I'm having a difficult time understanding what the meaning are with these two.
Is it correct if I have (P ⇒ Q) ∧ P and I say Q is a logical consequence. This means that whatever P may be T or F the result all comes down to what Q is?
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$\begingroup$Logical consequence means :
"every time the premise ($(P → Q) ∧ P$) is TRUE, also the conclusion ($Q$) is TRUE."
Logical equivalence means that the two formulas have the same truth value in every model.
$Q$ is a logical consequence of $(P → Q) ∧ P$ but the two are not logically equivalent.
An example of two logically equivalent formulas is : $(P → Q)$ and $(¬P ∨ Q)$. We can use a truth table to check it.
For details, see Logical consequence :
"is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is the consequence of the premises."
A premise is a statement that an argument claims will induce or justify a conclusion. In other words, a premise is an assumption that something is true.
A conclusion is the statement justified (in mathematics and logic : proved) by the premises of the argument.
$\endgroup$ $\begingroup$Welcome to MSE.
You refer to the modus ponens, a logical rule. Semantically, it says that if $P$ is true and $P\Rightarrow Q$ is true, then $Q$ is true.
The rule is also (equivalently) applied in proofs (syntax). If you can write down $P$ and $P\Rightarrow Q$ starting from the axioms and the hypothetical propositions, then you can write down $Q$.
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