:cool:

So to theorem 18. A given set of particular sentences cannot imply a sentence that is not particular relative to that given set.

In this version, a consequent of some set of particular sentences inherits particularity from that set. So this new version bypasses Vranas' objection that the Barrier says nothing about the particularity of the mixed conditional - it says that they can be derived only if the consequent is particular.

Identities between Names

In the discussion of tense fragility, the definition of stuff that was fragile and stuff that was past was the same. The difficulty with this is made evident in the section "Identities between Names" . That a=b should be true in both the past and the future, but appears to be true only in the past.

Russell makes the variations she does in order to formalise what in a natural language we might call an eternal temporal status. The structure she creates can accomodate a wider variety of tensed sentences, including those that survive both past and future switching, (eternal), those that survive past switching ("it will rain"), and those that survive future switching ("it rained").

That enables eternal truths that are not tautologies, such as a=b.

This is all part of the interplay between formal and natural languages, in which each is used to shed light on the other. Russell continues this with a brief discussion of an area of natural language that is difficult to formalise - using propositional attitudes as the example. She then looks at an example in which formalisation has moved to natural languages. Kripke developed his model theory for modal logic by having the individual constants keep their referent across possible worlds, which become the now familiar process of rigid designation used in natural languages. The point here is simply that formal logic can inform our usages in natural languages, and that natural language informs formal logic.

So Russell is suggesting that we might take the lesson of her formal approach and apply it back to natural languages.

So in Definition 27 she defines a particular sentence, not as being about some particular individual, but as not changing when further individuals are added to the situation. Her example is "Aristotle is a philosopher". This is about a particular individual, and so particular in a naive sense, but also, adding more individuals, philosophers or otherwise, will not change it, so it's particular in the way Russell would have us speak. But "Aristotle is the only philosopher" would become false were we to add Plato. So it is universal - a universal sentence being one that can be made false. by adding new individuals (Definition 28).

The definitions used in the formal version thus lend themselves to parsing in a natural language. A set of sentences is particular if, when the sentences are true, adding more objects does not make them false (Definition 29); and particular relative to some other set of sentences in the case in which the combination of those sentences are true, adding more individuals cannot make them false (30)

The barrier theorem 31 might be that given a set of particular sentences, no sentence that is not particular relative to that set may be a consequence of those sentences.