# Equivalent forms of “if p then q”

I'm wondering if "If p then q" is equivalent to "p unless q" or "p or not q" in regards to philosophy?

• Absolutely. You may always evaluate truth tables, but consider the truth conditions of the conditional; what does it mean for the implication to be true? It CANNOT be the case that p is true and q is false, yes? Hence (so to speak), ~(p and ~q) – Ryan Goulden Feb 4 '19 at 3:22
• We haven't learned truth tables or the symbolic meanings behind these conditionals so that's where I became confused. I'll try to take what you said and see what I find! Thank you – Katie Summers Feb 4 '19 at 3:26
• Something is wrong here. "If p then q" is p → q, "p unless q" is q → ¬p, and "p or not q" is classically equivalent to q → p. They are not even equivalent in elementary logic. – Conifold Feb 4 '19 at 5:46
• @Conifold Erm, no. "P unless Q" is not "Q--> ¬P"!!! For example: "He'll definitely be there unless he's ill" is not the same as "If he's ill, he won't be there"!!! :-) – Araucaria - Not here any more. Feb 5 '19 at 0:44
• @Araucaria Would ¬q → p work better? My sense of "unless" is very vague, I am afraid. – Conifold Feb 5 '19 at 1:14

In classical propositional logic, "if P then Q" is equivalent to "not P or Q" and to "not (P and not Q) and to "P only if Q". 'Unless' is taken to be equivalent to the inclusive 'or'. So in your two examples, "if P then Q" is not equivalent to "P unless Q" nor is it equivalent to "P or not Q".

In classical logic this kind of conditional is called 'material implication' and it is a truth function, which is to say that the truth of the conditional depends only on the truth values of P and Q. In practice, ordinary English conditionals do not always, perhaps do not often, behave like this. Conditionals usually express some kind of connection between P and Q, so their truth depends in some deeper way on what P and Q mean. Treating conditionals as truth functions runs into highly counterintuitive examples that are sometimes called the paradoxes of material implication. In reality, they are not paradoxes at all, just examples of conditionals that are not material implications.

Another point to notice is that conditionals often carry an implicature that the antecedent part is a precondition for the consequent part. For example, we hear a difference between:

1. If you learn to play the cello, I'll buy you a cello.
2. You'll learn to play the cello only if I buy you a cello.

or between

1. Mary will continue to love John unless he goes bald.
2. John will go bald unless Mary continues to love him.

These examples are from David Sanford's book "If P then Q".

• "if p then q" is logically equivalent to "p only if q". – Katie Summers Feb 5 '19 at 1:48
• @Araucaria The two are definitely equivalent. I'm not sure why you think there is a problem. "If P then Q" states that the truth of P is sufficient for the truth of Q, while "P only if Q" states that the truth of Q is necessary for the truth of P, which is the same. – Bumble Feb 5 '19 at 3:06
• Sorry, I was getting my Ps and Qs mixed up having been considering a similarlooking but totally different problem/comparison recently! I do apologise. (If you give your post a little edit - a comma or something - I`ll reverse my entirely erroneous downvote). Note to self, don´t comment late night from the pub. – Araucaria - Not here any more. Feb 5 '19 at 11:44
• (FWIW, I´ve been looking at when/whether "Only if P, Q" entails "If P, Q", which surprisingly perhaps is not the same in natural language as "Q, only if P" depending on the presence or absence of subject-auxiliary inversion in the main clause of the former example.) – Araucaria - Not here any more. Feb 5 '19 at 12:05
• @KatieSummers Yes, sorry. Was completely confusing myself with something completely different with that particular comment! :( – Araucaria - Not here any more. Feb 5 '19 at 12:05

The conditional `if P then Q` is not equivalent to `P unless Q`.

... Unless Q literally means : If not-Q then ...

In propositional logic, `P unless Q` can be translated : `If not-Q then P`.

Using Modus Tollens we can derive the following conditional : `If not-P then Q`.

So, `P unless Q` is not equivalent to `if P then Q`, but to `if not-P then Q`.

Always use examples :

• You breathe unless you are dead (P unless Q)
• If you are not dead then you breathe (if not-Q then P)
• If you do not breathe then you are dead (if not-P then Q)

Rule :

• Make the negation of what comes after the `unless` an antecedent of the conditional if..then.. (after if)
• Make what comes before `unless` a consequent of the conditional if..then.. (after then)
• Ah, you are right, thank you .. How didn't I see this mistake. Best ! I will correct my answer. – SmootQ Feb 4 '19 at 15:26
• Done, all i did is change change the place of not- , Thank you! – SmootQ Feb 4 '19 at 15:31
• It would be easier to make a universal rule: the verbiage after the term UNLESS gets the negation. So p unless q can be translated as if p then not q OR if q then not p (using the Transposition rule). – Logikal Feb 4 '19 at 18:39
• There is no need to rearrange the proposition. You can if you like but it is NOT a MUST. For instance, If you are my friend you will never hurt me unless I threaten your life can be written as. (F --> ~T) or (T --> ~F). F stands for the antecedent you are my friend and the consequent is I threaten your life. The swap is not required and that needs to be out in the open. The proposition makes sense as is without the switch. – Logikal Feb 5 '19 at 15:54
• @Logikal , I know it is not required, the reason I did that is because 1) it was easier for me to start with `(if not-Q then P)` as I am not an English native speaker, and I have to go through the translation in my language in order to find an easier logical way to think about it 2) , then I changed it to `(if not-P then Q)` as the question puts P in the antecedent . Best ! – SmootQ Feb 5 '19 at 16:40

if p then q should be equivalent to q unless not p

To extend on the answer below and give an example:

p = your're alive q = you breathe.

So: If (you're alive) then (you breathe) or: (You breathe) unless not (you're alive)

Or in more common words: Your breathe unless you're not alive.

There is another interesting thing you might consider. But there's a warning: The following thoughts are true in a programming context, but mey well be false in a philosopy context. So it is up to you to decide.

In programming you can replace "if p then q" by "p and q". This is because the term is evaluated from left to right. So if p evaluates to false, then q does not need to be evaluated. Because no matter to what q would evaluate, the result would be false anyway.

Now if you would consider this as "equivalent", then you may apply all kind of boolean algebra to it. So f.ex. negate inputs and outputs and replace the and with or.

But, as already said, in philosophy this may well be different.

• `q unless not p` is equivalent to `if not p then not q` and not `if p then q` . – SmootQ Feb 4 '19 at 10:32
• if p then q is equivalent to if not p then not q – Siegfried Feb 4 '19 at 10:39
• this is the truth table of `if p then q` , the first 2 digits are for p and q, the third for the conditional (00-1, 01-1, 10-0, 11-1) , this is the truth table for `if not p then not q` (00-1, 01-0, 10-1, 11-1) , there is a difference : Using modus tollens , if p then q is equivalent to (if not q then not p) – SmootQ Feb 4 '19 at 10:47
• I think "if p then q" would be equivalent to "if not q then not p". You might try putting this into a truth table. Here is one: web.stanford.edu/class/cs103/tools/truth-table-tool Input "(p=>q)<=>(~p=>~q)" doesn't give a result with all "T". But this one does: "(p=>q)<=>(~q=>~p)". – Frank Hubeny Feb 4 '19 at 10:47
• @Siefried `if and only if P then Q` is equivalent to : `(if P then Q) AND (if Q then P)` .. it is called a biconditional – SmootQ Feb 4 '19 at 12:22