# How can a material conditional feel unidiomatic, when LOGICALLY EQUIVALENT to another material condition that is intuitive?

Peter Smith's rhetoric question on p. 155 of his An Introduction to Formal Logic (2 edn 2021)

(As so often, however, ordinary language has its quirks: so can you think of examples where a proposition of the form if A then C seems not to be straightforwardly equivalent to the corresponding A only if C?)

is acknowledged on p. 86 by Ernest Lapore in Meaning and Argument An Introduction to Logic through Language (Revised 2 edn).

All that is being said is that ‘only if ’ statements are conditionals. To see the difference between distinguishing statements on the basis of logical and nonlogical respects, consider pairs (26)–(29).

1. If water is boiled, it evaporates.

2. Water is boiled only if it evaporates.

3. My pulse goes up only if I do heavy exercise.

4. If my pulse goes up, then I do heavy exercise.

The even-numbered statements are grammatical (or acceptable), but the odd ones seem peculiar. [emphasis mine] (27) affirms straightforwardly that water being evaporated is a necessary condition of water being boiled. Intuitively, (26) in English seems to be less strong. It affirms that water evaporates when boiled, but it seems neutral as to whether it has to. Nevertheless, differences between the odd- and even-numbered statements do not establish that the odd statements should not be symbolized as the same material conditional. Consider the truth conditions for each statement. (26)–(29) are all false under exactly the same conditions: their antecedent is true, and their consequents false. So we will adopt the convention of symbolizing the members of each pair identically in PL. Although, given our convention, ‘⊃’ adequately symbolizes both ‘if, then’ and ‘only if ’ in PL, we should be careful not to presume that these expressions are equivalent in every respect.

#### But Lapore doesn't expound in simple English why "the odd ones seem peculiar"? Why can 27 and 29 be syntactically infelicitous or unnatural, but not their logically equivalents?

I already know, and am NOT asking about, why P if Q ≡ Q if only P.

• Natural language "sensibility" is by definition not a rigorous concept... Maybe we have to organize a poll considering only native English speakers. Feb 8, 2023 at 8:46
• But, in general, the "logical form" expressed by mathematical formulas not always is a perfect model of natural language expressions. It is well known that "if P, then Q" as modeled by propositional calculus does not fit well with speaker's sensitivity: "if the Moon is made of green cheese, then...". Feb 8, 2023 at 10:15
• Because they are not, strictly speaking, logically equivalent, but why not is not expressible in simple English. The conditional of natural language is what is called indicative conditional, and the material conditional is only its closest truth-functional approximation. In some contexts the two come apart more than in others because the truth value of indicative conditionals is determined by context in addition to truth values of its terms. So in refined theories, the crude "equivalence" on the basis of truth conditions fails. Feb 8, 2023 at 10:15
• Welcome to SE. This is an intricate topic and not as easy to understand as some philosophers think. However, your questions at the end are misleading. Lapore does explain the difference between 26 and 27, but perhaps not in simple English. But the differences that he's getting at are not syntactic infelicity or being unnatural. He doesn't think that the logical versions of them are equivalent, just that the differences between them aren't important from the point of view of logic. I suggest that if you change these to ask for an explanation of what the differences are and why they are not imp Feb 11, 2023 at 17:56