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To emphasise the pertinence of the following to all other languages wherewith Logic can be communicated, I use grey font colour to denote the abstract lexemes.

Abbreviate Necessary Condition to NC, Sufficient Condition to SC. I already know that:

  1. If P, then Q.  (In French: 'Si P, alors Q.'   In German: 'Wenn P dann Q.')
    =  P only if Q.  ('P seulement si Q.'   In German: 'P nur wenn Q.')
    = P is a SC for Q.    = Q is a NC for P.

The above exhibits that if always precedes a SC, and only if a NC. So:

  1. Does the adverb only cause the logical difference between if and only if?

  2. If the answer to 6 is 'yes', then how does only do so?

My conjecture: 8. is that only causes the Adverbial only if to incorporate more Conditions.
9. matches the fact, and the picture below depicting, that any SC ⊆ its NC:

enter image description here

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    http://linguistics.stackexchange.com/questions/20206/is-the-logical-meaning-of-only-if-produced-by-the-adverb-only-that-widens-t?noredirect=1#comment47156_20206 compelled me to post here. –  Jan 03 '17 at 19:47
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    We may say that the presence of "only" reverse the "point of view": it acts as a sort of *contraposition*. **P→Q** can be read as : "if P, then Q", as "Q if P" and as "P only if Q". An alternative is to use "when" : "When P, then Q", "Q when P" and "P only when Q". Consider now "Q when P": if we have P, we are guaranteed that also Q holds. With "P only when Q", the "point of view" is reversed : we cannot have P without Q. – Mauro ALLEGRANZA Jan 03 '17 at 21:28
  • Related: *[Are "If P then Q" and "Q only if P" equivalent?](http://philosophy.stackexchange.com/q/28331/6692)* and the [Math.se posts](http://math.stackexchange.com/questions/617562/conditional-statements-only-if) its accepted answer [links to](http://math.stackexchange.com/questions/311192/how-does-if-p-then-q-have-the-same-meaning-as-q-only-if-p). – Dan Bron Jan 04 '17 at 21:14
  • Does this answer your question? [How can we reason about "if P then Q" or "P only if Q" statements in propositional logic?](https://philosophy.stackexchange.com/questions/4089/how-can-we-reason-about-if-p-then-q-or-p-only-if-q-statements-in-proposition) –  Feb 19 '23 at 20:11

2 Answers2

1

Here, the logical meaning is not "produced" by the adverb. The phrase "only if" is to be regarded as a single lexical item with an accepted meaning, the one you mention in your question.

Carl Masens
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In regards to

  1. Does the adverb only cause the logical difference between if and only if?

linguistically in this specific sentence "only" is what causes the difference between "if" and "only if" however I think that this raises a lot of the questions that come with using natural language to model formal logic (or vice versa, this example is slightly dated but see section 6). There are other ways to express the concept "P→Q" in English:

"P only if Q"

"P implies Q" / "if P, then Q"

"Q is a necessary condition for P"

"Q whenever P"

You can see that some of these do not use the term "only" so really we can't only focus on "only" in your example. Natural language is not modeled after formal logic and it is incredibly difficult to get the two perfectly in sync. Where the difference in "if" and "only if" comes from depends on if you're asking linguistically or logically. Linguistically, it comes from the word "only" which is, in English, the attempt we make to capture the essence of the logical principle we're discussing. If you mean logically, then the discussion needs to be shifted to a discussion about proof theory and formal syntax.

In regards to

My conjecture: 8. is that only causes the Adverbial only if to incorporate more Conditions.

the opposite is true. "Only" causes less conditions to apply and the point you made in 9 is backwards. If P is a sufficient condition for Q (P→Q) that means that there is a set of conditions for which Q is true and those include the conditions in which P holds. However there are also conditions in which Q is true when P does not hold. If P is a necessary condition for Q (Q→P) this means that the only conditions in which Q is true are conditions when P holds. This means that, in regards to P being a condition for Q, NC ⊆ PC. Reference for the last paragraph.

In summation, linguistically you are correct that "if" comes before a SC and "only if" comes before a NC. However, those exact expressions are not necessary to describe a material implication (→) (we can use other sentences, such as the ones I outlined above). This means that logically "only" is just an artifact of the natural language we're using that helps us identify an NC.

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