# Understanding this particular usage of "only if"

I am confused about the usage of "only if" in the below sentences:

Only if the two votes are in agreement will the deal go through.

Only if Ted and Kate both bring pies to the picnic, will Bill bring a birthday cake.

Fred: Yo, I hooked up with this gorgeous girl from work today. I'm wondering if I should start dating her seriously?

Only if I am motivated and don't get too tired do I diet.

The sentences from the examples follows this pattern: "only if P, Q." I learned from my propositional logic class that statements of the form "only if P, Q" precisely means "if Q then P." But sometimes I don't think this is the case in everyday conversations. My instinct tells me that all of these "only if" examples above mean will the same thing when the word "only" removed. So sometimes "only if P, Q" actually means "if P, Q." I am not sure about the function of the word "only."

Only means what it says; it emphasises that P is the only circumstance in which Q will happen.

"If the two votes are in agreement the deal will go through" could describe a routine procedure rather than a special case.

Bill might say "If you two are bringing pies, I'll bring a cake," and it would be an offer or suggestion rather than a strict condition.

"only if" sentences can often be usefully rephrased with negatives.

Only if the two votes are in agreement will the deal go through. <=> If either of the two votes are not in agreement then the deal won't go through.

And this could be further analysed, with positive statements and a necessary change in perspective:

If the deal does go through then the two votes must have been in agreement.

Now, from a formal "propositional calculus" standpoint, you can't deduce that "If the two votes are in agreement then the deal will go through."

Only if all five permanent members of the Security Council approve (or abstain) will the resolution be adopted. (But on its own, this is not sufficient, as eight votes by non-permanent members is also sufficient to prevent the adoption of a resolution)

Only if I am motivated and don't get too tired do I diet. <=> If I am either unmotivated, or I get too tired, I don't diet. <=> If I diet then I must have been both motivated and not too tired.

I think you may be confusing logical implication for semantic meaning. While you can logically rewrite sentences to form equivalent logic structures, it changes the meaning of the utterance.

For example, logically these two statements are equivalent:

• "Only if Ted and Kate both bring pies to the picnic, will Bill bring a birthday cake."
• "If Bill brings a birthday cake, Ted and Kate will have both brought pies to the picnic."

These statements are very distinct though, and anyone untrained in logic will struggle to figure out in the moment whether these statements actually mean the same thing. It's made even harder by the fact that you need to use the correct tenses to make these statements functionally equivalent. Logical equivalency is complicated and is not something native speakers are taking into account in normal speech.

While understanding these implications can help you craft more correct and interesting arguments, I think you may want to set it aside for the purpose of basic sentence interpretation.