Re: Universe nonemptyness assumption and Truth valuations

On Jun 5, 12:03 am, malc...@xxxxxxxxx wrote:

Does
the assumption that every structure has a nonempty universe forbid
certain truth valuations? For example any truth valuation assigning
any formula of the sort "There is some x such that x=x" the value
False.

David Ullrich makes a good point about the term 'tautology', which is
a term that comes up in your posts too. I use the word 'tautology' as
he does.

Meanwhile, I don't intend to insist on any standard meaning for 'truth
valuation', but I take it that in this context by a 'truth valuation'
you mean a function from the set of formulas into {0 1}. Furthermore,
the standard truth valuations would be those that agree with the
standard recursively defined functions where we first arrive at the
denotations of the terms then arrive at the truth value of the
formulas via the standard treatment of its connectives and quantifiers
(for formulas this requires a model and an assignment for the
individual variables, and is actually usually called 'satisfaction' or
lack thereof; for sentences this requires only a model and is called
'truth' or lack thereof called 'falsehood').

That said, yes, by considering only non-empty domains, certain truth
valuations would be non-standard if they don't agree with the standard
recursive definition. So, yes, an assignment of false to Ex x=x would
be precluded by any standard valuation (where '=' is given the
standard interpretation). Even more stark (by avoiding the issue of
whether '=' is interpreted in the standard way), let 'F' be any n-
place relation symbol of the language (n>0) (and every language must
have at least one predicate symbol). Then

Ex(Fx1..xn -> Fx1...xn)

does not evaluate to false under any standard evaluation.

We define formula to be a tautology if every truth valuation makes its
truth value True.

Again, I refer to Ullrich's point about this.

Under the assumption that we only allow nonempty
universes in our structures, would't this imply that we could weaken
the condition by restricting valuations to those allowed by such
structures?

That's what we do, aside from the matter of the word 'tautology'. A
validity is a formula that is satisifed by every model and every
assignment for the individual variables. If the formula is a sentence,
then assignments for the individual variables are irrelevent, so a
sentence is a validity iff it is true in every model, and, yes,
indeed, by "every model" we mean models with non-empty domain, as,
indeed, our very definition of a 'model' includes that the universe of
the model is non-empty.

If we allow non-empty universes, then that is quite a different
treatment and definition.

Note: By the way, Shoenfield (you said you're using Shoenfield, I
think I recall) says 'valid in the model' (or something like that)
where many other authors (and I think more usually) say 'true in the
model'. So don't let that confuse you. His 'valid in the model' does
NOT mean what the other authors mean by a 'validity' or 'valid
formula', which is a formula that is satisfied (or true, if a
sentence) in EVERY model.

MoeBlee


.

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