Having extreme difficulty understanding conditional statements.

Question

everyone. I'm having a very difficult time understanding conditional statements, and was hoping someone could help me understand them.

I took discrete math this last spring and remember struggling with them, but at some point had to let go and just say "it is what it is, I guess." I'm now taking a new course and conditionals are being thrown back in my face. I've watched several YouTube videos, read back over my discrete math textbook, did a google search and read the websites that came up in the first page and a half. Anything I do, I can't seem to grasp the meaning as a whole. Simply saying "if p, then q" or "p implies q," or "p is sufficient for q," or "q if p" does not make things clear.

I understand a conditional statement is a compound statement made of individual propositions that are either true or they aren't true. I can also see how compound statements such as "It's raining outside AND it's cloudy" could, as a whole statement, be true.

I've seen different definitions for what "if p, then q" mean, and they sometimes seem to contradict one another. So I'm going to ask it here. What is a conditional statement, and what does it mean?

Further, in a conditional statement:

1. Why does a true hypothesis and a true conclusion make the conditional true?
2. Why does a True hypothesis and a false conclusion make the conditional false?
3. Why does a false hypothesis and a true conclusion make the conditional true?
4. Why does a false hypothesis and a false conclusion make the conditional true?

Right now, I don't exactly know what questions I need to specifically ask to make it clear - all I know is I'm not getting it, and I'm exhausted with the run-around, lack of clarity, and simply not understanding. Hoping someone can help me.

Thank you and hope you're all well. 5.

Answer 1

Suppose that you’re told, If you score at least $85$ on the exam, you will get an $A$. There are four possibilities:

• you score at least $85$ and get an $A$;
• you score at least $85$ and don’t get an $A$;
• you score below $85$ and get an $A$; or
• you score below $85$ and don’t get an $A$.

Which of these outcomes contradict the statement? Certainly not the first one! The second one, however, definitely does: you scored at least $85$, but you did not get the promised $A$. Now what about the last two?

The thing to realize here is that taken literally, the statement says nothing about what will happen if you don’t score at least $85$; it merely promises you an $A$ if you do score at least $85$. In everyday contexts we generally don’t take such statements literally: we understand this one to mean (or at least very strongly imply) also that if you fail to score at least $85$, you will not get an $A$. But it does not actually say this, and in a formal logical context we interpret it literally. And since it says nothing about what will happen if you score below $85$, you can get any letter grade after scoring below $85$ without contradicting the statement: in that case there is nothing to contradict, because no prediction or promise was made. Since neither outcome — getting an $A$ or not getting an $A$ — contradicts it in this case, we say that it’s true.

In short, the truth values for these implications are based on the idea that the statement is false if and only if the facts actually contradict it. In the example that would be the case in which you scored at least $85$ but did not get the promised $A$. In general, for an implication $p\to q$, the only situation that absolutely contradicts it is the one in which $p$ is true, but $q$ is nevertheless false. It certainly isn’t contradicted when $p$ and $q$ are both true, and it makes no promises at all about $q$ when $p$ is not true.

There is one other difference between this logical implication and ordinary English usage. Ordinarily when we say that if $p$ is true, then $q$ is true, we are thinking of some causal connection between $p$ and $q$. In formal logic of the kind with which you’re dealing here we don’t care whether there’s any relationship between $p$ and $q$ at all: the truth value of the statement $p\to q$ is to be determined solely on the basis of the truth values of $p$ and $q$.

Answer 2

It is worth taking a second to point out that the logical conditional is a material conditional. A material conditional makes a statement about propositional universes : if you are in a universe where this proposition is true, then this other proposition is true in that universe. Material implication does not require any causal relationship between the two propositions. This can sometimes be confusing because in natural language an if-then construction almost always relates causally connected things. But not necessarily in logic. For instance, "if $1 = 1$, then my user name is Eric Towers" is a valid material conditional, but there is no evident causality between the tautology "$1 = 1$" and the vast ocean of complexity that led to my choice of user name. Logical/material conditional statements are not making claims of causal connection.

A conditional statement is a compound statement, containing two clauses, the antecedent (sometimes called the condition) and the consequent (sometimes called the conclusion). Syntactically, such a statement with antecedent $P$ and consequent $Q$ is written \begin{align*} P \implies Q \\ P \rightarrow Q \\ \text{if $P$ then $Q$} \\ \end{align*} and there are other syntactic forms, not listed here.

The semantics of the conditional $P \implies Q$ are "if you find yourself in a propositional universe where the proposition $P$ evaluates to true, then in that universe the proposition $Q$ is also true." Observe: If you find yourself in a propositional universe where $P$ is not true, then the conditional takes no position on whether the proposition $Q$ is or is not true -- the conditional stands mute when $P$ is false.

One verifies a conditional statement by the following general method:

• First (temporarily) assume the antecedent. This places you in a universe where the antecedent is true.
• Then, deduce the consequent.
• From the above two steps, conclude that the antecedent materially implies the consequent.
• Stop assuming the antecedent.

Although the above makes it sound like we must use the antecedent in the deduction of the consequent, this is false. If a consequent is true independently of the assumption, then it is materially implied by the assumption. (Such an independent cosequent is also materially implied by the negation of the assumption.) My user name is "Eric Towers". We may construct the valid conditional statement "If it is Tuesday, my user name is 'Eric Towers'." Any statement implies a true statement. Why? If you are in a universe where the antecedent is true, you are in a universe where the consequent is true -- because you are always in a universe where the consequent is true (regardless of the truth or falsity of the antecedent). This is where the first paragraph of this answer can cause some confusion -- irrelevancies imply truths but the implication claims no causal connection between the two clauses.

One refutes a conditional statement by the following general method: show that there is a universe where the antecedent is true and the consequent is false.

Examples:

• If $1 = 1$, then my user name is "Towers Eric"... We live in a propositional universe where $1 = 1$ is true and my user name is not "Towers Eric". So this conditional statement is false -- there is a propositional universe in which the condition is true but the conclusion is false. We (up to fussiness about the distinction between formal propositional systems and the real world) live in such a universe. The conditional leads to wrong conclusions in that universe.
• If $1 = 1$, then my user name is "Eric Towers" ... This conditional is true. My user name is "Eric Towers" in all universes, including universes where $1 = 1$ and also universes where $1 \neq 1$. (I promise that I have no understanding of universes where $1 \neq 1$.) So if you find yourself in a universe where $1 = 1$, you can confidently assert that my user name is "Eric Towers", with no risk of being incorrect.
• If $1 \neq 1$, then my user name is "Eric Towers"... Discussed in the previous bullet point. My user name is "Eric Towers" in all propositional universes, so if you find yourself in a universe in which $1 \neq 1$, you can confidently assert that my user name is "Eric Towers", with no risk of being incorrect.
• If $1 \neq 1$, then my user name is "Towers Eric"... There are two distinct ways to analyze this statement, depending on the nature of your propositional universes. If there is no universe such that $1 \neq 1$, then this conditional makes no assertion about my user name in any actual universe, so it is valid -- this is an instance of "'false implies anything' is valid" (see the principle of explosion) because the statement makes no actual assertion in any universe.
  
  If there are propositional universes where $1 \neq 1$ is valid (for instance any time one is using the above method of proof and has in the first step assumed "$1 \neq 1$") then, since my user name is not "Towers Eric", this conditional is invalid (and valid; keep reading). Note, however, that such a universe contains a contradiction, namely both "$1 = 1$" is tautologically true (assuming the usual semantics of "$1$" and "$=$") and its negation, "$1 \neq 1$" is also true. Again, from the principle of explosion, the conditional is both valid and invalid.

Answer 3

I'm not certain if this should be accepted as an answer, but I FINALLY found a video on YouTube that helped me understand things: https://www.youtube.com/watch?v=_uJ8rXDe6hs