Two popular yet nonequivalent definitions of conditional statements exist throughout the respectable and published literature on the topic, but only one of them is correct.
Under the first definition, a conditional statement is true if and only if the conclusion is true every time the hypothesis is true.
Under the second definition, a conditional statement is true if and only if it is not the case that: the hypothesis is true and the conclusion is false.
Only the first definition is correct. That’s our conditional statement.
Video Description:
Explains what really constitutes a conditional statement, despite references in numerous reputable works to an incorrect definition.
A conditional statement is not necessarily truth-functional. It’s truth is not necessarily determined by merely a truth table. A conditional statement is a modal claim. It is a necessary claim. A conditional statement is equivalent to it is necessary that: the hypothesis is false or the conclusion is true. It is equivalent to every time the hypothesis is true, the conclusion is also true.
Be on the lookout! Many reputable sources do not make the necessary distinction between what is actually a conditional statement, and what is not actually a conditional statement! These sources are ambiguous!
Among the tainted works include:
“Geometry” (2007) by Ron Larson, Laurie Boswell, Timothy D. Kanold, and Lee Stiff, published by McDougal Littell. The flaw seems to be contained to only Pages 94-95, where truth tables are discussed, and not the rest of the book.
“Discrete Mathematics and Its Applications, Sixth Edition” (2007) by Kenneth H. Rosen, published by McGraw-Hill. Definition 5 on Page 6, and the examples that follow, are the worst examples of the problem I have come upon. Watch out for this! It taints the entire book, although Rosen is usually able to circumvent any problems through the use of universal quantifications throughout the book.
“Language, Proof and Logic” (2008) by Jon Barwise and John Etchemendy, published by the CSLI (Center for the Study of Language and Information) Publications. Although clear warnings are given, they do seem to be incomplete.
A material implication is not a conditional statement. It is not deserving of the “If p, then q,” “p implies q,” “p only if q,” “q unless ~p,” “q whenever p,” “p is sufficient for q,” “q is necessary for p,” or any of the other wordings commonly associated with the conditional statement.
A material implication corresponds to Definition B in my video. As you can see though the video, Definition B does NOT correspond to the conditional statement. The common notion of what is a conditional statement is not equivalent to Definition B, the material implication. There is a clear difference between the two and many fail to make this distinction.
It clearly says in the Wikipedia article that a material implication p → q is “typically read ‘if p then q’ or ‘p implies q.’” As I’ve already pointed out, a material implication is not deserving of these wordings. They mean something else. Reading material implications that way introduces an huge ambiguity that isn’t even the case. The Wikipedia article is still wrong.
Furthermore, it’s not just the Wikipedia article that is wrong.
What this statement seems to claim is that the antecedent implies the consequent. There has to be some connection between the two terms for an implication to exist. No one reading this would think that such a connection exists.
p → q is a conditional if it is logically equivalent to ~(p * ~q) - by definition.
So, “it is false that (it is both monday and 5 + 5 does not equal 7)”. And indeed, that is false. It’s always false. It’s not a conditional - there is no connection between the two terms. Neither definition applies.
I disagree. An implication doesn’t have to involve some connection between the two parts. “If today is Monday, then 5 + 5 = 7” is a perfectly fine conditional statement. I’m surprised to be hearing that from you, Faust.
My thesis is that that definition is, while a popular one among the respectable literature on the topic, wrong. It’s not equivalent to another definition of the conditonal statement; one you yourself would probably agree with. That would be the first definition in my OP, or Definition A in my video.
Do you agree with that definition?
There are so many things wrong in this single quote I don’t even know how to begin.
You seem to be contradicting yourself. Keep in mind what you said earlier,
I don’t see anything involving a connection between the two terms in your definiton. It thus shouldn’t matter whether or not there is a connection. But you seem to contradict yourself by showing that it does matter,
Neither definition applies? Neither definition mentions a connection between the two terms at all! Where are you coming up with this?
Your post is so seriously flawed I’m questioning its very legitimacy.
browser, when reading your responses, I really don’t get the impression that you’re reading other peoples’ posts very carefully. “p → q is a conditional if it is logically equivalent to ~(p * ~q) - by definition.” ← your response to this was nonsense.
browser, I’m not sure where you’re getting your first definition from, but it might be slightly ambiguous. A conditional statement is defined as a statement that has two parts, an antecedent and a consequent - it’s an "if-then’ statement. Conditionals assert that the consequent is true whenever the antecedent is true, which is to say that there is a connection - logical, causal or definitional, between the antecedent and the consequent. Conditionals claim that where we find the antecedent, we’ll find the consequent. This is symbolised by ~(p * ~q), because this formulation states that it is not the case that we will find the antecedent (p) without finding the consequent (q).
It’s easy enough to make sentences in an “if-then” format that do not suggest a connection between the part between the “if” and the “then” and the part that comes after the “then.” If you believe that the day of the week has any bearing on math sums, you have other fish to fry before you get to formal logic itself.
Saying that the conditional “If p, then q” is logically equivalent to ~(p ^ ~q) is to say that the second definition in my OP is correct. It is to say Definition B in my video is correct. I’m saying that is wrong.
The conditional “If p, then q” is NOT logically equivalent to ~(p ^ ~q).
The conditional “If p, then q” is logically equivalent to IT IS NECESSARY THAT ~(p ^ ~q).
It is necessary in the logical sense, but not in the metaphysical sense. I have previously suspected that you are confusing logic with epistemology, which is what “modal” logicians regularly do. But the definitions you have a problem with are not, it appears, of modal logic. You cannot hold mathematicians to the criteria of a system in which they are not operating.
As classical logicians use the the implication sign the two expressions p → q is equivalent to ~(p * ~q). Adding “it is necessary” doesn’t change anything.
The first definition in my OP, and Definition A in my video, which are both correct, are both modal claims. The reason I don’t explicitly say so is because I don’t want to make my point seem more complicated than it actually is. I don’t need to state they are modal claims, involving the use of a whole different field of study. The conditional statements I’m talking about are so simple many learn them in high school geometry, often as freshman. There’s no need to complicate the simple here.
That is so false! Of course adding “it is necessary” changes things! It shows I’m saying the following proposition is true in every possible world! Saying merely ~(p ^ ~q) does not imply that at all! HeLLO!
So you believe all high school geometry books have it wrong about conditional statements? They’re just filling us up with nonsense. Every conditional statement found in them is just nonsense. Is that what you mean?
I think I know where you’d like to take this, but it’s still the case that no one would think that there is a connection between the day of the week and the sum of 5 and 5. Every conditional asserts some connection between the two simple statements contained therein, which is really what your first (incomplete) definition also asserts. The second definition is just a logical equivalence of the first, which your example does not refute.
My first definition asserts nothing about the hypothesis and the conclusion having some type of “special connection.” You are not taking my argument as it is. Propositions p and q could be ANY propositions, as ANY propositions will suffice to meet the needs of the definitions I have provided. Nowhere do I require a “special connection.” I quote my second citation,
And I agree. There is no reason for it to be otherwise.
My whole point is the the first and second definitions are NOT EQUIVALENT! My example DOES refute their equivalence! You don’t even seem to be paying attention to what you’re writing!
I think I said that the first definition is incomplete. if you wish to refute a definition that no logician is committed to, then be my guest. Perhaps you should write to the authors of these books. But beware - in mathematics, it is clear (to a mathematician) that the connection need not be spelt out, again, because math is a pure, or logical, or ideal language already, unlike spoken languages.
That’s why no cause and effect relationship need be spelt out in math, but no such relationship need be spelt out in English, for that matter. In math, every relationship is definitional. The required relationship between tow parts of a conditional may be definitional (not causal) in plain English as well. I don’t know why you included the quote about cause and effect. It’s not required by my claims so far.
I understand that you think the two are not equivalent. But you’re mixing up the purely mathematical use of material implication (which is really what we’re talking about) with the use by classical logicians in non-mathematical applications, with modal logic. Modal logicians do not agree about material implication, I agree, but their main focus has not been to refute it, but to devise a system for propositions that we cannot easily assign a truth value to. You’re fighting this war on the wrong battelfield.
At this point I wonder why this is important. Modal logicians do not stop at this point - it is where they begin. What is your point?
