Clarification about validity

[Guang Wu]

I just want some clarification regarding if a set is valid, satisfiable
and unsatisfiable.
If a set A is valid then all formulas in A for any valuation t are true.
If a set A is satisfiable there is some formula a such that for a given
valuation t a^t is true.
If a set A is unsatisfiable all formulas in A for any valuation t are false.
Are these correct?

[Curtis Bright]

Post by Guang Wu
If a set A is valid then all formulas in A for any valuation t are true.

Validity is not usually defined on sets, just on individual formulas.

Post by Guang Wu
If a set A is satisfiable there is some formula a such that for a given
valuation t a^t is true.

We actually have something stronger: a valuation t for which A^t is true
(a^t is true for ALL a in A).

Post by Guang Wu
If a set A is unsatisfiable all formulas in A for any valuation t are false.

Unsatisfiable means not satisfiable, so no valuations t with A^t true.
However, some individual formulas of A may be satisfiable.

Curtis

[Hao Chen]

Post by Curtis Bright

Post by Guang Wu
If a set A is valid then all formulas in A for any valuation t are true.

Validity is not usually defined on sets, just on individual formulas.

Post by Guang Wu
If a set A is satisfiable there is some formula a such that for a given
valuation t a^t is true.

We actually have something stronger: a valuation t for which A^t is true
(a^t is true for ALL a in A).

Post by Guang Wu
If a set A is unsatisfiable all formulas in A for any valuation t are false.

Unsatisfiable means not satisfiable, so no valuations t with A^t true.
However, some individual formulas of A may be satisfiable.
Curtis

So there is a difference between formular unsatisfiability and set
unsatisfiability? The definition for a WFF to be unsatisfiable is that
for every t, a^t=0 where a is WFF.

[Curtis Bright]

Post by Hao Chen
So there is a difference between formular unsatisfiability and set
unsatisfiability? The definition for a WFF to be unsatisfiable is that
for every t, a^t=0 where a is WFF.

Analogously, the definition for a set to be unsatisfiable is that for
every t, A^t=0 where A is a set.

Naturally there's some difference there, but the definitions do
correspond when A consists of a single formula.

Curtis

[Hao Chen]

Post by Curtis Bright

Post by Hao Chen
So there is a difference between formular unsatisfiability and set
unsatisfiability? The definition for a WFF to be unsatisfiable is that
for every t, a^t=0 where a is WFF.

Analogously, the definition for a set to be unsatisfiable is that for
every t, A^t=0 where A is a set.
Naturally there's some difference there, but the definitions do
correspond when A consists of a single formula.
Curtis

So a set can be considered as a intersection of WFFs? i.e. A^t is true
iff all WFF in A is true under t?

[Curtis]

Post by Hao Chen
So a set can be considered as a intersection of WFFs? i.e. A^t is true
iff all WFF in A is true under t?

Not sure exactly what you mean by the first question, but yes, that is
the definition of A^t for a set A.

Curtis

[Hao Chen]

Post by Curtis

Post by Hao Chen
So a set can be considered as a intersection of WFFs? i.e. A^t is true
iff all WFF in A is true under t?

Not sure exactly what you mean by the first question, but yes, that is
the definition of A^t for a set A.
Curtis

I meant that when it comes to some valuation t, A can be seen as an
"and-set". i.e. if one formula in A is false, then the entire set is
false under evaluation t; and the only way to get A to be true is that
all formulas in A are true under t.