# Have I totally misunderstood conditionals all along?

I've been talking to a friend of mine about modus ponens (and modus tollens) and I'm wondering if I have totally misunderstood something fundamental about conditionals. Here are the two opposing positions:

#1 Apply truth value to the conditional premise as a whole, as follows.

P1. If I drop a ball ('p'), then it will hit the ground ('q')
P2. I dropped a ball ('p')
C. Therefore it hit the ground ('q')

In this case, I assign P1 to be true, and therefore if p is true, then the consequent q is DETERMINED to be true, ('C. Therefore it hit the ground').

#2 Apply truth values to both p and q variables individually, as follows.

P1. If I drop a ball ('p'), then it will hit the ground ('q')
P2. I dropped the ball
P3. It did *NOT* hit the ground

In this case, I'm asserting that p is true, and q is false, and that these assertions DETERMINE that P1 is therefore false. Now, I understand that if P2 is actually true, then P1 or P3 must be false, that's fine.

My issue here is with how to treat the conditional. Is case #1 correct, where p determines q, or is case #2 correct, where p and q determine the truth value of the conditional?

My understanding was along the lines of case #1 - that we assign a truth value to the conditional as a whole - "yes, it's true that if I drop a ball it will hit the ground", and if I then assert p as true, then q becomes true. If I don't assert p to be true (or assert p to be false) then q is basically "undefined".

In the second case, I assign values for both p and q, and that determines the truth value of the conditional. The problem arises when I assert p to be false. That is, I did NOT drop the ball, and it may or may not have hit the ground. The truth table for "p => q", says that if p is false, then "p => q" is true. That makes zero sense to me, because I haven't dropped the ball - I have no reason to say that the conditional is true, because I don't have any evidence for it.

So in that case, what does it even mean to say that the conditional is true? Can I then go around saying "I've proven that if I drop a ball, it will hit the ground", even though I've never dropped a ball to test it?

Another thought I had was that maybe I'm conflating the "if/then" in a conditional with a "cause/effect" kind of relationship, and I'm not sure that's proper.

Hopefully someone can make sense of my ramblings and help me out.

• You have not "totally misunderstood" conditionals. There are just two+ different ones. The one used in everyday language is indicative conditional. It corresponds to your intuitions more or less, people tend to consider it meaningless when the premise is false. The one used in logic is called material conditional. It is defined for any truth values of the premise for technical convenience. It is true only in the sense that if you did not drop the ball then there is no counterexample. Feb 20 '21 at 7:23
• Thanks! I have some new terms to research :) I've also discovered the term "vacuous truth" which I guess applies to case #2 when p is false. I'm still unsure about whether case #1 - I assign a truth value to the conditional, or #2 - I assign truth values to BOTH p and q is correct. Feb 20 '21 at 9:12
• @RomanHoliday So do the answers in the linked question sufficiently address your concerns? I ask because if so, I would mark this question as duplicate so that people finding your question will automatically be pointed towards the already existing answers there. Feb 20 '21 at 10:00
• @PhilipKlöcking - Not exactly, no. My major concern is with the correct PROCESS. All my life, I've been doing case #1 - I apply a truth value to the conditional as a whole. If I assign the conditional as true, then what I'm saying is "if p is true then it DETERMINES that q is true". Case #2 assigns truth values to both p and q, and then that DETERMINES the truth value of the conditional. Which is correct? Feb 21 '21 at 1:07

What you stumbled upon is the classical "formal logic implication vs material implication" dilemma.

What formal logic does is just it formulates the relations between the truth values of the (formal-logically related) predicates it considers. Indeed, classical FOL system holds that from falsity follows whatever. And remember that it has nothing to do with the notions of cause and effect, it's just a syntax game, if you will. So, all the "formal logic implication" tells us about the cause and effect is that if you assign some truth values to the predicates of the sentence, the whole sentence will acquire some other truth value.

Now, "material implication" (or "a rule of inference") as the name suggests, is a relation more involved in our physical (material) world, because it denotes a sentence that includes implication (as opposed to "formal logic implication" denoting just the symbol, a relation between predicates). I suggest googling "the paradoxes of material implication".

But your question is not about logic, it's about the concept of proof, so a scientist could answer it better than a logician.

I can just say that assigning a truth value to a sentence is something totally different than proving the empirical causality it entails.

This has nothing to do with the problem of the material implication being an inadequate translation of conditionals.

The first expression is fine. Its logic is as you say the modus ponens (p → q) ∧ p ⊢ q, which just means the conditional "If both p → q and p are true, then q is true", which is obviously true.

However, the logic of the second expression is (p → q) ∧ p ⊢ ¬q, and this implication is simply false, and as such it is completely useless.

I suspect that what you were really after is the case where q is found to be false even as p is true. This, then, falsifies p → q.

Whatever the reasons for deciding that p, q and p → q are true or false, the modus ponens (p → q) ∧ p ⊢ q is true, but depending on the situation you may have to decide that for example p → q is false contrary to your initial belief. This is called belief revision and may be required each time some new fact crops up.

This is exactly what happens every time new empirical data come to falsify an accepted theory, including for example a scientific theory.

And, obviously, we shouldn't be fooled by the fact that in syllogistic arguments we normally assert that our premises are true. Asserting anything does not make it true. This is also why logic works so beautifully.

Unlike the material implication.

• The question, as I understand it, is about the notion of formal proof, and you're just talking about the syntax of basic logic... Maybe the idea of intuitionistic logic's constructive proof would qualify as an answer? Feb 20 '21 at 11:20
• @k-wasilewski The question does not specify "formal logic" and is explicitly about "conditionals". Also, my discussion could be couched in formal terms. I'm not sure why you bring up intuitionistic logic. My explanation seems sufficient. Is there anything you don't understand in it? I also think I answer the question. Or do you think I do not? Feb 20 '21 at 18:10
• Yeah, I can't find a precise answer to the question asked: "what makes a conditional true"... Regarding my answer and xomment: I was interested mainly in the thought of "if/else" vs "cause/effect". That's where I try to sketch the applications of material and formal implications, also the notion of a constructive proof. Feb 20 '21 at 18:29
• @k-wasilewski 1. 'the question asked: "what makes a conditional true"' But this is not the question asked. - 2.'"if/else" vs "cause/effect"' This is irrelevant to the question asked. 3. 'material and formal implications' There is just one kind of implication, namely the one that corresponds to the conditional. The material implication, despite its name, is not an implication. Your answer is nothing but waffle. You don't answer the question and you try to answer a different question, without success. I don't think anyone understands what it is you are trying to say. Feb 21 '21 at 10:35
• 1. You're correct, the question is: "what does it even mean to say that the conditional is true?". 2. You're correct, however, I explicitly stated: "I was interested more in the thought of [...]". I was not refering to the "question". 3. You're correct about the definitions and about the fact that I don't answer the question, but you're wrong about me trying to answer some other question. As I said: "That's where I try to sketch the applications of [...]". Did you even read this comment you're arguing with? Feb 21 '21 at 12:55

I recommend Vellenman's book how to prove it, I think it will clear up many doubts.

#1 Apply truth value to the conditional premise as a whole, as follows.

P1. If I drop a ball ('p'), then it will hit the ground ('q')
P2. I dropped a ball ('p')
C. Therefore it hit the ground ('q')

You must first understand the difference between validity and soundness.

Validity (From Wikipedia): In logic, more precisely in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

This means that in a valid reasoning as long as the premises are true, the conclusion must ALWAYS be true.

Soundness (From Wikipedia): In logic, more precisely in deductive reasoning, an argument is sound if it is both valid in form and its premises are true.

If the premises are not true, then it is not sound reasoning. So is your argument sound? No, since you could drop a ball and it won't necessarily fall to the ground. But it is a valid reasoning. That is, the proposition P1 is false, therefore the premises will never be true and therefore the conclusion could be true or false.

#2 Apply truth values to both p and q variables individually, as follows.

P1. If I drop a ball ('p'), then it will hit the ground ('q')
P2. I dropped the ball
P3. It did *NOT* hit the ground

Here you did not conclude anything but you could conclude what you want and the reasoning would always be valid, since if we apply modus tollens to P1 and P3 we get ~P2, then we have P2 and ~P2 a contradiction. In an implication where the premise will always be false then we can conclude anything, this is called vacuous truth as you said, there are many examples like this in set theory with the empty set.

For example, the fact that the empty set is included in all sets, but also that the intersection of two sets (without elements in common) is the empty set, or that the intersection between the universe (or any set) and the empty set is the empty set, or that the complement of the universe is the empty set. All these things can seem confusing if you don't know the vacuous truth.

But if you want to say that P3 is the conclusion, then the whole argument is a contingency or a fallacy, it is the same argumentative value as the fallacy of the converse. That means that the reasoning is NOT valid, since it is NOT a tautology (you can see it making a truth table).

The truth table for "p => q", says that if p is false, then "p => q" is true. That makes zero sense to me, because I haven't dropped the ball - I have no reason to say that the conditional is true, because I don't have any evidence for it.

I took a section from velleman's book to answer this:

Figure 1.16.

To help us fill in the undetermined lines in this truth table, let’s look at an example. Consider the statement “If x > 2 then x² > 4,” which we could represent with the formula P(x) → Q(x), where P(x) stands for the statement x > 2 and Q(x) stands for x² > 4. Of course, the statements P(x) and Q(x) contain x as a free variable, and each will be true for some values of x and false for others. But surely, no matter what the value of x is, we would say it is true that if x > 2 then x² > 4, so the conditional statement P(x) → Q(x) should be true. Thus, the truth table should be completed in such a way that no matter what value we plug in for x, this conditional statement comes out true. For example, suppose x = 3. In this case x > 2 and x² = 9 > 4, so P(x) and Q(x) are both true. This corresponds to line four of the truth table in Figure 1.16, and we’ve already decided that the statement P(x) → Q(x) should come out true in this case. But now consider the case x = 1. Then x < 2 and x² = 1 < 4, so P(x) and Q(x) are both false, corresponding to line one in the truth table. We have tentatively placed a T in this line of the truth table, and now we see that this tentative choice must be right. If we put an F there, then the statement P(x) → Q(x) would come out false in the case x = 1, and we’ve already decided that it should be true for all values of x. Finally, consider the case x = −5. Then x < 2, so P(x) is false, but x² = 25 > 4, so Q(x) is true. Thus, in this case we find ourselves in the second line of the truth table, and once again, if the conditional statement P(x) → Q(x) is to be true in this case, we must put a T in this line. So it appears that all the questionable lines in the truth table in Figure 1.16 must be filled in with T’s, and the completed truth table for the connective → must be as shown in Figure 1.17.

Figure 1.17.

Of course, there are many other values of x that could be plugged into our statement “If x > 2 then x² > 4”; but if you try them, you’ll find that they all lead to line one, two, or four of the truth table, as our examples x = 1, −5, and 3 did. No value of x will lead to line three, because you could never have x > 2 but x² ≤ 4. After all, that’s why we said that the statement “If x > 2 then x² > 4” was always true, no matter what x was! The point of saying that this conditional statement is always true is simply to say that you will never find a value of x such that x > 2 and x² ≤ 4 – in other words, there is no value of x for which P(x) is true but Q(x) is false. Thus, it should make sense that in the truth table for P → Q, the only line that is false is the line in which P is true and Q is false.

Can I then go around saying "I've proven that if I drop a ball, it will hit the ground", even though I've never dropped a ball to test it?

When you make a proof you must always assume that your premises are true, then you must say, If I would drop the ball then it should always hit the ground, if that does not happen even if it starts from a valid reasoning, it is not sound.