A question which still remains from last time is the order in which we apply the rules. Well, the general principles proposed in  appear quite plausible: rules are to be applied to phrases in ‘higher’ (dominating) clauses before they are applied to phrases in ‘lower’ (subordinate) clauses, and if we have two different phrases in the same clause, they are to be applied to the first one first. These principles can be overridden by the special rules of precedence associated with specific words. In particular, (G.any) has priority over (G.not), (G.if), “modal rules” and, as claimed in , also over (G.or). (G.some) has priority over (G.not), (G.each) has priority over other quantifier rules and over (G.and) and (G.or). (G.every) is not assumed to be subject to a special ordering rule.
I mentioned all of those mostly so that you can, if you wish, think them through and come up with examples / counter-examples. The only ordering principles relevant here are those pertaining to any.
Let’s look at some examples from  (the asterisk marks ungrammaticality):
If any members contributes, I’ll be happy.
If every member contributes, I’ll be happy.
* If Jane wins, anybody is happy.
If Jane wins, everybody is happy.
Jane can win any match.
Jane can win every match.
* Jane has won any match.
Jane has won every match.
Jane hasn’t won any match.
Jane hasn’t won every match.
* Mary believes that Jane will win any match.
John has not dated any girl.
John has not dated every girl.
* Any girl has not been dated by John.
Every girl has not been dated by John.
I included a plenty of examples for a reason – try analyzing some of them for yourself to see how (G.any/every) along with the rules for preference determine the meaning of these sentences. Some of them are straightforward, some of them probably need some commentary. The difference in meaning between sentences like “Jane can win any/every match” is seen as a matter of talking about all possible matches vs. talking about actual matches only. Perhaps a clearer example is seen in :
Any candidate will be considered on his merits.
Every candidate will be considered on his merits.
There is clearly a difference in meaning between, again involving a difference in the range of quantification. Either we first choose a particular future (the actual one) and quantify over individuals in it, or we first quantify over individuals in various possible futures and claim that all of them will be considered on their own merits (or would be if such a future were to occur). We can say that every candidate happens to be left-handed, but we can’t really say that any candidate happens to be left-handed. On the other hand, “any” is not assumed to have any special precedence over “intensional operators” like “hope”, hence in sentences like “Mary hopes that Jane will win any/every match” Hintikka claims that “the choice of the ‘world’ is absorbed to the intensional operator and hence precedes the choice of an individual involved either in (G.every) or (G.any)”.
You may have noticed a regularity in the un/grammaticality of these any constructions. In fact, studying such constructions led Hintikka to formulate the following principle, which he calls the any-thesis:
The word ‘any’ is acceptable (grammatical) in a given context X – and Y – Z if and only if an exchange of ‘any’ for ‘every’ results in a grammatical expression which is not identical in meaning with X – any Y – Z.
This is a well-formed condition, as the rules of game semantics for English unambiguously determine whether the sentences are equivalent or not (even when one of them is ungrammatical according to this rule). I’ll give you some time to ponder on the linguistic implications of this thesis. Meanwhile, a technical note: in , Hintikka relaxes the condition that the resulting expression be grammatical, as it can fail to be for completely unrelated reasons (for example not being a suitable antecedent for a pronoun, as in “if Bill has every donkey, he beats it”).
The obvious thing to notice is that in defining grammaticality in terms of meaning, it goes against the classical generativist idea of syntax (grammar) being independent of semantics. But even if semantics is taken into account, it is usually in the form of some simple, algorithmic rules. That is, we still have an algorithm which can tell us whether any given sentence belongs to the set of grammatical English sentences or not (such a set is then called “recursive”). In contrast, the any-thesis would imply that the set of grammatical English sentences is not recursive. Indeed, it is not even recursively enumerable! (That would mean having an algorithm which is guaranteed to confirm that a grammatical sentence indeed is grammatical but which is not guaranteed to terminate if it is not.)
Why? Well, the set of theorems of first-order predicate logic is not recursive (assuming we have at least one two-place predicate), and having a procedure to determine the logical equivalence of sentence of the form “if any X, then Y” and “if every X, then Y” would enable us to create a decision procedure for first-order predicate logic (details in ). So the set of grammatical sentences of the fragment of English we are considering is not recursive. What’s more, if we have a recursive test for sentences which would be grammatical if it were not for the any-thesis, then it cannot even be recursively enumerable – since there is a recursive enumeration of the ungrammatical sentences of our fragment (why?), it would contradict our weaker result, as a set is recursive iff it and its complement (in a recursive set) are both recursively enumerable (intuitively speaking, one could run both enumerations in parallel and every sentence is bound to pop up in one or the other). By a similar argument, it follows that the whole set of grammatical English sentences is not recursively enumerable either.
But this flatly contradicts the basic principles of generativism and what Hintikka calls the “recursive paradigm”. If the thesis is true, it follows that grammar cannot be conceived as based on generative rules as Chomsky would have us believe. And even if you disagree with this particular thesis (i.e. you disagree with some ordering principle proposed by Hintikka), it demonstrates that you cannot a priori exclude such rules from consideration, so there doesn’t appear to be any a priori reason to assume that the generativist enterprise can be successful and restrict ourselves to generativist accounts only, as we cannot rule out the existence of some similar semantic-based criterion of grammaticality. At the very least, the Chomskian must not only present valid counter-examples against the any-thesis, but also convincing arguments that such rules simply cannot operate in natural languages.
There is much more to discuss about game semantics and Hintikka, but I think I’ll end my attempt at an introduction here, for now at any rate. I hope it has been at least somewhat informative. If anyone’s interested in Hintikka’s polemic with Chomsky, check out article  (available in its entirety on Google Books). For a bit more about the “recursive” vs. “strategic” paradigm, try  (partly available online).
As for my opinion about this whole thing, I dunno. On the one hand, it seems unrealistic to assume that people would judge that grammaticality of complicated sentences including any by testing, on some level, their equivalence with the corresponding every construction. On the other hand, the concept of grammaticality partly breaks down when we’re dealing with sufficiently complicated sentences, so the set of “all” grammatical English sentences is to a large extent an artifact of the theory anyway, a matter of how we choose to extrapolate from the actual data which consists of grammaticality judgements of relatively simple sentences. In line with this, one way of interpreting the result (assuming the thesis holds for sentences of “normal” length) which is not completely unrealistic psychologically would be to assume that we do have a procedure which raises a red flag if it notices that the any construction is equivalent to the corresponding every construction and does so quite reliably for constructions of reasonable complexity and to extrapolate our notion of grammaticality from this. And Hintikka’s way of extrapolating seems quite elegant.
References (all papers by Hintikka):
 Quantifiers in Natural Languages: Some Logical Problems (1977), in Game-Theoretical Semantics
 On the Any-Thesis and the Methodology of Linguistics (1980), in Paradigms for Language Theory and Other Essays
 Rejoinder to Peacocke (1979), in Game-Theoretical Semantics
 Paradigms for Language Theory (1990), in Paradigms for Language Theory and Other Essays