Consider a statement like, "If London is in France, then London is in Asia."
AIUI, in classical/proportional logic, this is "true" because the antecedent is false. This (tenously) makes sense to me, along with the fact that $ P \rightarrow Q \Leftrightarrow ¬P \vee Q $. I'm not asking why vacuous truths are considered true, since there's endless discussion of that already.
However, the statement I opened with is clearly unsound, even if it is true. It would be incorrect, even in the context of a hypothetical/"possible world"/etc, to infer London actually being in Asia from it being in France. You cannot apply modus tollens and actually get $\{ P, (P \rightarrow Q) \} \rightarrow Q$ despite the conditional being true.
My problem with this is that my understanding of the "formal" part of formal theory is that we can do logical manipulations regardless of the specific content of the statement, as long as they have the correct forms. $\{P, (P \rightarrow Q)\} \rightarrow Q$ should always work, regardless of the specific content of the statements $P$ and $Q$. However, clearly, this doesn't work in this case.
I (think I) understand the distinction between syntatic inferences and semantic consequence, so I understand that just because $P \rightarrow Q$ doesn't mean that $P \Rightarrow Q$, but in that (this) case, why do we treat syntatic manipulations as "inferring" anything at all? Or is my understanding of formalism somehow incorrect?
IOW, what is the meaning of $\rightarrow$ as an "implication" – even in a syntatic, formalist, single-model sense – if we cannot always apply modus tollens to the result?
Best Answer
Short answer. You are misled by a confusion between :
(1) The conjunction ((P--> Q) AND P) logically implies Q
(CORRECT READING OF MODUS PONENS)
(2) If (P --> Q) is true and P is also true, then P logically implies Q.
( ERRONEOUS READING OF MODUS PONENS)
Let me try to locate precisely where is the problem.
You seem to believe this about modus ponens:
(1) P --> Q is a machine waiting to be activated
(2) "P is true" activates the machine
(3) as soon as the the machine is activated, P " infers" Q.
So , you locate the " inferential" activity in (1) , that is in P--> Q.
But, in modus ponens, the " inferential step" is not located in the conditional " P-->Q" , but in the arrow of the the whole BIG conditional.
[ ( P-->Q) & P ] ==> Q
Note : the antecedent is not (P--> Q) but the whole conjunction
What is " always true" ( tautological, valid ) in a modus ponens is not the first arrow, but the second one. ( See the truth table below). In other words, what is valid/ logical is not (P --> Q) but the link ( the relation) between the antecedent , that is the conjunction [ ( P-->Q) & P ] and the consequent , that is, Q.
If you say :
(1) London is in France --> London is in Asia
(2) London is in France
(3) Therefore, London is in Asia
you do not mean that " London is in France" ( logically) implies " London is in Asia" in case "London is in France" is true
What you mean is that
IN CASE [ the conditional (London is in France --> London is in Asia) was true and the proposition London is in France was also true ]
THEN ( logically) the proposition London is in Asia would also be true.
So, in modus ponens, the conditional that belongs to the antecedent is and always remains an ordinary material conditional.
It is the central material implication that also qualifies as logical implication ( since the whole big sentence is a tautology).
Hope it helps.
Note : More on the distinction between material implication and logical implication , see Lipschutz, Set Theory, ( at archive.org).