I saw this post by Mark Lance over at New APPS and he brought up one of the issues that I have recently been concerned with: What is a logical domain? He said:
So our ignorance of our domain has implications for which sentences are true. And if a sentence is true under one interpretation and false under another, it has different meanings under them. And if we don’t know which of these interpretations we intend, then we don’t know what we mean.
I am inclined to think that this is a really serious issue…
When we don’t know what we, ourselves, mean, I regard this as THE_PHILOSOPHICAL_BAD, the place you never want to be in, the position where you can’t even speak. Any issue that generates this sort of problem I regard as a Major Problem of Philosophy — philosophy in general, not just of its particular subject.
A little over a year ago I was trying to integrate probability and logic in a new way. I developed indexed domains in order that different quantifications ranged over different values. But then I said:
(aside:
I prefer to use an artifact of Independence Friendly logic, the dependence indicator: a forward slash, /. The dependence indicator means that the quantifier only depends on those objects, variables, quantifiers or formulas specified. Hence
Яx/(Heads, Tails)
means that the variable x is randomly instantiated to Heads or Tails, since the only things that Яx is logically aware of are Heads and Tails. Therefore this too represents a coin flip, without having multiple domains.)
I used the dependence slash to indicate the exact domain that a specific quantification ranged over. This localized the domain to the quantifier. About a week after publishing this I realized that the structure of this pseudo-domain ought to be logically structured: (Heads, Tails) became (Heads OR Tails). The logical or mathematical domain, as an independent structure, can therefore be completely done away with. Instead a pseudo-domain must be specified by a set of logical or mathematical statements given by a dependence (or independence) relation attached to every quantifier.
For example:
∀x/((a or b or c) & (p → q))…
This means that instantiating x depends upon the individuals a, b or c, that is, x can only be a, b or c, and it also can only be instantiated if (p → q) already has a truth value. If ((p → q) → d) was in the pseudo-domain, then x could be instantiated to d if (p → q) was true; if ¬d was implied, then it would be impossible to instantiate x to d, even if d was implied in some other part of the pseudo-domain. Hence the pseudo-domain is the result of a logical process.
The benefit of this approach is that it better represents the changing state of epistemic access that a logical game player has at different times. You can have a general domain for things that exist across all game players and times that would be added to all the quantifier dependencies (Platonism, if you will), but localized pseudo-domains for how the situation changes relative to each individual quantification.
Moreover, the domain has become part of the logical argument structure and does not have an independent existence, meaning fewer ontological denizens. And, to answer the main question of this post, every domain is completely specified, both in content and structure.
I’m inclined to call this logic Domainless Independence Friendly logic, or DIF logic, but I really also like EIFL, like the French Tower: Epistemic Independence Friendly Logic. Calling this logic epistemic emphasizes the relative epistemic access each player has during the logical game that comes with the elimination of the logical domain.
