Why are the inverse and converse of an implication not equivalent?

[NoahsArk]

If we have the statement: "If we prepare, we'll win the war", then according the rules of the truth table for this implication, this statement is only false if we prepared and still lost the war. This is what I'm having trouble with about implication. I understand that the only way to falsify this statement is the case where we prepared and still lost. But say we didn't prepare and lost. According to the rules of implication, we'd say that this fact makes the initial statement true.

As another poster said in an older discussion on this forum from 2012, we have an "innocent until proven guilty standard"? My questions are 1) Why do we have this standard in logic? If we didn't prepare for the war, and lost the war, then the we don't know what would have happened if we prepared for it. In the legal system it makes sense to have an innocent until proven guilty standard since the consquences of putting someone in jail for life are so serious that we don't want to do it unless there is strong proof of their guilt. In logic, though, I don't see why this is the case. Someone could have committed a crime even though there is no proof.

My other question is 2) why is it that logical statements must either be true or false and not just "unknown"? If we didn't prepare for the war and lost, then the statement "If we prepare, we'll win the war" is not made false. Why do we default, though, to making it true? Why not default to making it "unknown". While we haven't proved the statement false, we haven't proved it true. We may just as well default to making it false. Does the law that statement have to be either true or false have something to do with technology and the fact that switches must be either on or off in order for machines to do logic? Or, is it something more fundamental?

Thanks.

 

[DaveC426913]

I understand that the only way to falsify this statement is the case where we prepared and still lost. But say we didn't prepare and lost.

According to the rules of implication, we'd say that this fact makes the initial statement true.

My other question is 2) why is it that logical statements must either be true or false and not just "unknown"? If we didn't prepare for the war and lost, then the statement "If we prepare, we'll win the war" is not made false.

I am not wise in the ways of formal logic, but informally, I concur with your view.

The conditions 'we didn't prepare for the war and lost' have no bearing on the statement ' If we prepare, we'll win the war '.

Is it at all possible you are misinterpreting the rules of logic as you've read them? Can you provide a reference?

 

[Dale]

With deductive logic you start with a premise that you are “given” and then you see what you can conclude. If you are given the premise “if we prepare then we win” then the following statements may all be true “we prepared and won”, “we didn’t prepare and lost”, and “we didn’t prepare and won”. Only the following statement cannot be true “we prepared and lost”.

So if we prepared we can deduce that we won, and if we lost we can deduce that we didn’t prepare. In contrast, if we didn’t prepare, we cannot deduce whether or not we won. Similarly, if we won we cannot deduce whether or not we prepared.

 

[WWGD]

This is called the paradox of material implication. A bit long to elaborate on here ( and also I am rusty in my logic).

 

[Stephen Tashi]

My questions are 1) Why do we have this standard in logic?

It's a matter of convenience. This is difficult to understand in the case of isolated propositions, but easy to understand if we look at the case of quantified propositional functions. The sentence "if 5 > 4 then 5 > 2" is a proposition. The sentence "if x > 4 then x > 2" is a propositional function because it contains a variable, "x" and is neither True nor False unless "x" is assigned a particular value.

In mathematics, a propositional function is often turned into a proposition by precceeding it with the quantifiers "there exists" or "for each". For example, "For each (real number) x, if x > 4 then x > 2" is a True proposition.

Suppose someone claims the above proposition as False. To prove it is False, he asserts that the case x = 3 shows it is False. We would not consider that the case x=3 demonstrates the falsity of "For each (real number) x, if x > 4 then x > 2". So, effectively, we consider the statement "if 3 > 4 then 3 > 2" to be True instead of being an example where the "For each x" condition doesn't work.

Instead of making the rule that causes "if 3 > 4 then 3 >2" to be a True statement, we could imagine other approaches to formal logic that define different versions of the "For each" quantifier or the "if ...then..." structure and try to implement a concept of "when applicable". Perhaps some logicians have tried to do this. However, the standard and most convenient approach in contemporary mathematics is to define the meaning of "if...then..." propositions in such a way that makes them True when the "if..." part is False.

My other question is 2) why is it that logical statements must either be true or false and not just "unknown"?

People have created formal Logics where propositions can have values other than "True", "False". However, these logics have not (so far) been found useful for doing mathematics. (Whether they are useful for fields like Politics, Ethics, Economics etc. is probably as controversial as those fields themselves.)

 

[FactChecker]

If we have the statement: "If we prepare, we'll win the war", then according the rules of the truth table for this implication, this statement is only false if we prepared and still lost the war.

What is wrong with that? If there is not a single counter-example, the statement is true.

This is what I'm having trouble with about implication. I understand that the only way to falsify this statement is the case where we prepared and still lost. But say we didn't prepare and lost. According to the rules of implication, we'd say that this fact makes the initial statement true.

That is not the contrapositive of the initial statement. The initial statement is true if we can say that "If we did not win, then we did not prepare."

 

[PeroK]

If we have the statement: "If we prepare, we'll win the war", then according the rules of the truth table for this implication, this statement is only false if we prepared and still lost the war. This is what I'm having trouble with about implication. I understand that the only way to falsify this statement is the case where we prepared and still lost. But say we didn't prepare and lost. According to the rules of implication, we'd say that this fact makes the initial statement true.

As another poster said in an older discussion on this forum from 2012, we have an "innocent until proven guilty standard"? My questions are 1) Why do we have this standard in logic? If we didn't prepare for the war, and lost the war, then the we don't know what would have happened if we prepared for it. In the legal system it makes sense to have an innocent until proven guilty standard since the consquences of putting someone in jail for life are so serious that we don't want to do it unless there is strong proof of their guilt. In logic, though, I don't see why this is the case. Someone could have committed a crime even though there is no proof.

My other question is 2) why is it that logical statements must either be true or false and not just "unknown"? If we didn't prepare for the war and lost, then the statement "If we prepare, we'll win the war" is not made false. Why do we default, though, to making it true? Why not default to making it "unknown". While we haven't proved the statement false, we haven't proved it true. We may just as well default to making it false. Does the law that statement have to be either true or false have something to do with technology and the fact that switches must be either on or off in order for machines to do logic? Or, is it something more fundamental?

Thanks.

I think you are missing the point, somewhat. Mathematical logic deals with well-defined statements, not hypothetical questions that have no answer.

Also, legal "logic" is not necessarily based on a well-defined set of self-contained statements, so I'd be careful drawing comparisons with that.

You ought to confine your attention to statements that can be formally tested. This, in itself, is a good exercise: to distinguish between statements that have a clear logical structure and ones that don't. Examples include:

Everyone who has prepared has won

No one who has prepared has lost.

There exists a case where someone prepared and won.

There are no cases where someone prepared and lost.

These are all things that you can test and assign true or false to. Assuming, of course, you have all the required information.

In short, I'd focus on getting to grips with logic in clear well-defined scenarios and steer clear of cases where you get tangled up in not knowing what something is supposed to mean. Especially if this is preparation for pure mathematics.

 

[NoahsArk]

Is it at all possible you are misinterpreting the rules of logic as you've read them? Can you provide a reference?

If statement A = "We prepared", and statement B = "we won". In that case the truth table given for "if A then B" $(A \implies B)$ would be:\begin{array}{|c|c|c|c|}
\hline A & B & A \implies B\\
\hline t & t & t\\
\hline t & f & f \\
\hline f & t & t\\
\hline f & f & t\\
\hline
\end{array}B

All the material I've seen so far on logic, including Cousera's "Introduction to Mathemactical Reasoning", is stating some variation of this. Only When A is true and B is false is $A \implies B$ false. My issue is with the last two rows of the table. The fact that they don't falsify the statement $A \implies B$ should not make the statement true in my mind.

In contrast, if we didn’t prepare, we cannot deduce whether or not we won. Similarly, if we won we cannot deduce whether or not we prepared.

This is why the last two rows of the table are difficult for me to understand. They are saying if we didn't prepare and we won, or if we didn't prepare and we lost, we can conclude that had we prepared we would have won.

However, the standard and most convenient approach in contemporary mathematics is to define the meaning of "if...then..." propositions in such a way that makes them True when the "if..." part is False.

I'm trying to understand why it's the most convenient. In the example you gave with if x > than 4, than x > 2, when x = 3 then the first statement is false and the second statement is true, but I don't see an advantage in having a convention where we say that the initial statement is true.

You ought to confine your attention to statements that can be formally tested.

That helps part of my confusion since I was not sure what kinds of statements the rules applied to. The examples you gave look to the past so they can be tested, whereas my example deals with something that might happen in the future. Also, the example with x being > than 4, unlike the war example, is something that can be deduced. E.g. we know that if x is greater than 4, it's also greater than 2, but we don't know from pre existing knowledge that if we prepare for the war we'll win. Even in the well defined scenarios, though, I don't see how we can reach the conclusion that if A is false and B is true, that therefore when A is true B is true too.

 

[PeroK]

Even in the well defined scenarios, though, I don't see how we can reach the conclusion that if A is false and B is true, that therefore when A is true B is true too.

That's called being vacuously true. Let's take a example:

If $x^2 = -1$ then $x = 0$. Which is true, of course. Assuming we are dealing with real numbers. The logic is that if that statement is false, then we need a counterexample. And, given that $x^2 = -1$ is not possible, then we won't be able to find a counterexample.

You might want to argue that that statement is meaningless and has no place in mathematics. But, that is problematic. Often, you don't know at the outset whether a condition ever holds. And, often you can prove something by assuming it's false and, by arguing logically, reach a known contradiction. This is called proof by contradiction.

If you aren't allowed to apply logic to a set that may be the empty set, then you're a bit stuck in these cases.

A neat example might be to prove that $\sqrt 2$ is irrational. We start by assuming that $\frac{m^2}{n^2} = 2$ for whole numbers $n, m$ with no common factors. We argue that $m^2 = 2n^2$, hence $m^2$ is even, hence $m = 2k$, and $2k^2 = n^2$, hence $n$ is even, hence $m$ and $n$ have $2$ as a common factor. A contradiction as we assumed they had no common factors.

The conclusion is that no such $m, n$ exist. Proof by contradiction. QED

But, wait a minute, if no such $m, n$ exist, then how have we been doing the above mathematics? The whole line of argument, it seems, was all vacuously true, as $m, n$ never existed in the first place.

So, it looks like logical reasoning based on the empty set is perfectly valid and very useful.

In any case, there is nothing to fear from the case of vacuous truth. It never jumps up and bites you and actually it's quite a tame beast!

 

[Dale]

They are saying if we didn't prepare and we won, or if we didn't prepare and we lost, we can conclude that had we prepared we would have won.

No, you are misunderstanding the table. You are thinking of inductive reasoning not deductive reasoning. You are thinking “I have made observations about preparing and winning, can I generalize those observations into a logical rule?”

That is not what is being described here. This table is about deductive reasoning where you have some logical rule or truth already known and use those to find other logical rules.

This table is saying that if we know as a logical truth (not merely an observation) that we don’t prepare then the logical rule “if we prepare then we win” (IPTW) is true. If we know as a logical truth that we prepare then we cannot deduce if IPTW is true.

If we know as a logical truth that we win then IPTW is true. If we know as a logical truth that we lose then we cannot deduce if IPTW is true.

 

[Stephen Tashi]

I'm trying to understand why it's the most convenient. In the example you gave with if x > than 4, than x > 2, when x = 3 then the first statement is false and the second statement is true, but I don't see an advantage in having a convention where we say that the initial statement is true.

By "the initial statement", do you mean "if 3 > 4 then 3 > 2"?

In mathematics, the quantifier "for each" is used to indicate there are no exceptions. If each member of the collection of statements (propositions) that can be formed by substituting a number for x in the sentence (propositional function) " if x > 4 then x > 2 is True then it must be that the particular case "if 3 > 4 then 3 > 2" is a True proposition.

my example deals with something that might happen in the future.

You are considering propositional functions or some other form of non-proposition instead of propositions. Using common language, it's actually very difficult to state examples of propositions! A proposition is a sentence that is (right now!) definitely true or definitely false. By contrast, a common language sentence like "if we prepare then we will win the war" can be interpreted as having components like "we prepare" that are not (now) definitely True or definitely False. Also a sentence like "if we prepare then we will win the war" can be interpreted as a generality that applies to several wars, several times in history, or even several "we" (nations).

(Even in mathematical writing, people often use the convention that the "for each" quantifier is to be understood when if...then... sentences are stated. For example , somone who writes the sentence "If x > 0 then 3x > 0" may mean to say "For each real number x, if x > 0 then 3x > 0". )To think clearly about the Truth value of "if A then B", you have to think clearly that A and B represent propositions, not propositional functions or non-propositional sentences that have some yet to be determined truth value. The clearest examples of propositions come from mathematics. "5 > 4" is specific enough to be a proposition. "We prepare" is the type of example used in logic textbooks, but it invites confusion. To understand it as an example of a proposition in Logic, you must think of it as referring to a specific incident that either did or did not happen. Don't think of "we prepare " as phrase that does not now have a truth value or as a propositional function whose truth value depends on which "we" or which time is used to interpret the sentence.

There are versions of Logic called "temporal logics" that deal with statements whose truth value varies in time or becomes definite at a particular time. The type of Logic you are studying deals with propositions, not more general kinds of sentences that appear in temporal logic.

 

[Mark44]

I understand that the only way to falsify this statement is the case where we prepared and still lost. But say we didn't prepare and lost. According to the rules of implication, we'd say that this fact makes the initial statement true.

It might be helpful to think of an implication as a sort of contract. Let's say that I am selling a car that you want to buy, and that we agree on a price of $500 (it's not a very new car, but I have bought numerous cars in the past for this amount).

If you give me $500, then I will give you the car.

There are four possibilities:

1. You give me $500 and I give you the car - no problem.
2. You give me $500, but I don't give you the car - this violates the contract.
3. You don't give me $500, but I give you the car anyway - not a violation of the contract.
4. You don't give me $500, and I don't give you the car - also not a violation of the contract.

The only scenario in which you have a valid complaint is #2, when you pay me the money, but I don't give you the car.

 

[stevendaryl]

All the material I've seen so far on logic, including Cousera's "Introduction to Mathemactical Reasoning", is stating some variation of this. Only When A is true and B is false is $A \implies B$ false. My issue is with the last two rows of the table. The fact that they don't falsify the statement $A \implies B$ should not make the statement true in my mind.

It seems that you want to say that $A \implies B$

• is true if both $A$ and $B$ is true.
• is false if $A$ is true, but $B$ is false.
• is undefined otherwise.

You could certainly make up a logic where there are three truth values: true, false and undefined, and give such a "truth table" to implication. However, it would make mathematical reasoning a lot more complicated, and no additional benefit.

Let's just take a trivial mathematical fact: "If $x$ is greater than 1, then $x^2 > x$". I would call that a mathematical fact about real numbers. (Or if you want to be picky, it becomes a mathematical fact when you universally quantify: $\forall x\ (x \gt 1) \implies (x^2 \gt x)$

Under your proposed meaning of $\implies$, the statement $(x \gt 1) \implies (x^2 \gt x)$ is only true if $x \gt 1$. Otherwise, it's undefined. But that means that almost no universal statement would be true. Sort of the whole point of proving things in mathematics is because after you prove it, you know that it's true.

 

[Jarvis323]

Why not default to making it "unknown". While we haven't proved the statement false, we haven't proved it true. We may just as well default to making it false. Does the law that statement have to be either true or false have something to do with technology and the fact that switches must be either on or off in order for machines to do logic? Or, is it something more fundamental?

It's not something fundamental, it's a choice for a particular mathematical system that happens to be the one we mostly use.

There is also three valued logic, where statements can be true, false, and unknown.
https://en.wikipedia.org/wiki/Three-valued_logic.

Or more generally, many valued logic.
https://en.wikipedia.org/wiki/Many-valued_logic

There is also quantum logic, which is a bit strange.
https://en.wikipedia.org/wiki/Quantum_logic

Computer languages sometimes use three valued logic. For example, logical expressions in SQL can evaluate to { true, false, unknown }.

In Mathematics, you start with a foundation, which includes a language to form valid mathematical statements under a given logic (doesn't have to be boolean), and some axioms, which are some statements you take for granted to have determined values. With that, a mathematical universe is defined, and you can explore it.

Do we need three valued logic? Maybe. It's useful sometimes for programming languages. I haven't explored it enough to be able to say much about the fundamental benefits.

 

[Dale]

Do we need three valued logic? Maybe.

Haha. Was that intentional?

 

[Jarvis323]

Haha. Was that intentional?

Not at first, but I did recognize it once I typed it. lol

 

[Mark44]

It seems that you want to say that
$A \implies B$

• is true if both $A$ and $B$ is true.
• is false if $A$ is true, but $B$ is false.
• is undefined otherwise.

I disagree with the third point. The only scenario in which the implication is false is when A is true but B is false. In all other cases, the implication is defined to be true.

 

[stevendaryl]

I disagree with the third point. The only scenario in which the implication is false is when A is true but B is false. In all other cases, the implication is defined to be true.

The "you" in "It seems that you want to say that.." wasn't YOU. It was the original poster. He wanted a different definition.

 

[NoahsArk]

I've thought a lot about these responses last night and this morning. I'm starting to think that one reason for my confusion was a misunderstanding of the definition of $A \implies B$. Maybe this statement should not be defined as "A implies B". The word "implies" is starting to sound like a misnomer. $A \implies B$ has nothing to do with causation from what I'm seeing. I think a lot of my confusion would go away if $A \implies B$ were defined as "It's not the case that when A is true, B is false". @PeroK if in your example, "if $x^2$ = -1, then x= 0" I can accept that this is true if we take the word "implies" out of the definition of $A \implies B$. @Mark44 your car example also makes sense if we take the word "implies" out of the definition.

A and B are both premises and $A \implies B$ is the conclusion. Is it true that we are not even concerned with the truth of the premises, and that we are only concerned that the premises lead to the conclusion? One example that I thought of after reading some of the examples in the replies was the statement: "if x > 3, then $x^2$ is greater than 12". A = x > 3 and B = $x^2$ is greater than 12. The truth table for this statement would be:
t,t t
t,f f
f,t t
f,f t

When A is between 3 and around 3.4, $A \implies B$ is false, and we are in column two of the truth table. Because the statement is talking about all values of x, it is clearly false as a general statement. But there are particular cases of A and B that will lead to the statement being true. So, is it true that, in addition to not being concerned about whether the premises of A and B are true, we are not concerned about whether or not the statement itself $A \implies B$ is true, so long as the premises, assuming they are true, lead to the conclusion $A \implies B$? All elephants are green, Tom is an elephant, therefore Tom is green is a logical statement despite both premises and the conclusion being false. @Dale I think this is what you were getting at when you said "if we know as a logical truth and not just as an observation".

But that means that almost no universal statement would be true.

What do you mean by this? Can you give an example of a universal statement that would not be true using a definition of $A \implies B$ where you get undefined values? In your example, $(x >1) \implies (x^2 > x)$, I am assuming that this is a "universal statement", and that it's true, even if we use a definition of $A \implies B$ where the statement is undefinied for cases where A is false. It would be undefined only in cases where $x \leq 1$, but in all other cases would be true.

@PeroK I did not fully understand the proof that $\sqrt 2$ is an irrational number. I think I understood the general idea of the example, though- that you can prove something true by proving the opposite of it can't be true. In this example I think you proved that $\sqrt 2$ could not be a rational number, so therefore it must be rational. The opposite of the statement $A \implies B$ would be the case of a true A value and a false B value. So I think what you showed is that if we can prove it's impossible for A to be true while B is false, then we proved the statement is always true.

Why $A \implies B$ is defined in a way which means that false values of A lead to true values of $A \implies B$ is still something I'm not sure about. @Jarvis323 and @Stephen Tashi you mentioned other systems of logic, for areas outside of math, where we could use a different definition. I'm still not sure why in math we use this definition and why it's convenient for math. Is there an example where having an undefined value creates some problem?

Even a more basic problem I'm having is why we have the $\implies$ operator to begin with? What are the applications it's used for? Just like calculus was invented for a specific purpose, I'm assuming so were logical operators. I understand how having the AND and the OR operators are useful, but even then I only understood it after reading part of the book "Code, the Hidden Language" which is about the basics on how computers use on/off switches to do logic. I can't imagine how George Boole thought of this idea in the 1800s where there doesn't seem to have been any application for them.

 

[PeroK]

@PeroK I did not fully understand the proof that $\sqrt 2$ is an irrational number. I think I understood the general idea of the example, though- that you can prove something true by proving the opposite of it can't be true. In this example I think you proved that $\sqrt 2$ could not be a rational number, so therefore it must be rational. The opposite of the statement $A \implies B$ would be the case of a true A value and a false B value. So I think what you showed is that if we can prove it's impossible for A to be true while B is false, then we proved the statement is always true.

I would encourage you to learn logic via some useful mathematics, like the proof of the irrationality of $\sqrt 2$. That is a perfect example of some simple mathematics and some key elements of mathematical logic.

If you want to learn logic for its own sake, then fine. But I, for example, have never studied logic, other than through pure mathematics.

Even a more basic problem I'm having is why we have the $\implies$ operator to begin with? What are the applications it's used for?

All mathematics involves logic. And one thing implying another is fundamental. Take the proof of the quadratic formula:
$$ x^2 + bx + c = 0 \implies x^2 + bx = -c \implies x + bx + (\frac b 2)^2 = (\frac b 2)^2 - c$$ $$ \implies (x + \frac b 2)^2 = \frac {b^2}{4} - c \implies x + \frac b 2 = \pm \sqrt { \frac {b^2}{4} - c } \implies x = -\frac b 2 \pm \frac 1 2 \sqrt {b^2 - 4c } $$ $$ \implies x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$$ You simply cannot do any mathematics without it!

 

[Mark44]

A and B are both premises and
$A \implies B$ is the conclusion.

The usual terminology is that A is the hypothesis and B is the conclusion. $A \implies B$ is the implication.

Is it true that we are not even concerned with the truth of the premises, and that we are only concerned that the premises lead to the conclusion?

Ordinarily we are concerned only with a hypothesis that is true. The truth of the implication then depends only on whether the conclusion is also true.

One example that I thought of after reading some of the examples in the replies was the statement: "if x > 3, then $x^2$ is greater than 12". A = x > 3 and B = $x^2$ is greater than 12. The truth table for this statement would be:
t,t t
t,f f
f,t t
f,f t

When A is between 3 and around 3.4, $A \implies B$ is false, and we are in column two of the truth table.

If $3 \lt x \le 3.4$ the implication is true, because $3.4^2 = 11.56 < 12$. If you extend the interval upward, then for some x in that interval, $x^2 > 12$. The hypothesis you wrote is not specific enough to lead to the conclusion you wrote. IOW, for some values of x, the implication is true, but for others, it is false.

 

[Mark44]

Why A⟹B is defined in a way which means that false values of A lead to true values of A⟹B is still something I'm not sure about.

This is by definition.

 

[Mark44]

In your example, (x>1)⟹(x2>x), I am assuming that this is a "universal statement", and that it's true

Yes, this is a universal statement. The full version would be $\forall x \in \mathbb R \backepsilon x > 1, x^2 > x$
Here $\forall x \in \mathbb R \backepsilon x > 1$ is the hypothesis, and $x^2 > x$ is the conclusion.
The $\forall$ symbol means "for each" or "for all" and the $\backepsilon$ symbol means "such that."

 

[stevendaryl]

What do you mean by this? Can you give an example of a universal statement that would not be true using a definition of $A \implies B$ where you get undefined values? In your example, $(x >1) \implies (x^2 > x)$, I am assuming that this is a "universal statement", and that it's true, even if we use a definition of $A \implies B$ where the statement is undefinied for cases where A is false. It would be undefined only in cases where $x \leq 1$, but in all other cases would be true.

To say $\forall x \Phi(x)$ is to say that $\Phi(x)$ is always true (for any value of $x$). If $\Phi(x)$ is sometimes undefined, then it's not always true.

So if you want to say

$(x >1) \implies (x^2 > x)$

is not true, but is undefined when $x = 0$, then it's not always true. So $\forall x\ (x >1) \implies (x^2 > x)$ is not true (under your interpretation of $\implies$.

To say that $A$ implies $B$ is not at all about causality. It's about deduction.

"If your goldfish is floating upside down, then he is dead"

That doesn't mean that floating upside down will kill your goldfish, it just means that his floating upside down is enough information for you to deduce that he is dead.

I think that what you're really getting at is usefulness, or relevance, or something. If you want to know whether $B$ is true, or not, then knowing $A$ implies $B$ isn't helpful in the case where $A$ is false. But if I'm trying to prove $x^2 \gt x$, there are lots of true statements that are unhelpful. For example, "Paris is the capital of France" isn't helpful in this case, but it's true.

 

[PeroK]

I would encourage you to learn logic via some useful mathematics ...

I've just had a look at Introduction to Abstract Algebra by Thomas Whitelaw. The first chapter is Sets and Logic and covers all this in a straightforward way. It has a section of about three pages on the implies sign.

There's no rubbish like "If we prepare, we'll win the war." That's like nothing I've ever seen before in a maths book. I don't see how you can learn mathematical logic from that. It just confuses the issue, IMHO.

 

[Jarvis323]

@Jarvis323 and @Stephen Tashi you mentioned other systems of logic, for areas outside of math, where we could use a different definition. I'm still not sure why in math we use this definition and why it's convenient for math. Is there an example where having an undefined value creates some problem?

Different logics are not necessarily outside of mathematics. Different mathematical logics are just more experimental, and perhaps underdeveloped. Three valued logic is much more complicated than Boolean logic and tends to introduce some problems and limitations that make it unattractive as a foundation for mathematical systems. There has been an enormous struggle, that is not over, just to understand the simpler foundations we've been using thus far.

Maybe in a thousand years, a new or existing formulation of many valued logic will somehow revolutionize mathematics.

One of the issues, according to the answer linked below, is that we lose tautologies. $p\implies p$ is unknown if $p$ is unknown.

https://math.stackexchange.com/questions/891912/three-valued-logic-as-foundation

In math, you build on previous results, and it's nice to have a simple world where the things you've deduced are 100% certain, so you can build more on top of that without introducing more and more complicated dependencies hiding underneath everything.

Maybe a more complicated logic than Boolean logic will turn out to be better as a foundation for mathematics in some sense, but at the same time maybe it will be too complicated for people to effectively work with.

 

[Stephen Tashi]

$A \implies B$ has nothing to do with causation from what I'm seeing.

In Logic, that is correct. The concept of causation is used in everyday speech but it is not a precise concept. There can be lots of philosophical and metaphysical debate about its meaning. So the standard form of mathematical Logic does not give a definition for "A causes B".

I think a lot of my confusion would go away if $A \implies B$ were defined as "It's not the case that when A is true, B is false".

That can be a helpful way to think. In symbols $(A \implies B) \iff ( \lnot (A \land \lnot B)$). Whether we take this as a theorem or a definition is a matter of style. In most texts $(A \implies B)$ is defined by giving its truth table.

A and B are both premises and $A \implies B$ is the conclusion. Is it true that we are not even concerned with the truth of the premises, and that we are only concerned that the premises lead to the conclusion?

Thinking about Logic requires thinking in a sophisticated manner and using precise concepts. What would you mean by "lead to the conclusion"? Trying to sort that out can lead to studying the difference between the concepts of "A can be derrived from B" versus "A implies B". Perhaps you have read articles about a situation where there are True statements in a mathematical system that cannot be proven ( derrived) within that system.

One example that I thought of after reading some of the examples in the replies was the statement: "if x > 3, then $x^2$ is greater than 12". A = x > 3 and B = $x^2$ is greater than 12. The truth table for this statement would be:
t,t t
t,f f
f,t t
f,f t

If you are studying propositional logic, you need to think about propositions.

You aren't giving an example of a proposition. What you call a "statement" is a propositonal function. It is not a propositional function where the variable x represents a proposition, so it is not a propositional function defined by a truth table. If the variable is "x" then the function must be defined by showing a table of its value as "x" takes various values.

So, is it true that, in addition to not being concerned about whether the premises of A and B are true, we are not concerned about whether or not the statement itself $A \implies B$ is true, so long as the premises, assuming they are true, lead to the conclusion $A \implies B$?

You have hit upon an important thought! Logic is not the study of truth. Logic is the study of reliable methods of reasoning. Reliable methods of reasoning lead to True conclusions when the premises of the arguments are True. What happens when the premises of the arguments are not True is not the focus of Logic.

In common language, people say a statement is "logical" when it is factual or plausible. But mathematical Logic is not the study of scientific facts.

All elephants are green, Tom is an elephant, therefore Tom is green is a logical statement despite both premises and the conclusion being false.

It's a valid syllogism if that's what you mean by "logical statement". Notice that this syllogism is not descibed by using only propositions. It does not have the pattern ( known as modus ponens) of :
$A \implies B$
$A$
========
therefore $B$

To represent the concept "All elephants are green" requires the use of a quantifier, so we must use a quantifier and a propositional functions

$\forall x ( E(x) \implies G(x))$
$E(t)$
=======
therefore $G(t)$

@Stephen Tashi you mentioned other systems of logic, for areas outside of math, where we could use a different definition. I'm still not sure why in math we use this definition and why it's convenient for math. Is there an example where having an undefined value creates some problem?

Try to write some simple mathematical proofs using a version of logic where statements can be neither True nor False and you will begin to see the problems.

 

[NoahsArk]

I think I finally understand, and that I was getting unnecessarily confused before. I can see that there are good reasons for the convention that implications are true when their hypothesis are false.

For example, maybe a certain medicine, "Medicine A", when taken by a patient will be dangerous if not also taken with Medicine B. We might have a hundred patients who have written down what medications they've taken. If they say that they've taken Medicine A in combination with anything other than Medicine B, we identify them as at risk. If they didn't take Medicine A, then we will mark them as safe whether or not they took Medicine B or something else. Safe = true, and at risk = false. If the patients save their answers into a computer, a program could decide if they are safe or not.

The word "implies" or "implication" is what threw me off I think. We are not saying that taking Medicine A implies that the patient also took Medicine B. Even the words "if", "then" are confusing. We are not saying "If a patient takes Medicine A, then they also took Medicine B. $A \implies B$ is a tool for determining whether or not a certain combination of facts is present.

So the medicine example is like the $500 car example. The war example is bad because $A \implies B$ is not about making predictions.

Do I have the right idea now?

 

[PeroK]

For example, maybe a certain medicine, "Medicine A", when taken by a patient will be dangerous if not also taken with Medicine B. We might have a hundred patients who have written down what medications they've taken. If they say that they've taken Medicine A in combination with anything other than Medicine B, we identify them as at risk. If they didn't take Medicine A, then we will mark them as safe whether or not they took Medicine B or something else. Safe = true, and at risk = false. If the patients save their answers into a computer, a program could decide if they are safe or not.

This I don't follow. I'd still like to see you focus on some clear mathematical examples.

For example, for numbers greater than $2$:
$n$ is prime $\implies n$ is odd.
$n$ is even $\implies n$ is not prime.

But,
$n$ is odd $\nRightarrow n$ is prime.

 

[NoahsArk]

This I don't follow. I'd still like to see you focus on some clear mathematical examples.

Medicine A might help with joint pain but create a risk of seizures, and Medicine B might reduce the risk of seizures. So, when someone takes Medicine A, they should also take Medicine B or they are at risk. The table is:

1. Took Medicine A, Took Medicine B - Safe (T)
2. Took Medicine A, Didn't Take Medicine B- Risk (F)
3. Didn't take Medicine A, Took Medicine B- Safe (T)
4. Didn't take A or B- Safe (T)

Numbers 3 and 4 are examples of false hypothesis but true overall implications. Before I was stuck on why we'd want true results where A was false.

To me the medicine example is closer to a math example then the war one, and I just want to make sure I'm getting the general idea. I will also try and work with math examples, but right now I can't think of any where false "A"s lead to a useful result.

 

[PeroK]

Medicine A might help with joint pain but create a risk of seizures, and Medicine B might reduce the risk of seizures. So, when someone takes Medicine A, they should also take Medicine B or they are at risk. The table is:

1. Took Medicine A, Took Medince B - Safe (T)
2. Took Medicine A, Didn't Take Medicine B- Risk (F)
3. Didn't take Medicine A, Took Medicine B- Safe (T)
4. Didn't take A or B- Safe (T)

Numbers 3 and 4 are examples of false hypothesis but true overall implications. Before I was stuck on why we'd want true results where A was false.

To me the medicine example is closer to a math example then the war one, and I just want to make sure I'm getting the general idea. I will also try and work with math examples, but right now I can't think of any where false "A"s lead to a useful result.

I still don't get it. Sorry.

 

[Mark44]

I can see that there are good reasons for the convention that implications are true when their hypothesis are false.

Or when both hypothesis (premise) and conclusion are false.
A simpler way to describe an implication is to state the condition that makes an implication false; i.e., when the hypothesis is true but the conclusion is false.

 

[Mark44]

Medicine A might help with joint pain but create a risk of seizures, and Medicine B might reduce the risk of seizures. So, when someone takes Medicine A, they should also take Medicine B or they are at risk. The table is:

1. Took Medicine A, Took Medicine B - Safe (T)
2. Took Medicine A, Didn't Take Medicine B- Risk (F)
3. Didn't take Medicine A, Took Medicine B- Safe (T)
4. Didn't take A or B- Safe (T)

This is really the long way around. What it shows is the situation for $\neg A \vee B$ (not A or B), which is equivalent to $A \Rightarrow B$. To convince yourself of the equivalence of the two expressions, construct the truth tables for each and notice that they are the same. $\neg A$ ("not A") means the logical negation of A; flipping T for F or F for T.

 

[NoahsArk]

@Mark44 yes, I see that these two expressions are equivalent. $\neg A \lor B$ is a concise way to capture the meaning. It made me think that this expression and the expression $A \implies B$ should be called a "NOT OR" gate or a "NOT OR" connector. That would keep the term in line with other connectors like the AND, OR, and XOR connectors, and, I think would remove a lot of the confusion caused by calling it an implication connector. The expression, from what I am understanding from the replies and other references, has nothing to do with the common meaning of implication.

 

[PeroK]

The expression, from what I am understanding from the replies and other references, has nothing to do with the common meaning of implication.

I'm not sure what else it can mean. Implication in common use is similar to the mathematical meaning.

$A \implies B$ means "If $A$, then $B$"

That should be a plain matter, IMHO.