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What is the correct definition for logical consequence?

I came across a note which says,

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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.

Hasini
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    You can define logical consequence using the syntax: for that you establish some rules of inference that will tell you when you can conclude a logical sequence. Other approach would be via semantics: given propositions $p, q$, we say that $q$ is a logical consequence of $p$ if and only if $q$ is true whenever $p$ is true (of course one would have to have a notion of truth). Side note: this two approaches are (in a sense) equivalent. – Air Mike Feb 06 '24 at 03:04
  • Thank you @AirMike But in the second definition it says check for the situations where entire Knowledge base is true and in those situations B has to be true. But according to first when Knowledge base is false and B is true also, the overall result is true. That's where I don't understand. – Hasini Feb 06 '24 at 03:42

1 Answers1

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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.

Bram28
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  • Thank you @Bram28 But in the second definition it says check for the situations where entire Knowledge base is true and in those situations B has to be true. But according to first when Knowledge base is false and B is true also, the overall result is true. That's where I don't understand. – Hasini Feb 06 '24 at 03:41
  • @Hasini Since a conditional is only False when the 'if' part is true and the 'then' part is false, that means that if the 'if' part is false, the whole conditional is true. – Bram28 Feb 06 '24 at 04:20
  • Thank you @Bram28 okay. it is ok with the first definition, but isn't it giving a problem when compared with the second definition? Because it says to check for the places where only if part is true, but according to first we are checking places where the if part is false too? – Hasini Feb 06 '24 at 04:30
  • @Hasini I just explained that in my previous Comment! If the 'if' part is false, the whole conditional is true. – Bram28 Feb 06 '24 at 22:24