20140512BRUSSELS"Applying Metaphysics of Truth Tables to The Color Exclusion Problem"

Applying Metaphysics of Truth Tables

to The Color Exclusion Problem

Marcos Silva [email protected]

Brussels - CRISSP Group – 12.05.2014

Three main goals:

1. To submit the idea of reading the rise and fall of the Tractatus through the rise and fall of its notation. The focus will be on the Color Exclusion Problem (no incompatibility, but negation; not nature, but logic of colors). (Origin of truth table? No. Solution of CEP? No. The articulation of both is the important step!) In 6.3751, it must be a contradiction (“p and not-p” X “RTP & BTP”). This is consistent with logicism.

2. To convince you that truth tables constitute a special exegetical key. It embodies the tractarian image of logic so we can see from it many features of the Tractatus and its limitations as well. If the problem is a problem with his logic, it shoulxd also be a problem for his notation. To read off some of the main tenets in the Tractatus from truth tables for good and bad represents an exegetical gain.

3. This sheds light on how secularized our use of a “metaphysical” tool can be and how philosophical overloaded a technical use actually is.

Outline:

Some historical and philosophical motivations!

I. A image of logic

II. Secularized Use of Truth tables

III. Truth table as a exegetical key (for good and bad!)

a) some positive aspects (metaphysical use of truth tables)

b) (three) negative aspects (Color Exclusion Problem!)

IV. Conclusion (with further questions about “mutilation” and “part-whole relationship”)

Metaphysical use: Truth tables should show the essence of language and world. Rules of manipulation in a notational system capture the (unique and absolute) logical structure. One logical space! A (robust) philosophical program! Secularized use: current-day use in a (defined) formal system. Technical use: totally independent of philosophy! (Post, 1923) We do not need to talk about essence of language or of world to explore this logical machinery.

1) Distinction between sagen and zeigen (representation without negation nor falsehood) 2) Distinction between metaphysical use and secularized use

Two distinctions:

Secondary literature on truth tables: Shosky (1997), Anellis (2004, 2011), Beziau (2012). Their question: Who was the first one to think about truth tables? Frege, Peirce, Schröder, Russell, Wittgenstein, Post, Łukasiewicz? Our question: Is it really relevant to investigate, who was the first one to think about truth tables? In which sense this historical approach could advance philosophical discussion? 1. Problem: Truth table was not at his disposal! Wittgenstein really thought that truth table was his Notation! (SRLF, “our symbolism”, p.169, “our notation”, p.170-1; WWK „meine Notation“, p. 80 „in meiner Notation“ p.92, „mein Zeichensystem“ p. 80) Ramsey too! (1923, 1927) 2. Problem: Truth table: a mathematical device? A logical technique? A notation! An alternative notation. More adequate, clearer than the Russelian notation (WWK, p. 80 und p. 92; and Appendix III Notebooks 14-16) 3. Problem: Color Exclusion Problem through his notation. Missing the point! Examples, SRLF p. 170 and Länge WWK p. 92 (see von Wright, 1996). “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.“ (p.169)

On a secularized use:

“We do not have to fear that the contemplation of symbols will lead us away from things, but, on the contrary, it conducts us to their inner side.” Leibniz, Letter to Tschirnhauss, 1678

“Ramsey neither understands the value that I set on a particular notation nor the value that I set on a particular word, because he does not see that a whole point of view [Anschauungsweise] of an object is expressed through it, the angle from which I now consider a thing. The notation is the ultimate expression of the philosophical intuition [Philosophischen Anschauung].“ Wittgenstein, Wiener Ausgabe I, NACHLASS, 1929

Leibnizian intuition:

On a metaphysical use:

TLP 3.324, 3.325: “So entstehen leicht die fundamentalsten Verwechslungen (deren die ganze Philosophie voll ist). Um diesen Irrtümern zu entgehen, müssen wir eine Zeichensprache verwenden, welche sie ausschließt, indem sie nicht das gleiche Zechen in verschiedenen Symbolen, und Zeichen, welche auf verschiedene Art bezeichnen, nicht äußerlich auf die gleiche Art verwendet. Eine Zeichensprache also, die der logischen Grammatik – der logischen Syntax – gehorcht. (Die Begriffsschrift Freges und Russells ist eine solche Sprache, die allerdings noch nicht alle Fehler ausschließt).“ Irony in the History of Philosophy! Russell/Frege (1901) → Ramsey/Wittgenstein (1923) How dramatical these discussions were → First (natural) reaction: To fix and rescue the program! SRLF: „appropriate symbolism“ p. 163; „clear symbolism“, p. 163; „special symbolism“ p.166; „perfect notation“ (p. 170) WWK: „Es ist so: Syntax und Zeichen arbeiten immer gegeneinander. Was die Zeichen leisten, geht auf Kosten der Syntax, und was die Syntax leistet, geht auf Kosten der Zeichen. Ich kann sagen: Ein Zeichensystem von richtiger Mannigfaltigkeit macht die Syntax, überflüssig. Ich kann aber ebensogut sagen: Die Syntax macht ein solches Zeichensystem überflüssig. Ich kann ja auch ein unvollkommenes Zeichensystem verwenden und die Regeln der Syntax hinzufügen„ (p.80)

Digression! Relationship between a symbolism and Syntax: TLP (1918), SRLF (1929), WWK

(1929-32)

I. But, what is logic in the tractarian period? 5.551 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.) 2.0121 (…) Logic treats of every possibility, and all possibilities are its facts. (...) Letter to Russell from Norway, 1913 (Apendix III Notebooks 14-16) “I want to repeat what I wrote about logic in my last letter, putting it in a different way: All propositions of logic are generalizations of tautologies and all generalizations of tautologies are propositions of logic. There are no logical propositions but these. (I consider this to be definitive.) [Dies halte ich für definitiv].” (Tagebücher 14-16, p. 127). “...the great question is now: How should a notation be constructed, which will make every tautology recognizable as a tautology in one and the same way? This is the fundamental problem of logic. [Dies ist das Grundproblem der Logik!]“ (Tagebücher 14-16, p.128)

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II. Secularized use of truth table

Shosky’s distinction (1997) between mathematical device and logical technique makes sense (legitimate): Diagrammatical or tabular representation (device) of truth functionality and of truth conditions (technique) Definition of logical operators Determination of Semantic Equivalence (too weak for TLP, with two

propositions have the same truth conditions they are not equivalent, they are the same proposition! pq = not (p and not q)

Consistence-proofer of some sets of propositions Validity and Invalidity of some arguments Algorithmic power for propositional calculus (Method to effectively determine logical validity, decidability)

III. The importance of Truth Table: It is an exegetical key. It embodies an image of logic! a)positive aspects (metaphysical use of truth tables)

• Originally a metaphysical instrument: to show the essence of Logic (TLP 5.511, 6.12, 6.124) • It is an alternative Notation!, i.e., 1) a propositional sign TLP 4.442 (Satzzeichen, quotation marks!); 2) a symbolic language (Zeichensprache) which obey logical syntax; 3) it should block logical errors and nonsense (TLP 3.325). Irony in the History of Philosophy. Criticism on Frege and Russell. It should be impossible to judge a nonsense (TLP 5.5422) • Truth table shows all possible combinations of complex propositions, and thus of Sachverhalte. Birth certificate! (TLP 4.3, 4.31, 4.442, 5.101)

Let’s take a look at an example!

• Combinatorial neutrality (exhaustion) • Compositionality (not decision whether p is complex or not, but

method of logical decomposition based on truth-functionality) • Full analysis (to hit a bottom) • Bipolarity (criterion for meaningfulness: contingency!) • Avoid nonsenses and pseudo-propositions (Scheinsätze, TLP

3.225) • Behavior of tractarian negation (contradiction, neutral switcher of

discrete states! Nothing is being added! sein Grundgedanke!) • Propositions constituted (uniquely) by names (To substitute ‘aRb’

by p, 4.22, 4.24) • Objective criterium for the difference between logic and empirical

propositions (Frege and Russell) • Logical independence of elementary propositions (neither

tautologies nor contradictions at the atomic level)

IV. b) three negative aspects: Problems with limited expression capacity. To put in a simple way: a problem with the truth-table notation means a problem with the Tractatus as well. 1. Quantification, interaction of quantification (Infinity and exponentialization) 2. Negation is always propositional → lack of sensitiveness. But it is important so. it functions to invert sense or truth conditions (TLP 5.2341) p T,F ~p F,T ~~p T,F 3. Color Exclusion Problem. In TLP, it is no problem. But in 1929, after Ramsey (1923). “It is, of course, a deficiency of our notation that it does not prevent the formation of such non-sensical constructions, and a perfect notation will have to exclude such structures by definite rules of syntax. These will have to tell us that in the case of certain kinds of atomic propositions described in terms of definite symbolic features certain combinations of the T's and F's must be left out. Such rules, however, cannot be laid down until we have actually reached the ultimate analysis of the phenomena in question. This, as we all know, has not yet been achieved.” (Wittgenstein, Some Remarks on Logical Form, 1929, p.171) cf. PB 79 p.107

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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 phenomenological rules. At this time, the problem is 1) neither with the truth value in the last column (no falsehood, no null, but nonsense!); 2) nor with the connective “and” (WWK, p. 80), 3) nor with an exclusive disjunction, since an inclusive disjunction cannot be used either; 3) nor with color-system (WWK, p. 80). 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 sensitive.

A is red A is blue A is red

and A is

blue

T T F

T F F

F T F

F F F Some lines have to be ruled out, taken away, blocked, “mutilated”. (Mutilation, Von Wright, 1996). Some combinations must be ad hoc blocked. Dramatic turn! We must add up rules to restrict logical space. In this sense, mutilations capture some logical patterns, such as: 1) contrariety and 2) subcontrariety.

p q

T F

F T

F F

p q

T T

T F

F T

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…

T F

F T

F F

IV. Conclusion with four further questions: 1) What does it mean to cancel/delete/eliminate (streichen) a truth table´s line? „What happens if these two propositions are R P T and B P T? In this case the top line "T T T" must disappear, as it represents an impossible combination“ (SRLF, p.170) As we saw, new combinations should be given by some special phenomenological rules! Time, space (see Russell 1903) and synthetic a priori. So, what does it really mean „to disappear“ und „to be left out“? An enlargement/expansion (Erweiterung) or a restriction/limitation (Begrenzung) of the Tractatus´ Logic? Both! “Logical paradox”? Not really, if we understand what a rule is: To add up rules means in this period to limit/to restrict the Tractatus´ Logic! It makes no sense to expect rules, when we allow (erlauben) every possibility. To have rules we need constraints. „This possibility is not possible“. To have rules means to limit/to restrict the „Spielraum“ (logical space)! Cf. TLP 3.325 The Grammar (phenomenology) and the tractarian logic must be separated! 2) How can they be combined? What is the kind of relation which we have between these two “logics”? At this time was a part-whole relation! „As a summary you could say that the truth-functional connection of propositions forms only one part of syntax [einen Teil einer Syntax]. The rules I laid down at that time are now restricted by the rules [eingeengt durch die Regeln] that originate from the inner syntax of propositions and prohibit propositions from ascribing different co-ordinates to reality. All truth-functions that are not forbideen by these rules are permitted.“ (WWK, p.80, mein kursiv, WWK 81 too! Methode der Abbildung der welt!)

3) Ok, but what does that all mean for Wittgenstein? That means inter alia that he was trying to rescue the tractarian program in this period! The program was unfortunately not complete! There is a gap there to be filled up with additional rules. (Ramsey, 1927) 4) And for Philosophy of Logic, should we accept as a result that we have different Logics? Maybe. I would say yes. The Principle of Excluded Middle, for example, would be a sharp criterion. 1) For any p, p or its negation is true. If “a is blue” deny “a is red”, then we must accept that both can be false. 2) Some indetermination (Ex. “my t-shirt is not green” or “that table is not 3 meters long). But at this time Wittgenstein really did not see this problem in this perspective. Different systems are indeed combinable! Many parts of a whole. Eg. tractarian logic and phenomenology (many logical spaces) and the whole (one logic!)

A last mea culpa by Wittgenstein (Jan, 1930) for illustrating the part-whole relation: “I laid down rules for the syntactical use of logical constants, for example ‘p.q’, and did not think that these rules might have something to do with the inner structure of propositions. What was wrong about my conception was that I believed that the syntax of logical constants could be laid down without paying attention to the inner connection of propositions. That is not how things actually are. I cannot, for example, say that red and blue are at one point simultaneously. Here no logical product can be constructed. Rather the rules for the logical constants form only a part of a more comprehensive syntax about which I did not yet know anything at that time. [Die Regeln für die logischen Konstanten bilden vielmehr nur einen Teil einer umfassenden Syntax, von der ich damals noch nichts wusste.]“ (mein kursiv, vgl. SRLF, p.171 „certain kinds of elementary propositions“ and last paragraph as a phenomenological project; WWK, p. 76 Dort wo die Sätze voneinander unabhängig sind, bleibt alles in Kraft!; WWK, p.80 „nur einen Teil einer Syntax“; „PB 83, p.111, „ein Teil der Grammatik über diese Wörter, aber nicht die ganze“, Wittgensteins kursiv)

Obrigado pela atencao.

Thank you for your attention.

Literature RAMSEY, Frank. Critical Notes to Tractatus Logico-Philosophicus. Mind 32, (1923, pp. 465-478). _____. Facts and propositions. Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 7, Mind, Objectivity and Fact (1927), pp. 153-206. SILVA, Marcos. Muss Logik für sich selber sorgen? On the Color Exclusion Problem, the truth table as a notation, the Bildkonzeption and the Neutrality of Logic in the Collapse and Abandonment of the Tractatus. PHD Thesis - Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, 2012. ______. On Degrees of Exclusion within and among Systems. Revista Argumentos. Fortaleza, 2013 ______. Wittgenstein, Cores e Sistemas: aspectos logico-notacionais do colapso do Tractatus. Revista Analytica. Rio de Janeiro. 2011. ______. Holismo e Verofuncionalidade: sobre um conflito logico-filosófico essencial. Philosophos, Goiânia, 2013. ______. Sobre a fragmentação do espaço lógico. Revista Brasileira de Filosofia. São Paulo. 2014. Forthcoming. VON WRIGHT, Georg Henrik. On Colour: a logic-philosophical Fantasy. In Six Essays in Philosophical Logic. Acta Philosophica Fennica. Vol. 60, Helsinki, 1996. (pp. 9-16). WITTGENSTEIN, Ludwig. Philosophische Bemerkungen. Werkausgabe Band 2. Frankfurt am Main: Suhrkamp, 1984. ______. Some Remarks on Logical Form. Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 9, Knowledge, Experience and Realism (1929), pp. 162-171 Published by: Blackwell Publishing on behalf of The Aristotelian Society. ______. Tractatus Logico-philosophicus. Tagebücher 1914-16. Philosophische Untersuchungen. Werkausgabe Band 1. Frankfurt am Main: Suhrkamp, 1984.

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