Definition for logical consequence

Question

What is the correct definition for logical consequence?

I came across a note which says,

It makes me think that what we have to do is check whether the compound proposition Knowledge base -> B is a tautology.

But according to the definition in How to prove logical consequence?

I feel that I need to check only the places where (Knowledge base, that is, A1 and A2 and ...and An, is havving the truth value "True". If the corresponding truth value for B then is also "True" then it is a logical consequence?

Please help me to clarify this idea.

Thanks a lot in advance.

Answer 1

These two ideas are equivalent:

Since a conditional $A \to B$ is false if and only if $A$ is true and $B$ is false, we get:

$(A_1 \land A_2 \land ... \land A_n) \to B$ is a tautology

iff

it is impossible for $(A_1 \land A_2 \land ... \land A_n) \to B$ to be false

iff

it is impossible for $A_1 \land A_2 \land ... \land A_n$ to be true and $B$ to be false

iff

it is impossible for all of $A_1$ through $A_n$ to be true and $B$ to be false

iff

whenever $A_1$ through $A_n$ are all true, $B$ will be true as well

So you can use either definition of logical consequence.