Haldo fora, I bought a discrete math book awhile ago since I was amused with the fun symbols in the introduction . Ended up being a really good decision. I'm studying logic from the book, and it's done a very good job so far, although I need some more justification for the definition of [imath]q \rightarrow p[/imath].
When explaining the concept of an argument, it feels very strange to say an argument is true when one of its premises isn't met. I understand it mechanically, and I'm getting by with it qualitatively by understanding it returns false whenever q is false while p is true. But this isn't always a satisfying explanation.
First off, if an individual is trying to illustrate something is true, and one of their assumptions in their argument is wrong, isn't their argument false? Since it isn't the case that their conclusion will necessarily be true. This argument feels different then a theorem, since showing the theorem is happening seems to be just as important as the theorem itself.
Secondly, I'm understanding [imath]p \wedge \neg q \Leftrightarrow p \rightarrow q[/imath] by observing that statements that are always false, have a contradiction in them, and that this contradiction is logically equivalent to a logical implication.
[math]p \rightarrow q
\Leftrightarrow \neg p \vee q
\Leftrightarrow \neg ( p \wedge \neg q )[/math]
Adding the extra or to say [imath]p \wedge \neg q \rightarrow \mathbf{F}[/imath] Only seems to confuse the matter given how I'm understanding the implies, since I value it for returning false when q is false when p is true. Adding the implies, just feels like a roundabout way for guaranteeing the premises are always false. However when I read it outloud, it sounds like the premises are trying to imply no contradiction in particular, just some contradiction. Also reading that a list of premises imply a false statement is unintuitive. Some advice on how to approach explaining these concepts, or thinking about these concepts would be very helpful. Thanks in advanced !
Good way to understand imply statement.
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 jestingrabbit
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Re: Good way to understand imply statement.
polymer wrote:First off, if an individual is trying to illustrate something is true, and one of their assumptions in their argument is wrong, isn't their argument false? Since it isn't the case that their conclusion will necessarily be true. This argument feels different then a theorem, since showing the theorem is happening seems to be just as important as the theorem itself.
The way you translate that part of an argument into logic is to say [imath]p \wedge (p\to q) \Rightarrow q[/imath] and that's true. Just [imath]p\to q[/imath] doesn't prove anything.
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 silverhammermba
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Re: Good way to understand imply statement.
polymer wrote:[imath]p \wedge \neg q \rightarrow \mathbf{F}[/imath]
You're setting yourself up for confusion with that sort of notation. The conventional way of translating that statement into English would be "If p is true and q is not true, then false is true." Using the constants true and false in logical statements is sort of a waste of time since they can always be evaluated and simplified. See the following:
[imath](p \rightarrow T) \Leftrightarrow T[/imath]
[imath](p \rightarrow F) \Leftrightarrow \neg p[/imath]
[imath](p \wedge T) \Leftrightarrow p[/imath]
[imath](p \wedge F) \Leftrightarrow F[/imath]
[imath](p \vee T) \Leftrightarrow T[/imath]
[imath](p \vee F) \Leftrightarrow p[/imath]
Overall I'm having a really hard time understanding your wording since you have a sort of atypical way of describing logical statements.
Surprisingly not. As jestingrabbit was sort of pointing out, every formal logical argument is essentially of the form, "If all of these statements are true then the following statements are true." So translating the following into English,polymer wrote:First off, if an individual is trying to illustrate something is true, and one of their assumptions in their argument is wrong, isn't their argument false? Since it isn't the case that their conclusion will necessarily be true.
jestingrabbit wrote:[imath]p \wedge (p\to q) \Rightarrow q[/imath]
we have "If the statement p is true and the statement 'When p is true, q is true' is true, then q is true." Which is a true logical statement regardless of the truth of p or [imath]p \rightarrow q[/imath]. That's the very basis of proofs by contradiction. Such proofs depend on the fact that one can start with an assumption that we know is false and then construct a valid logical argument using it!
Re: Good way to understand imply statement.
Sorry, I agree I did word my questions funny. Let me try again...
I understand a logical argument is correct if when all of its premises are true the conclusion must be true. This is embodied in logical statements like [imath](p) \wedge (p \rightarrow q) \Rightarrow q[/imath]. If however some student argued that polluted water fountains make kids sick, principles help sick kids, and that their water fountains were polluted. If the fountains weren't actually polluted, and the kid was telling people the principle will fix them on the basis that they are polluted, then he's wrong. This type of argument is slightly different then a proof for some theorem, because theorems(as I understand them) don't care whether their conclusions are true or not, they only care if they're true given various assumptions. A theorem isn't wrong if all of its assumptions aren't met, a theorem is only wrong if a conclusion is false for a given set of premises. I feel like I understand this distinction, and that there is a distinction, although I didn't know if there was a more formal term or mindset for this distinction. What I want to write is [imath](p \rightarrow q) \wedge p \wedge ( (p \rightarrow q) \wedge p \Rightarrow p)[/imath], with the idea that if the whole statement is true, then p is necessarily true, but if the whole statement is false then p isn't necessarily true, and the "argument over the truth of p" is false. I'm sure my attempt at wrestling this with mine and others intuition has been dealt with before though, so I was curious what your thoughts were regarding arguments that wish to show some statement is true, but fail because one of their assumptions are false.
silverhammermba: See I thought [imath]p \rightarrow q \Leftrightarrow p \wedge \neg q \rightarrow \mathbf{F}[/imath] was confusing too. That's just how my book presented it, so I figured it was a convention. I prefer to think [imath]p \rightarrow q \Leftrightarrow \neg (p \wedge \neg q)[/imath]. With the idea being, that if I have a statement like [imath](p \rightarrow q) \wedge p \Rightarrow q[/imath] then I know the equivalent statement [imath]\neg ((p \rightarrow q) \wedge p \wedge\neg q)[/imath] is a tautology, and that consequently it's negation [imath](p \rightarrow q) \wedge p \wedge\neg q[/imath] is a contradiction. Seeing the list of propositions as a contradiction works better in my head. I just didn't know if there was a way of thinking about [imath]((p \rightarrow q) \wedge p\wedge\neg q) \Rightarrow \mathbf{F}[/imath] that I wasn't getting.
Hopefully that explanation was clearer...I feel like I have a pretty good understanding of how the implies statement works, I can't answer every question people throw at me though. So I'm exercising my understanding further.
I understand a logical argument is correct if when all of its premises are true the conclusion must be true. This is embodied in logical statements like [imath](p) \wedge (p \rightarrow q) \Rightarrow q[/imath]. If however some student argued that polluted water fountains make kids sick, principles help sick kids, and that their water fountains were polluted. If the fountains weren't actually polluted, and the kid was telling people the principle will fix them on the basis that they are polluted, then he's wrong. This type of argument is slightly different then a proof for some theorem, because theorems(as I understand them) don't care whether their conclusions are true or not, they only care if they're true given various assumptions. A theorem isn't wrong if all of its assumptions aren't met, a theorem is only wrong if a conclusion is false for a given set of premises. I feel like I understand this distinction, and that there is a distinction, although I didn't know if there was a more formal term or mindset for this distinction. What I want to write is [imath](p \rightarrow q) \wedge p \wedge ( (p \rightarrow q) \wedge p \Rightarrow p)[/imath], with the idea that if the whole statement is true, then p is necessarily true, but if the whole statement is false then p isn't necessarily true, and the "argument over the truth of p" is false. I'm sure my attempt at wrestling this with mine and others intuition has been dealt with before though, so I was curious what your thoughts were regarding arguments that wish to show some statement is true, but fail because one of their assumptions are false.
silverhammermba: See I thought [imath]p \rightarrow q \Leftrightarrow p \wedge \neg q \rightarrow \mathbf{F}[/imath] was confusing too. That's just how my book presented it, so I figured it was a convention. I prefer to think [imath]p \rightarrow q \Leftrightarrow \neg (p \wedge \neg q)[/imath]. With the idea being, that if I have a statement like [imath](p \rightarrow q) \wedge p \Rightarrow q[/imath] then I know the equivalent statement [imath]\neg ((p \rightarrow q) \wedge p \wedge\neg q)[/imath] is a tautology, and that consequently it's negation [imath](p \rightarrow q) \wedge p \wedge\neg q[/imath] is a contradiction. Seeing the list of propositions as a contradiction works better in my head. I just didn't know if there was a way of thinking about [imath]((p \rightarrow q) \wedge p\wedge\neg q) \Rightarrow \mathbf{F}[/imath] that I wasn't getting.
Hopefully that explanation was clearer...I feel like I have a pretty good understanding of how the implies statement works, I can't answer every question people throw at me though. So I'm exercising my understanding further.

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Re: Good way to understand imply statement.
When explaining the concept of an argument, it feels very strange to say an argument is true when one of its premises isn't met.
I think your confusion here is semantic. Propositions (including the premises and the conclusions of arguments) are true or false. An argument, per se, is not true or false but valid or invalid. When you talk about an argument being true, I think what you have in mind is the conclusion of the argument being true. An argument with a false premise is technically valid, but its conclusion may be false. The validity of an argument implies the truth of its conclusion only in the case that all the premises are true. So while an argument with a false premise is valid, it is also vacuous  it does not establish the truth of its conclusion, and a rational agent will not be persuaded by it.
Re: Good way to understand imply statement.
polymer wrote:When explaining the concept of an argument, it feels very strange to say an argument is true when one of its premises isn't met. I understand it mechanically, and I'm getting by with it qualitatively by understanding it returns false whenever q is false while p is true. But this isn't always a satisfying explanation.
First off, if an individual is trying to illustrate something is true, and one of their assumptions in their argument is wrong, isn't their argument false? Since it isn't the case that their conclusion will necessarily be true. This argument feels different then a theorem, since showing the theorem is happening seems to be just as important as the theorem itself.
I'm no expert on this, but most people would agree that this is true for every real number x:
x < 0 implies x < 1
It is therefore true for x = 2:
2 < 0 implies 2 < 1
Re: Good way to understand imply statement.
Yes, that is correct. The way to read "X implies Y." is "Suppose X, then Y." So, you are effectively saying, "Suppose that 2<0, then 2 < 1." which is a true statement since the first clause is false.mark999 wrote:I'm no expert on this, but most people would agree that this is true for every real number x:
x < 0 implies x < 1
It is therefore true for x = 2:
2 < 0 implies 2 < 1
 t1mm01994
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Re: Good way to understand imply statement.
Time for the Ifstatement crosstable!
You have a minor premise A, and a major premise B, and C is the entire ifstatement. so C: if A, then B is your logic statement (sorry if I got the terms wrong, this last part should have made it clear).
We now have 4 cases:
A is true A is false
B is true C is true C is true
B is false C is false C is true
So, if the minor premise (A) is false, that yields the entire statement to be true, no matter what gibberish the last part is.
The statement "If 2+2=5, then my mom ate the universe", is true, since obviously, the minor premise is false.
The statement "If my mom at the universe, then 2+2=4" Is true too, because again, the minor premise is false. The fact that 2+2 is in fact 4, does not matter.
Sorry for the "my mom ate the universe, but I had to come up with something that made no sense and I'm not that good in logic symbols.
You have a minor premise A, and a major premise B, and C is the entire ifstatement. so C: if A, then B is your logic statement (sorry if I got the terms wrong, this last part should have made it clear).
We now have 4 cases:
A is true A is false
B is true C is true C is true
B is false C is false C is true
So, if the minor premise (A) is false, that yields the entire statement to be true, no matter what gibberish the last part is.
The statement "If 2+2=5, then my mom ate the universe", is true, since obviously, the minor premise is false.
The statement "If my mom at the universe, then 2+2=4" Is true too, because again, the minor premise is false. The fact that 2+2 is in fact 4, does not matter.
Sorry for the "my mom ate the universe, but I had to come up with something that made no sense and I'm not that good in logic symbols.

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Re: Good way to understand imply statement.
There's specific logical terminology designed for understanding things like this.
The following:
is a valid argument because, not to complicate things too much, the logic works. That is, it works on a purely abstract basis  before you plug in specific statements for P and Q.
However, it's only sound if premises 1 & 2 are actually true.
So, the following:
is a valid argument, because the conclusion follows from the premises, but not a sound one, because at least one of the premises isn't true. In this case, both the premises. And a valid argument with a false premise gives a false conclusion  not because the logic is wrong, but because the premises are wrong. Garbage in, garbage out.
Note that this:
is also valid, but not sound  this time, even though premise 1 is true, premise 2 is still false, hence the conclusion is false.
Whereas, the following:
is both valid (because the logic works) and sound (because the premises are true).
So when you said true or false instead of valid/invalid and sound/unsound you are confusing the issue, because it's not clear whether you mean the logic is "correct" or the facts are "correct". And, as we all know, if you get the wrong facts to begin with, no matter how correct your logic is, you can't prove the right conclusion from the wrong facts. (You can coincidentally arrive at the right conclusion, but not by proving it correctly from the logic).
"True" / "False" = facts, or in logical argument terms, premises
"Valid" / "Invalid" = whether your logic is correct or not
"Sound" / "Unsound" = whether you have the correct combination of true facts and valid logic.
The following:
Code: Select all
Premise 1: If P then Q
Premise 2: P
Conclusion: Therefore , Q
is a valid argument because, not to complicate things too much, the logic works. That is, it works on a purely abstract basis  before you plug in specific statements for P and Q.
However, it's only sound if premises 1 & 2 are actually true.
So, the following:
Code: Select all
Premise 1: If 2+2=5, then my mother is the President of the USA
Premise 2: 2+2=5
Conclusion: My mother is the President of the USA
is a valid argument, because the conclusion follows from the premises, but not a sound one, because at least one of the premises isn't true. In this case, both the premises. And a valid argument with a false premise gives a false conclusion  not because the logic is wrong, but because the premises are wrong. Garbage in, garbage out.
Note that this:
Code: Select all
Premise 1: If 2+2=4, then my mother is the President of the USA
Premise 2: 2+2=4
Conclusion: My mother is the President of the USA
is also valid, but not sound  this time, even though premise 1 is true, premise 2 is still false, hence the conclusion is false.
Whereas, the following:
Code: Select all
Premise 1: I am Ellen's son
Premise 2: If I am a woman's son, then that woman is my mother
Conclusion: Ellen is my mother
is both valid (because the logic works) and sound (because the premises are true).
So when you said true or false instead of valid/invalid and sound/unsound you are confusing the issue, because it's not clear whether you mean the logic is "correct" or the facts are "correct". And, as we all know, if you get the wrong facts to begin with, no matter how correct your logic is, you can't prove the right conclusion from the wrong facts. (You can coincidentally arrive at the right conclusion, but not by proving it correctly from the logic).
"True" / "False" = facts, or in logical argument terms, premises
"Valid" / "Invalid" = whether your logic is correct or not
"Sound" / "Unsound" = whether you have the correct combination of true facts and valid logic.
Re: Good way to understand imply statement.
Thanks for the suggestions, particularly the new terminology. Words are good, they can be very helpful. I'll try and use the extra terminology more in the future when explaining things, finding some means to specify why an argument isn't correct(or "sound") is important. Was just trying to see logic statements as true of false propositions, and figured that there standard way to structure a proposition to represent whether a truth value was "necessarily true." I'll obsess over this less, thanks for the bits of advice.
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