Thursday, September 4, 2014
In 1976, when I delivered the John Locke Lectures at Oxford, I often spent time with Peter Strawson, and one day at lunch he made a remark I have never been able to forget. He said, "Surely half the pleasure of life is sardonic comment on the passing show". This blog is devoted to comments, not all of them sardonic, on the passing philosophical show.
The modal logical interpretation of mathematics and Tarskian truth definitions; intentionality does not presuppose language; fully objective looks
Today’s Post deals with two communications from friends (the first of them a couple of years old) + a correction to a slip I made in Different Looks (August 15).
The modal logical interpretation of mathematics and Tarskian truth definitions
Friend 1, S.W. suggested that the modal logical interpretation of mathematics that I favor has the disadvantage that Tarskian truth definitions are not possible, and so the “distribution of truth values” over the sentences of the (modal logical) language goes unexplained. Here is my reply:
The belief that Tarski’s technique doesn’t work with modal logic is quite mistaken. The reason that it is mistaken is that Tarskian “satisfaction” is defined inductively (I would write, “recursively”, except that term has been preempted by computer science). The clause for existential quantification (simplifying greatly for expository purposes) tells us in effect that an object a satisfies a monadic formula (Ey)F(x,y) just in case (Ey)F(a,y), the clause for negation tells us that a satisfies –F(x) just in case a does not satisfy F(x), and the clause for disjunction tells us that a satisfies (F v G) just in case a satisfies F or a satisfies G. If a modal primitive is added to the language, say the symbol ◊, then the appropriate clause will read: a satisfies ◊F just in case ◊(a satisfies F). Here is a word example: take (F(x) to be (Ey)(x loves y). Interpret ◊ as physical possibility (or, alternatively, sociological possibility), and take a to be Alice. Then “a satisfies ◊F” says that Alice satisfies “it is possible x loves somebody", “◊(a satisfies F)” says that it is possible that Alice satisfies "x loves somebody" and these two formulas have the same truth condition, namely that in some possible world there is a person whom Alice loves. In mathematical jargon, “satisfies” commutes with ◊.
A Tarskian truth-definition for a non-modal language shows how truth and falsity depend on the extensions of the predicates of the language; but equally so, a Tarskian truth-definition for a modal language shows how truth and falsity depend (in the case of the modal formulas) on the possible extensions of the predicates, which is what one should expect. If the first sort of truth-definition counts as explaining the distribution of truth values over the formulas of the language, so should the second.
In the case of languages for a bounded part of mathematics, an inductive (or “recursive”) definition of truth via “satisfaction” can be replaced by an explicit definition in a language with a ‘bigger’ ontology of sets via a technique due to Frege, and this bigger language can, in turn be given a modal-logical equivalent. When the language formalizes all of extant mathematics, then the inductive definition has to suffice. But, unless one’s purpose is to code inductive definitions into set theory, there is no reason to want the explicit definition; inductive definitions are perfectly mathematically kosher. And, in any case, the impossibility of an explicit Tarski-style truth definition for the whole language of mathematics applies to both the “objectual” form of the language and the “modal” form; neither scores a point against the other here.
Does intentionality presuppose mastery of a language?
This question was put to me by friend 2, S.C., and my reply was quite short:
I think that the mental states of all animals are capable of representing simple features of the world such as shape and distance (and, for animals with color vision) color, and that in the case of chimpanzees. (see http://biohorizons.oxfordjournals.org/content/3/1/96.full) it is plausible that there is already full fledged intentionality. Language vastly enlarges the range of contents our mental states are able to have, but I don't think language is necessary for there to be intentionality at all. (Further down the evolutionary ladder, e.g., the case of dogs, there are what I once called “proto-concepts” - one could also speak of “proto-intentionality”.)
Correcting a slip in “Different Looks”
In that post (August 15), I wrote, “Looks are dispositional properties of objects” (which is what I believe). But later in the same post, describing a class of Intersubjective looks that I called “fully objective looks” I wrote, “As for the “ontology” of such objective looks: when the look of a color (hue) in a particular situation can be displayed by a photograph or a painting, the “objective look” is simply the color shown by the photograph or painting; the “look” of one hue can sometimes be a different hue.” The problem with this is that hues are dispositions to affect light, not people, on the physicalist view of color that I endorsed, but looks are dispositions to cause people to enjoy certain qualia, in the case of subjective looks, or to cause the visual system to produce certain representations in the case of Intersubjective looks, including “fully objective looks”. So what I should have written is, “As for the “ontology” of such objective looks: when the look of a color (hue) in a particular situation can be displayed by a photograph or a painting, the “objective look” is a representation (in the brain of the viewer) of the color shown by the photograph or painting; the “look” of one hue can sometimes be a mental representation of a different hue.”
 For my most recent explanation of the “modal logical interpretation of mathematics”, see chapter 11, “Set Theory: Replacement, Realism, and Modality”, of my Philosophy in an Age of Science.
 See Tyler Burge’s Origins of Objectivity, and for a short paper on the human case that draws on Burge’s fundamental work, Ned Block’s “Seeing-As in the Light of Vision Science,” Philosophy and Phenomenological Research, 2014.
 “Full-fledged” in the sense that the notion of a “mistaken belief” is available to the animal's “theory of mind”.
 See chapter 2 of my Renewing Philosophy, particularly 28ff.
 For my use of “visual system” and “representation” see Burge, Origins of Objectivity. Note that on my view, and I believe Burge’s as well, different people’s visual systems may produce the same representation without it necessarily being the case that they enjoy the same qualia.