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

[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.

 

[Mark44]

@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.

I don't believe there is such a thing -- at least I've never heard of one, but there is such a thing as a NOR gate, which as a logical expression would be written as $\neg(A \lor B)$. Notice that this is different from $\neg A \lor B$. See NOR gate - Wikipedia

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.

 

[stevendaryl]

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

I wouldn't say it has "nothing to do with" the common meaning of implication. If you make a claim such as "If Alice goes to the store, Bob will go with her.", the only way that anyone would say that your claim was definitely false is if Alice goes to the store and Bob does not. The other three cases (1. Alice and Bob both go, 2. Alice doesn't go, but Bob goes anyway, 3. Alice doesn't go, and neither does Bob) don't make the claim false.

So the material implication is a kind of double-negation of the common meaning: $A \implies B$ is true exactly when the common meaning is not definitely false.

 

[NoahsArk]

I appreciate the responses. I think I have the general idea now. Hopefully it starts to sink in more after going over more examples.

 

[ST Mannew]

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.

If we don't prepare, then we can't win the war.

 

[Mark44]

If we don't prepare, then we can't win the war.

Your implication is the inverse of the implication you quoted, but doesn't necessarily follow logically from that implication.

Given an implication $p \Rightarrow q$ that we assume is true, we can form three other implications.
$\neg p \Rightarrow \neg q$ -- this is the inverse of the implication above.
$q \Rightarrow p$ -- the converse of the implication above.
$\neg q \Rightarrow \neg p$ -- the contrapositive of the implication above.

Of the latter three implications, only the contrapositive is equivalent to $p \Rightarrow q$. The inverse and converse are not equivalent.