Jean-François Monteil, ancien maître de conférences de linguistique générale à l’Université Michel de Montaigne de Bordeaux
Adresse électronique :
Jean-francois.monteil@neuf.fr
Les deux sites associés :
mindnewcontinent
https://mindnewcontinent.wordpress.com/
mindnewcontinentfrenchcornerLes sujets abordés et les articles publiés:KNOLmnc 0 Sites and topics – mindnewcontinentKNOLmnc Liste et classification des KNOLs mncX KNOLmnc 0 Diffusion |
If the symbol (l) represents a priori necessity, (l) p w ~p means that the fact p and the fact not-p are a priori necessarily contradictory. On the one hand, they are necessarily in-compatible in reality, on the other hand they cannot be both excluded from reality. Hence the fact p ≡ ~~p. That means that the fact p is the fact excluding the fact not-p as the fact not-p is the fact excluding p. The author of this remark refers the potential reader to An inquiry into meaning and truth, chapter 20 by Bertrand Russell and to what is devoted to the said chapter entitled The law of excluded middle in the following papers:
–KNOLmnc 1 To defend his views about modal logic and strict implication, Jean-François Monteil utilizes the chapter of Bertrand Russell’s An inquiry into meaning and truth entitled The law of excluded middle. To Mind a Quarterly Review of Philosophy
–KNOLmnc 1 Modal logic. The three ingredients of strict implication: L (p ≡ Lq). To Mind a Quarterly Review of Philosophy.
To my mind, the twentieth chapter entitled The law of excluded middle, constitutes a sort of climax in the celebrated An inquiry into meaning and truth. In light of Tarski and thanks to the use of the logical hexagon of the Frenchman Robert Blanché in modal logic, a lot of problems raised by Russell in his book and particularly in the twentieh chapter can be solved. Tarski said: the proposition “Snow is white” is true, if and only if snow is white. One may conclude that instead of saying the proposition p is true, one must say that the fact p is certain and symbolize the certainty of the fact p by Lp. If we are in a position to assert: ‘It snowed on Manhattan Island on the first of January in the year 1 Anno Domini’, the fact p in question must be symbolized by Lp, to be read It is a certain fact that it snowed on Manhattan Island on the first of January in the year 1 Anno Domini. If we are in a position to assert: ‘It did not snow on Manhattan Island on the first of January in the year 1 Anno Domini’, the fact not-p in question must be symbolized by L~p, to be read : It is a certain fact that it did not snow on Manhattan Island on the first of January in the year 1 Anno Domini. If we are in a state of ignorance concerning the two contradictory facts p and not-p, in other words, if we are unable to assert ‘It snowed on Manhattan Island on the first of January in the year 1 Anno Domini’ as well as ‘It did not snow on Manhattan Island on the first of January in the year 1 Anno Domini’, we experience a fact, the fact that neither p nor not-p is certain. This third fact can be symbolized by ~L~p & ~Lp, both the certainty of the fact not-p and the certainty of the fact p are excluded. I emphasize here that the third fact I mention must be given as much importance as the facts Lp and L~p we are led to consider when we are in a state of knowledge. The third fact is the fact we have to envisage when we are in a state of ignorance. It corresponds to what is called the bilateral possible. ~L~p, the non-certainty of the fact not-p is equivalent to the possibility of the fact p to be symbolized by Mp, ~Lp, the non-certainty of the fact p is equivalent to the possibility of the fact not-p to be symbolized by M~p. There exist three situations corresponding to the case envisaged by Bertrand Russell in the chapter 20 of his An inquiry into meaning and truth and entitled The law of excluded middle. One of three things, either Lp the certainty of the fact p or L~p the certainty of the fact not-p or Mp & M~p the possibility of both p and not-p to the extent that both are non-certain. In any of the three situations, the law of excluded middle is preserved. This law can be represented thus: (l) p w not-p. The facts p and not-p are necessarily, by definition ( this is the meaning of the symbol (l) here used) contradictory. They are incompatible and they cannot be both excluded of reality.
If what is said above is true, the truth-table must be rejected.
The proposition p envisaged by Russell is true, if and only if the homonymous fact p it makes known is certain and can be symbolized by Lp.
The proposition not-p envisaged by Russell is true, if and only if the homonymous fact not-p it makes known is certain and can be symbolized by L~p.
Now, neither the fact p nor the fact not-p is certain in the situation envisaged by Russell. About them, we are in a state of ignorance.
We can keep asserting (l) p w ~p because unshakable is the law of excluded middle, which does not depend on experience, always precedes it and constitutes its a priori condition. Safely can we say, whatever our knowledge may be: Necessarily, it is one thing or the other, either it snowed on Manhattan Island on the first of January in the year 1 Anno Domini or it did not snow on Manhattan Island on the first of January in the year 1 Anno Domini. Whatever the state of things may be, whether it is known which of the two facts is the case or not, p and not-p are contradictory facts. They are in-compatible, they cannot be both excluded.
In the situation described by Russell, we are in a state of non-knowledge to be represented by ~L~p & ~Lp, neither the fact not-p nor the fact p is certain or by Mp & M~p, the bilateral possible. For if both facts are non-certain, they are both possible.
Taking Veritas adaequatio rei et intellectus, the scholastic definition of truth, seriously and adhering to the correspondence theory of Tarski, I think that the propositions p and not-p identified with the assertive sentences It snowed on Manhattanare… and It did not snow on Manhattan…are both false. « p true » if one speaks in terms of proposition is nothing else than Lp certainty of the fact p if one speak as I think we must. « not-p true » is nothing else than L~p certainty of not-p. If one wants to go on speaking in terms of proposition instead of speaking in terms of fact as I think we must, the proposition which could be deemed true in the situation evoked by Russell is Maybe, it snowed on Manhattan Island on the first of January in the year 1 Anno Domini, maybe it did not. As the fact referred to is Mp & M~p, only the statement Maybe, it snowed on Manhattan Island on the first of January in the year 1 Anno Domini, maybe it did not can realize the adaequatio rei et intellectus, the correspondence of a thing to the intellect.
The author of these lines thinks that the solution of the Russellian problem renders possible a consistent formula of strict implication:
KNOLmnc 1 Modal logic. The three ingredients of strict …
x
x
Useful documents
To Mind a Quarterly Review of Philosophy. The non-mark, the paradoxical sign of Jean-François Monteil. The chain reaction leading to the formula of strict implication.
The hypothesis of the systematic non-mark and the crucial importance thereof was presented in 1989. The paradoxical sign renders possible a rational approach to the French verbal system. It also throws some light on the nature of the so-called affirmative and negative propositions constituting a pair of contradictories. The negative proposition is the marked contradictory, the affirmative the unmarked contradictory. From the start, and before speaking of the hypothesis itself, I indicate the two main results of its use. (1) Thanks to the substitution of the logical hexagon for the square of opposition, an understanding of the relationship between the system of six propositions apprehending quantification in natural language and the underlying logical system is now possible.
Useful links:
KNOLmnc 1 On the Aristotelian Organon. Importance of the chapter 7 of De Interpretatione (or Peri Hermeneias), second book of the Organon
KNOLmnc 1 A German exception: the translation of On Interpretation by Professor Gohlke. His tenth note on indeterminate propositions
KNOLmnc 1 The logical universal affirmative and the two natural universals affirmative
KNOLmnc 1 From the deficient square of opposition to Blanché’s hexagon. The triangle of Indian logic as a simplification of the latter. The rationalization of the scholastic symbolization.
KNOLmnc 1 To the British Society for the History of Philosophy.The logical square of Aristotle or square of Apuleius. The logical hexagon of Robert Blanché in Structures intellectuelles. The triangle of Indian logic mentioned by J.M Bochenski
L p ≡ Lq is most probably the formula of strict implication. It says that a fact p implies a fact q strictly, if the fact p is equivalent to the certainty of the fact q. The formula definitively disposes of the two unpleasant paradoxes inherent to the so-called material implication. The paradoxical sign leads to an exact view of the relation of contradictoriness, the latter to the replacement of the square of opposition by the hexagon of Robert Blanché. Applied to modal logic, the hexagon shows the existence and importance of the bilateral possible. The notion conducts to the formula of the strict implication of a fact q by a fact p: L p ≡ Lq A fact p strictly implies a fact q, if it is established that the fact p is equivalent to the certainty of the fact q.
More useful links:
KNOLmnc 1Modal logic. The three ingredients of strict implication. Calcutta
KNOLmnc 1 Crucial importance of the bilateral possible M(p), the third contrary fact represented in the logical hexagon of Robert Blanché applied to modal logic.
FR MNC (I) Modal logic. The final touch in defining the strict implication of a fact q by a fact p. If the bilateral possible Mp & M~p is to be symbolized by L ( p w ~p) , the formula of strict implication is L (p ≡ Lq).
FR MNC (II) Modal logic. The final touch about strict implication L (p ≡ Lq). The two facts containing it.
FR MNC (III) Modal logic. The three ingredients of L (p ≡ Lq). Unconditional certainty and conditional certainty.
FR MNC (IV) Modal logic. The three facts to be considered: L (p ≡ Lq), the strict implication of q by p and the two facts containing it, namely, L ((p & Lq) w (~p & M(q)) on the one hand and L ((p & Lq) w (~p & L~q) on the other.
KNOLmnc 1 To defend his views about modal logic and strict implication, Jean-François Monteil utilizes the chapter of Bertrand Russell’s An inquiry into meaning and truth entitled The law of excluded middle.
FR MNC I Contingency and necessity. The a priori conditions of experience. Something about Bertrand Russell’s An inquiry into meaning and truth.
FR MNC II Contingency and necessity. The a priori conditions of experience. Interpretation of the formula M L(p) v M Mp v M M~p
The substitution of the logical hexagon of Robert Blanché for the square of opposition, whose origin is to be found in On Interpretation, chapter 7 puts an end to a curse impeding the understanding of the relationship between a system of six propositions of natural language and the underlying logical system. Robert Blanché – Wikipedia, the free encyclopedia https://www.google.com/?hl=en&gws_rd=ssl
newsletter
mindnewcontinentfrenchcorner 
Laisser un commentaire