artikuliert. Konfiguriert oder nicht konfiguriert. Dual structures or dichotomies. Roughly, just one negative proposition in relation to p.

The contradiction! b) Primitive Incompatibility (Material exclusions). To deny p means to relate it with some contrariety. Roughly, we may have several (organized in systems), maybe infinite alternatives. At least, more then two.

The contrariety?!!?!? Which is the contrary of “this is blue”, “this is three 3 long”, “This is a dog”? Problem: Do we have to think of a exclusion without negation?

3) Logical: If the logical organization of colors represents a problem for the tractarian logic, it should represent a problem for tractarian notation too. “the top line must disappear” “certain combinations of the T´s and F´s must be left out” (SRLF, p.170-1) “Wegfall der ersten Linie” (WA I, p. 58) “eine Reihe einfach durchstreichen, d.h. als unmöglich betrachten” (id.ib) “ich muss die ganze obere Reihe durchstreichen” (id.ib) “die ganze Linie ausstreichen” (id. p.59) “die obere Linie streichen” (id.ib.)

Let‘s pin down Wittgenstein‘s logical problem at this time (1929) and other concerns: “The mutual exclusion of unanalyzable statements of degree contradicts an opinion which was published by me several years ago and which necessitated that atomic propositions could not exclude one another. I here deliberately say “exclude" and not "contradict", for there is a difference between these two notions, and atomic propositions, although they cannot contradict, may exclude one another. (…) Our symbolism, which allows us to form the sign of the logical product of "R P T" and "B P T" gives here no correct picture of reality”. (SRLF, p.168-9)

Note! 1) He did recognize that the notion of exclusion should be broader than the one of contradiction! Both should not be taken as the same. Moreover 2) he also criticizes his symbolism!

Part I A image of Logic or how colors challenge this image of logic

(Wittgenstein 1918)

I.

What is logic in the tractarian period ?

(Neutral) Formal logic, truth-functionality, extensionality, tautology and contradiction (based on the notion of a unique and absolute logical space) 5.551 (attractive image of logic!) “Our fundamental principle is that every question which can be decided at all by logic can be decided without further trouble. (And if we get into a situation where we need to answer such a problem by looking at the world, this shows that we are on a fundamentally wrong track.)” See Wittgenstein´s letter to Russell from Norway, 1913 (logic = tautology!) My point: The logical organization of colors taught the young Wittgenstein that he had to start looking at the world in order to do logic! This represents the first step outside of his formal tractatarian logic. Even if we accept the (romantic and highly misleading) idea of full analysis, Wittgenstein’s tractarian (extensional) operators will not do the work. This represents the recognition of a dramatic limitation. 8

Part II Colors as a logical problem!

Colors as a logical problem! (the form of a contradiction!? Where is the negation!?) p.~p The point a in the visual field is blue and the (same) point a is red (6.3751) No point in the visual field can be both blue and red The table over there is 3 meters long and the (same) table over there is 4 meters long No object can be both 3 meters long and 4 meters long Now is 25 degrees Celsius and now is 26 degrees Celsius A particular moment cannot have two temperatures Flamengo has lost yesterday and Flamengo has won yesterday A soccer team cannot both lose and win simultaneously. The animal over there is a cat and the (same) animal is a dog. No animal can be both a cat and a dog. What is going on? Where is the form of a contradiction!? Where is the negation!? Contrarieties! And… Rules! No law, no axiom, but “in a certain sense a tautology” (SRLF). Ok, but why rules? Rules restrict a Spielraum! Let’s try to capture contrarieties with tractarian notation! (Wittgenstein 1929 e Von Wright 1996)

Part III (some philosophical motivation for) systematic mutilation of truth table (Wittgenstein 1929 e Von Wright 1996)

III. General leitmotif: If the logical organization of colors represents a problem for his logic, it should represent a problem for his notation too. Exploring the distinction between contradiction and contrarieties using truth tables. Truth table (1918) X Truth table (1929) 1918! p

q

p.q

T T

T

T F

F

F

T

F

F

F

F

John is scientist

John is logician

John is scientist and John is logician

A is red

A is blue

A is red and A is blue

T

T

T

T

T

F

T

F

F

T

F

F

F

T

F

F

T

F

F

F

F

F

F

F

But, where is the negation and the contradiction? Is there a exclusion without negation? The tractarian answer: NO! In line with a logicist tradition: 1) A complex proposition has a unique logical form (here the contradiction); 2) It is hidden (not visible in its grammar); 3) It is very complex; 4) It must be discovered!

Truth table (1929)! Technically it is not a big deal, but it is philosophically momentous. He keeps the Russelian idea of full analysis but talks about laying down some rules. At this time, the problem is 1) neither with the truth value in the last column (no falsehood, no null, but we have some nonsensical construction!) Note! The exclusion is not nonsensical, but W’s representation itself!; 2) nor with the connective “and” (WWK, p. 80), 3) nor with an exclusive disjunction, since an inclusive disjunction cannot be used either; 4) nor with color-system (WWK, p. 80). 5) Things as redish-green, or transparent white are not relevant here. The problem is with the scheme itself, with the free distribution of truth values! The combinatorial procedure has to follow some rules. It has to be contextually sensitive. A is red A is 3m long Now it‘s 28°C hardness, sound, ETC…

A is blue A is 4m long Now it‘s 29°C Volume, sound, ETC…

A is red

A is blue

A is red and A is blue

T

T

F

T

F

F

T

F

F

T

F

F

T

F

F

F

F

F

Note! Some lines have to be ruled out, taken away, blocked, “mutilated”. (Mutilation, Von Wright, 1996). Some combinations must be ad hoc blocked. Dramatic turn! To impose restriction of truth tables means to impose restrictions on truth functionality, extensionality and so other typical (classical) tractarian features.

We must add up rules to restrict logical space. Rules are constraints (Spielraum). In this sense, mutilations may nicely capture some other logical patterns, such as: 1) contrariety; 2) subcontrariety; 3) contradiction. p

q

p

q

T

F

T

T

F

T

T

F

F

F

F

T

p

q

T

F

F

T

So, is it a problem with propositional logic?

No! Predicates as simple extension will not do the work either. “… if ‘f(x)’ says that x is in a certain place, then ‘f(a).f(b)’ is a contradiction. But what do I call ‘f(a).f(b)’ a contradiction when ‘p.~p’ is the form of the contradiction? Does it mean that the signs ‘f(a).f(b)’ are not a proposition in the sense that ‘ffaa’ isn’t? Our difficulty is that we have, nonetheless, the feeling that here there is a sense, even if a degenerate one (Ramsey)” MS112

This clearly shows that both, Wittgenstein and Ramsey, were discussing about Color Exclusion Problem and challenges for formal logic at this time. Intense debate with Wittgenstein about this subject shows that Ramsey was aware of the ad hoc restriction of truth-functionality! No laws, principles, or axioms etc. But rules!

IV) Two forms of mutilation? TLP 3.325 In order to avoid these errors, we must apply a symbolism which excludes them , by not applying the same sign in different symbols and by not applying signs in the same way which signify in different ways. A symbolism , that , is to say, obbeys the rules of logical grammar – of logical syntax.

Two remarks:

1) Relation between a symbolism and philosophical errors. 2) Identification between syntax and grammar

Color Exclusion Problem will challenge this identification. My hypothesis: Actually we could say that we have a conflict between the tractarian syntax and the grammar.

Two forms of mutilations!? TLP 3.325 identification between syntax and grammar P

Q

T

T

T

T

F

F

F

T

F

F

F

F

Restriction of a combinatorial procedure through a non tractarian grammar! P

Q

T

F

F

F

T

F

F

F

F

Grammatical procedure, the first line is not there to be restricted!

Obrigado pela atencao! Thanks for your attention!