No-p

En mi post sobre certeza modal apuntaba que una proposición de la forma Np “no está comprometida” a una proposición de la forma p. Creo que esta aserción es por lo menos oscura.

Lo que quería decir con eso puede replantearse así:

1. Np no implica p,
2. Np implica Np

Hay que distinguir entre ambos factores. Voy a discutir solamente el primero por ahora.

Que Np no implique p es el núcleo de la aserción de que Np no esté comprometida a p. Podría darse una explicación en términos de la noción de fuerza lógica, que mencionaba entonces, pero hay una dificultad: p tampoco está comprometida a Np. p no implica Np (p implica NNp…). No tiene sentido decir que p es más fuerte que Np; Np podría se igualmente más fuerte que p. Ambas posibilidades llevan a un absurdo; la única opción es que p y Np tengan la misma fuerza lógica.

Recuerdo la idea del Wittgenstein más temprano de que las proposiciones son bi-polares (“Notes on Logic”, 1913). Verdad y falsedad son los polos de la proposición. p es una variable proposicional. La proposición que representa puede ser o no verdadera. Asimismo, el compromiso de una proposición a otra sólo puede darse cuando el valor de verdad de la proposición queda determinado; si “p” es verdadera o falsa, es decir. En ese caso (y sólo entonces) puede la proposición implicar algo. p implica Mp, si p es verdadera. [¿Qué representan las tautologías, entonces? La forma de inferencias, pero no inferencias.]

Una proposición de la forma Np implica lo inverso que una proposición de la forma p. Si una proposición de la forma Np es verdadera, la proposición p es falsa.

EDIT: On retrospective to what I wrote last year (!) on negation, I think I should reconsider my assumption of assertion and negation being modalities. The option would be to devise a structure where all modalities would, so to say, hang from assertion and negation.

On modal certainty

When we evaluate modal propositions (that is, any proposition), we can be certain about them and, from that, be certain about some others. By example, if I evaluate the proposition “the mousse is tasteless”, I can infer from it the proposition “it is possible that the mousse is tasteless”, and be as certain about this second proposition as about the first.

We regard modalities as sequences of monadic operators. We have N for negation, M for possibility and L for necessity (I use polish notation, so is Np). Those operators form propositions out of propositions. Given a proposition p, we can build series of propositions by applying those operators on p (series containing Np, NNp, Mp, Lp, MMp, MLp…). We will say any item on those series is more or less committed to p. On the meaning of commited in this modal sense we must point

• Any proposition p is committed to itself.
• Any proposition Mp is less committed to p than p.
• Any proposition Lp is more committed to p than p.
• Any proposition Np is not committed to p.

This modal notion of commitment is grounded on that of logical strength. We say a proposition p is stronger than another r if:

• Cpr []

but

• NCrp []

Given a predicate S of two arguments for “logically stronger than”, defined as a conjunction of the theses given above, SLpp and SpMp (note those are usual assumptions in modal logic, though they are not necessary).

We’ll say the truth of a proposition is grounded on some other truth. In the cases given above, stronger propositions serve as ground for weaker ones. Still, we must have some reason to accept the stronger propositions. That ground can be something different to yet stronger propositions. I see my keyboard in front of me; that is ground for assertion of the proposition “that keyboard is in front of me” (never mind the indexicals). From there, I can assert that “it is possible that that keyboard is in front of me”. My starting proposition wasn’t grounded on a stronger one (namely on “it is necessary that that keyboard is in front of me”). Likewise, the proposition “it is possible that that keyboard is in front of me” could be grounded on something different to my commitment to there being a keyboard in front of me. What could serve as ground for such an assertion?

I’d be tempted to say possibilities (possible states of affairs) are grounded on actualities (actual states of affairs). That isn’t the point I want to make here, however.

If Np, we need some ground for assertion of Mp, and we could have. However, can we have, in that case, certainty about Mp? Generally: can we have certainty about a weaker proposition than a proposition p if we have Np? In principle, yes. To have a ground for a weaker form than p, we need only have ground for that form; we need not ground for p. This much we know formally. However, can say we can have ground for that form in all cases that Np? Can we know this a priori?

For one, we can say we can have ground for that weak form in cases when we can have ground for p. This is trivial: having ground for saying we can have ground for p is to have ground for Mp, a weaker form than p. Obviously, this isn’t enough for saying we can always have have ground for the weak form if Np: nothing is said about there being any ground for Mp necessarily.

We can state the problem this way: we seem to be unable to be necessarily certain about counterfactuals. However, as a form like Mp is less committed to p than p itself, less information seems to be needed for its assertion than we need for assert p. What we could regard as possible seems to be more certain for us than what we could regard as actual.