$p \to q$ vs. $p$ only if $q$

Question

The question I wanted to ask is exactly as discussed on Math SE here. What's bugging me is why the first conversion from $p \to q$ to $p$ only if $q$ doesn't make sense in everyday speech.

Here's the accepted answer on that page:

"I win the lottery only if I give you \$1 billion" is equivalent to "if I don't give you \$1 billion, then I won't win the lottery", and the latter statement is the contrapositive to "if I win the lottery, then I will give you \$1 billion."

If you have doubts about why the contrapositive of a statement is an equivalent statement, have you tried making a truth table?

My problem is, sure the converted statement makes sense when I consider its equivalent and then contrapositive, but why doesn't it make sense in the original form?

Does this disconnect point to a fundamental difference between Boolean Logic and Human Language that human beings are not able to keep apart (like in my case) without adequate practice? Perhaps it's a question more related to philosophy or psychology, in which case I apologize for posting it here.

Another explanation I came across is this one. Here the statements "I'll stay home tomorrow only if I'm sick" and "If I stay home tomorrow then I'm sick" make perfect sense. I think that might be because in this example, the cause comes after the effect, whereas in the earlier example, the cause comes before the effect.

Honestly, I'm hopelessly confused at this juncture, and any guidance is much appreciated.

Answer 1

I think that the idea of causation is the (or at least a) culprit here. In "most" implications in natural language, the implication is due to one thing causing another:

If I eat thumbtacks, I will be unhappy.

Unfortunately, this doesn't always line up with how the formal implication is interpreted. For instance, let's look at "I will only give you a billion dollars if I have won the lottery." Then:

• This sentence is equivalent to "If I (will) give you a billion dollars, then I (will have) won the lottery."

• But there's no way that me giving you a billion dollars somehow causes me to win the lottery! In fact, the only reasonable causal mechanism here is the other way around: if I win the lottery, that causes me to - at the very least - be able to give you a billion dollars (hm! big lottery here :P).

The crucial test of how to interpret an implication is: Suppose I tell you $p\implies q$. Under what circumstances do you know I was lying to you? Clearly, the only way you can be certain that I lied to you is if $p$ is true and $q$ is not.

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This raises the question of how to express causality in formal logic. One approach to this is via modal logic. In (propositional) modal logic, we have

• A set of propositional constants, $p, q, r, . . .$ representing the basic things which can be true or false.

• A set of worlds, representing possible states of affairs.

• An accessibility relation $Sees(\cdot,\cdot)$ between worlds: $u$ is accessible from $v$ (written "$Sees(v, u)$," or "$v$ sees $u$") if $v$ is "possible", from the point of view of $u$.

• Our language is then that of propositional logic, together with two new symbols (modalities) "$\Box$" and "$\Diamond$", read as "It is necessary that" and "It is possible that," respectively.

• A satisfaction relation, telling us which propositions are true at which worlds. This basically amounts to an atomic satisfaction relation, telling us which propositional letters are true at which worlds, and is then extended to all propositions via induction: for instance, $w\models p\vee q$ if $w\models p$ or $w\models q$. The only new bit is how to define satisfaction for the operations $\Box$ and $\Diamond$; the rule is $$w\models \Box\varphi\iff \forall u(Sees(w, u)\implies u\models\varphi)$$ and dually $$w\models\Diamond\varphi\iff \exists u(Sees(w, u)\wedge u\models\varphi)\iff w\models\neg\Box\neg\varphi.$$

There are many different kinds of modal logics, and ways of interpreting them; for instance, we might consider worlds to be moments in time, and the relation $Sees(w, u)$ to be "$u$ is a later time than $w$." For instance, suppose $p$ means "I have won the lottery" and $q$ means "I have given you a billion dollars." We can express "the only way I'm going to give you a billion dollars is if I have won the lottery" as $$p\iff \Diamond q.$$ Note that this lines up with our usual understanding that me winning the lottery causes me to be able to give you a billion dollars. Meanwhile, at any given world, we have $q\implies p$, reflecting the truth-functional picture of things.

Answer 2

$p\to q$ , $\neg p \vee q$, $\neg q\to \neg p$,

"$p$ is sufficient for $q$", "if $p$, then $q$", "$p$ only if $q$", "$q$ if $p$", "$q$ whenever $p$", "$q$ or not $p$", "$q$ is necessary for $p$", "not $p$ if not $q$", ...

What's bugging me is why the first conversion from $p→q$ to $p$ only if $q$ doesn't make sense in everyday speech.

Implication clauses in natural language are based on temporal cause-and-effect concepts rather than purely logical-connectives.   Our brains are wired to look for causal relations.   It takes practice and effort to avoid thinking that way.

Answer 3

Personally I've been having the same issues between the language aspect and the mathematical writing. from what I understand now: When it is "if and only if" would be the implies in both directions, whilst "only if" would be q only if p which is equivalent to p implies q. I agree that there is a problem between the boolean logic and the human language but when trying to work these out I would recommend doing a truth table. I hope I have been of some help.