Some material conditionals are not valid inferences, so how does $\rightarrow$ infer anything?

Question

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?

Answer 1

I disagree with your conclusion that the rule $\{P,P\rightarrow Q)\}\vdash Q$ does not apply here. (By the way, this rule is called "modus ponens". "Modus tollens" is the rule $\{P\rightarrow Q,\lnot Q\}\vdash \lnot P$.)

The statement "If London is in France, then London is in Asia" is true under a particular meaning of the non-logical words involved ("London", "France", "Asia", maybe even "is in"...). It is certainly not true under every interpretation of these words. So, as you correctly observe, letting $P = \text{London is in France}$ and $Q = \text{London is in Asia}$, the implication $P\rightarrow Q$ is not valid (i.e. true under every interpretation of the non-logical words involved).

But no matter what interpretations these words have, the entailment $\{P,P\rightarrow Q\}\vdash Q$ is sound.

This entailment says that if $P$ is true and $P\rightarrow Q$ is true, then $Q$ is true. Ok: Imagine $$P = \text{London is in France}$$ is true. And let's also imagine $$(P\rightarrow Q) = \text{If London is in France, then London is in Asia}$$ is true. Well, we're forced to conclude that London is also in Asia, i.e. $Q$ is true.

You write "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." Can you explain why you feel this way? Can't you imagine a world in which $P$, $P\rightarrow Q$, and $Q$ are all true (e.g. a world in which London is a city in France, which is a country in Asia)?

Answer 2

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)

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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).

Answer 3

Modus Ponens is and always is a valid inference. The definition of validity is that if the premises are true, then the conclusion must be true as well. And, as the truth-table below indicates, it is the case that if the premises are true, then the conclusion is true as well (the premises are true only in line 1, and in that line the conclusion is true as well):

\begin{array}{cc|ccc} P&Q&P&P\to Q&Q\\ \hline T&T&T&T&T\\ T&F&T&F&F\\ F&T&F&T&T\\ F&F&F&T&F\\ \end{array}

Now, remember that the lines in a truth-table reflect all possible worlds, only of of which is our world. And, in the case where $P$ is 'London is in France' and $Q$ is 'London is in Asia', both $P$ and $Q$ are false, and hence our world is represented by line 4. The other $3$ lines represent imaginary worlds where London might be in France and/or London might be in Asia.

Now, here is an important definition: An inference is sound if and only if it is both valid, and has all true premises.

So, notice that in line $1$, you have both premises true, and indeed for that world, the conclusion is true as well. As such, we can say that in world $1$ we can soundly infer that London is in Asia from the premises that 'London is in France', and 'If London is in France, then London is in Asia'.

In our world, however, the premises are not true. So, this argument is not sound for our world. The argument is still valid though. Indeed, validity is not relative to any specific world, as opposed to soundness.

In sum: in our world, the argument is valid .. as it is in any world. In our world, the argument is not sound, however. Therefore, we certainly can't conclude from the argument that the conclusion is true ... because we can only do that if we know the argument is sound .. which it isn't.