Firstorder validity and consequence
In the preceding chapters we have discussed the notions of logical truth, logical
consequence, and logical equivalence.
We also discussed the notions of tautology,
tautological consequence, and tautological truth.
In this chapter we want to discuss the notions of firstorder validity, firstorder
consequence, and firstorder equivalence.
But first, a review of the earlier terms:
A logical truth (logical necessity) is a sentence that is a logical conseqence of any set of
premises, even the empty set.
It is impossible for a logical truth to be falsified.
a = a
SameShape(a, a)
SameSize(b, b)
Logical consequence describes a situation in which a sentence is true in virtue of a given
set of premises.
If a sentence is a logical consequence of some set of premises, then it is
impossible for that sentence to be false if all the premises are true.
SameShape(a, b)
Cube(a)
Cube(b)
Logical equivalence describes a situation in which two sentences have the same truth
value in all possible circumstances.
If two sentences are logically equivalent, it will be
impossible for one to be true while the other is false.
LeftOf(a, b)
⇔
RightOf(b, a)
BackOf(c, d)
⇔
FrontOf(d, c)
There is no precise method to test for logical truth, consequence, and equivalence.
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 Fall '06
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 Logic, consequence

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