In my local supermarket, there is a notice with a recall, in the end it says:
This warning does not imply that the damage was caused by the producer, manufacturer, importer, or distributor
My first thought was that there are two formal readings of this sentence:
- $\neg(W \implies (P \lor M \lor I \lor D))$
- $\neg(W \implies P) \land \neg(W \implies M) \land \neg(W \implies I) \land \neg(W \implies D)$
The idea here is that the first line means the damage is caused by some entity outside of the chain $\{P,M,I,D\}$ (as implied by the warning) while the second line would be the weaker statement that there is no single entity amongst the chain for which the warning implies that it caused the damage (but the warning might still imply that one element in the chain must have caused the damage, i.e. the chain might still represent the entirety of causal agents for the damage).
However, in classical logic the two sentences are equivalent. So I wonder if this would be a good use case of intuitionistic logic, or is there some other formal language or some other translation into propositional logic better suited?
The two statements are also equivalent in intuitionist logic.
For suppose (1) holds. Now suppose $W \implies P$. Then $W \implies (P \lor M \lor I \lor D)$; contradiction. Thus, we must have $\neg (W \implies P)$. The same applies to the other 3 conjuncts.
Now suppose (2) holds. Suppose $W \implies (P \lor M \lor I \lor D)$. Suppose $W$. Then $P \lor M \lor I \lor D$. Without loss of generality, suppose $P$. Then $W \implies P$; contradiction. Thus, $\neg W$. Then $W \implies P$; contradiction.
One possible way of interpreting this statement is in terms of modal logic. Perhaps it is saying $\neg \square (W \implies (P \lor M \lor I \lor D))$.