Modus Ponens
1. If P then Q
2. P
Ergo,
3. Q
1a. If it rains then the ground will be wet
2a. It rains
Ergo,
3a. The ground will be wet
If the premises 1 and 2 are true, it's impossible for the conclusion, 3, to be false. — TheMadFool
What makes it impossible that 3 is false? — Metaphysician Undercover
1. If P then Q
2. P
Ergo,
3. Q — TheMadFool
3 has to be true; no possible world exists where 1 and 2 are true with 4 false. — TheMadFool
As for temporal aspects of sufficient and necessary conditions and causality, we can forgo discussion on them for they muddy the waters. — TheMadFool
The form of premise 1, as a conditional statement, is crucial to the validity of the conclusion, as what is used to determine the truth table. For example, if the premise was changed to a biconditional, the truth table would be different. — Metaphysician Undercover
I say "within" means a necessary part of the definition, "necessary" being relative to that specific logical proceeding. — Metaphysician Undercover
Logical consequence (also entailment) 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 entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?" — Wiki
. . .Relevance logicians have attempted to construct logics that reject theses and arguments that commit “fallacies of relevance”. Relevant logicians point out that what is wrong with some of the paradoxes (and fallacies) is that the antecedents and consequents (or premises and conclusions) are on completely different topics. . . . — SEP
Syntactic consequence does not depend on any interpretation of the formal system.
. . .
A formula A is a semantic consequence of a group of premises G where the set of the interpretations that make all members of G true is a subset of the set of the interpretations that make A true. — Wiki with some liberties
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