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Logical Relations
The most important idea in logic:
Validity of an argument.
Logical Relations
The most important idea in logic:
Validity of an argument.
Validity is a logical relation
between statements that make up an argument.
Logical Relations
The most important idea in logic:
Validity of an argument.
Validity is a logical relation
between statements that make up an argument.
Namely that assuming premises are T
the conclusion has to be T.
Entailment
Statement A entails statement B
iff
It is not possible for A to be T and B to be F.
Entailment
Statement A entails statement B
iff
It is not possible for A to be T and B to be F.
iff
The argument A | B has no counterexample.
Entailment
Statement A entails statement B
iff
It is not possible for A to be T and B to be F.
iff
The argument A | B has no counterexample.
iff
The argument A | B is valid.
Entailment
Statement A entails statement B
iff
It is not possible for A to be T and B to be F.
iff
The argument A | B has no counterexample.
iff
The argument A | B is valid.
Summary: Entailment is “one premise” validity.
Entailment
More Generally:
we can speak of a group of statements entailing another:
A, B, C entails D
iff
A, B, C | D is a valid argument.
Testing Entailment
To show A entails B ...
with a table:
There is no A=T, B=F row.
with a proof:
Given A, prove B.
with a tree:
The tree for A, -B closes.
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