A discussion on the Conditional

Question

I was reading about the Conditional from a book on Real Analysis by Terence Tao and I saw him put forward so many different ways of saying "If A, then B." He elaborated on the meaning of "If A, then B." and talked about its truth value for all possible truth values of A and B. He also talked about vacuous truths. One can read about the Conditional on page 312.

To make sure that I grasped his words, I put forward my thoughts about the Conditional.

Is the following discussion on the Conditional valid?

If A is a statement and B is a statement, the Conditional($\implies$) from A to B is written as A $\implies$ B, which in words is written as, "If A, then B."

The following statements have the same meaning:

If A, then B; If A is true, then B is true; When A, then B; When A is true, B is true; B is true when A is true; B is true if A is true; B is true whenever A is true; Whenever A is true, B is true; 'A is true' is sufficient to conclude that 'B is true', A is sufficient for B.

An example will help you see that the above statements mean the same.

"If it is raining, then the sky is cloudy."

The fact that it is raining is sufficient to conclude that the sky is cloudy. In other words, when it is raining, the sky is cloudy. To put it in another way, the sky is cloudy when it is raining. Let's put it in yet another way(one last time). Whenever it is raining, the sky is cloudy.

A digression: What does "X means Y." tell you, where X and Y are some statements?

It tells us that Y is the meaning of X. In other words, I can fully explain X by telling you Y. Realize that Y is another way of saying X. Some people find it easier to grasp X while others find it easier to grasp Y. In the end, they grasp the same information. Both X and Y share the same truth value(either true or false, but not both).

Let me elaborate on "If A, then B."

When A is true, "If A, then B." means that B is true. What is now evident is the fact that when A is true, "If A, then B." does not mean that B is false. In other words, when A is true, "If A, then B." is false when B is false.

When A is false, "If A, then B." tells us nothing about B being true or false.

Let me give you an example.

"If there is one person in the room, then the room is not empty."

When it is not the case where there is one person in the room, can you be certain about the room being empty or not? You cannot. There could be no one in the room, in which case it is empty. There could be two or more persons in the room, in which case the room is not empty. We cannot be sure about the room being empty or not just by knowing that it is not the case where there is one person in the room and "If there is one person in the room, then the room is not empty."

We realize that when A is false, "If A, then B." gives us no information about B being true or false. In the case where A is false, "If A, then B." is true, but vacuous(i.e., does not give us any new information beyond the existing fact that A is false). We have to live with this non-intuitive concept of the conditional being vacuously true when the antecedent is false. No example will show you that when A is false, "If A, then B." tells you whether B is true or false.

In summary, "If A, then B." is not the case where A is true and B is false.

When I discussed the different ways of writing "If A, then B.", I skipped the statement "A is true only if B is true."

Let's talk about this new statement now.

Let B be false. In this case, A is false. The statement "A is true only if B is true." can never get us to the case where A is true and B is false. The only other case it can get us to is "If A, then B."

Therefore, another way of saying "If A, then B." is "A is true only if B is true."

Furthermore, "A is true only if B is true." is the same as saying "B being true is necessary for A being true." or "B is necessary for A."

It seems like you are discussing different ways of saying $A\implies B$ ? Formally, the definition $$A\implies B\equiv \lnot A\lor B$$ is enough — Maximilian Janisch, Apr 12 '20 at 08:59
@MaximilianJanisch Terence Tao elaborates on "if A, then B" and helps us see that it is the case ¬A ∨ B. I tried to say a little more than he did in my post. I put up his book in my post. You could take a look what he has to say. — R004, Apr 12 '20 at 09:23
See this post for links to many similar posts about the conditional. — Mauro ALLEGRANZA, Apr 12 '20 at 10:23
@MauroALLEGRANZA I have edited the post. I shouldn't have used the same variables A and B there. — R004, Apr 12 '20 at 13:29
"When A is true, "If A, then B." means that B is true." Maybe it is more correct to say: "When $A$ and "if $A$, then $B$" are true, also $B$ must be true". This (IMO) is the "gist" of the truth-functional definition of the conditional connective: the link with the ubiquitous inference rule of Modus Ponens. — Mauro ALLEGRANZA, Apr 13 '20 at 07:58

score 1 · Answer 1 · answered Apr 14 '20 at 01:04

You are correct regarding everything you mention in this discussion, however, as people have mentioned in the comments, the definition of $A\implies B$ as $\lnot A\lor B$, is sufficient to understand the conditional.

It could be helpful to mention most of the common definitions of the conditional that you missed in your discussion. This, together with your question, will provide you with a more complete list of definitions that could be a useful future reference.

The definition of $A\implies B$ via truth table is:

\begin{array}{ccc}&\lnot A&A\\\lnot B&\color{blue}{1}&\color{red}{0}\\B &\color{blue}{1}&\color{blue}{1}&\end{array}

From this table we see that $A\implies B$ is only false when A is true and B is false. We also see that an alternative definition for $A\implies B$ is $\lnot A\lor B$.

Notice that $\lnot B\implies \lnot A$ will produce the same true table. This is known as the principal of transposition, from the statement "If A, then B" we can conclude "If not B, then not A".

A good example can be found using your sentence "If there is one person in the room, then the room is not empty". From this we can conclude that "if the room is empty, then there is not one person in the room".

A discussion on the Conditional

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