Friday, August 3, 2018

World shuffling and quantifiers

Let ψ be a non-trivial one-to-one map from all worlds to all worlds. (By non-trivial, I mean that there is a w such that ψ(w)≠w.) We now have an alternate interpretation of all sentences. Namely, if I is our “standard” method of interpreting sentences of our favorite language, we have a reinterpretation Iψ where a sentence s reinterpreted under Iψ is true at a world w if and only if s interpreted under I is true at ψ(w). Basically, under Iψ, s says that s correctly describes ψ(actual world).

Under the reinterpretation Iψ all logical relations between sentences are preserved. So, we have here a familiar Putnam-style argument that the logical relations between sentences do not determine the meanings of the sentences. And if we suppose that ψ leaves fixed the actual world, as we surely can, the argument also shows that truth plus the logical relations between sentences do not determine meanings. Moreover, can suppose that ψ is a probability preserving map. If so, then all probabilistic relations between sentences will be preserved, and hence the meanings of sentences are not determined by truth and the probabilistic and logical relations between sentences. This is all familiar ground.

But here is the application that I want. Apply the above to English with its intended interpretation. This results in a language English* that is syntactically and logically just like English but where the intended interpretation is weird. The homophones of the English existential and universal quantifiers in English* behave logically in the same way, but they are not in fact the familiar quantifiers. Hence quantifiers are not defined by their logical relations. I’ve been looking for a simple argument to show this, and this is about as simple as can be.

Philip Rand said...

Are these mappings area preserving?

Philip Rand said...

If the mappings are area preserving. The result is most interesting.

Alexander R Pruss said...

If by "area" you mean volume in probability space, then we can suppose that.

Philip Rand said...

No. Area preserving is more fundamental than probability preserving. It can be used to describe the probability dynamics & more.

It is the result one gets with the homophones that is quite interesting.