Logical truth and logical consequence

Question

I understand the concept of logical consequence, for example:

1.All persons are human

2.I am a person

Conclusion: I am human.

If 1 and 2 are true, conclusion must be true.

My question is about logical truth

would this be a logical truth for example?

1. My name is Drx
2. Drx is batman

conclusion: 100 = 100

As I understand it the conclusion is true no matter what the premises are. But my definition says:

A logical truth is a sentence that is a logical consequence of any set of premises. That is, no matter what the premises may be, it is impossible for the conclusion to be false.

Is the conclusion above really a logical consequence of its premises even when the premise doesnt have anything to do with the conclusion?

Sorry if this got confusing, if the above is not a logical truth please give me an easy example :D

Thanks!

Answer 1

With logical truth today we mean a valid formula, i.e. a formula that is true in every interpretation.

In propositional logic, a valid formula is called: tautology.

Examples:

$A \to (B \to A) \ , \ A \lor \lnot A$, etc.

Tautologies can be identified via truth table algorithm.

In general, the link between logical consequence and "logical truth" is the following:

if the formula $C$ is logical consequence of the formulae $A$ and $B$, i.e.

$A, B \vDash C$,

then $(A \land B) \to C$ is valid.

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With your example:

1.All persons are human

2.I am a person

Conclusion: I am human

we have that: "If all persons are human and I am a person, then I am human" is a logical truth.

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The formula $x=x$ is a (first-order) axiom for equality and thus is valid.

Every instance of it, like e.g. $100=100$ and "Socrates=Socrates" is a logical truth.