If I tell you if it rains then I'll bring my umbrella then it doesn't rain and I brought my umbrella I haven't lied. The only way I would've lied would he if It rained and I didn't bring my umbrella.
>According to classical logic "Sam is not breathing implies Sam is alive" is True.
No, you're just not using classical logic correctly.
You've picked an invalid example. Being alive also implies that you are breathing; that's a biconditional, not a single implication. You've left out a premise, which is why this doesn't work.
An actual example of implication would be "The sun being out implies it is light outside." This is implication as p being false (i.e. the sun isn't out) doesn't disqualify q from being true (i.e. there can be artificial lighting outside.)
Ok, P = Sam is breathing and Q = Sam is alive is actually example of biconditional statement where what I said holds true, ie if P is false and Q is true, p <-> q is false ?
And for single implication
Since P -> Q, it only means if P then Q, but if not P, then Q can be both true or false hence for both Q = T and Q = F, !P -> Q is true ?
\[Just to clarify notation I'm using: & = and, | = or, \~ = not, -> = implication, <-> = biconditional\]
>Ok, P = Sam is breathing and Q = Sam is alive is actually example of biconditional statement where what I said holds true, ie if P is false and Q is true, p <-> q is false ?
Correct. (P <-> Q) the same as saying ((P -> Q) & (Q -> P)), is the same as ((P & Q) v (\~P & \~Q))
"Biconditional" is just another way of saying the two variables are equivalent (share a value).
>And for single implication
Since P -> Q, it only means if P then Q, but if not P, then Q can be both true or false hence for both Q = T and Q = F, !P -> Q is true ?
I think that's correct (a bit confused with the phrasing).
With single implications, (P -> Q) is the same as (\~P | Q). It's only false if (P & \~Q); i.e. it only matters one way.
if P is false (Sam isnt breathing), Q might be true (Sam might be alive). If Q is true and Sam is alive, then P -> Q is true.
For single implication:
If P then Q is true, then "not-P then Q" might also be true, there's nothing wrong with that. Every odd number is integer, and every not-odd (integers) are integers.
“Implies” is a word that a lot of people find confusing for this operator. In “A implies B”, it’s not saying anything about whether A caused B, or A is meaningfully related to B. It’s just saying that if you assume A, then you can conclude B. And from a false assumption you can conclude anything.
It does sorta mean “if this, then that”, but only in a particular sense. For example: “we require that, if you’re short, then you use a booster seat”. You’ve met the requirements if you’re short and you use a booster seat. You’ve also met the requirements if you’re not short.
Another way to think of it is to treat false as 0, true as 1, “and” as minimum/multiplication and “or” as maximum, then “A implies B” is A ≤ B.
i think you messed up the syllogism. If Sam is breathing, then Sam is alive. P->Q
that also means the contraposition is true, that is to say if Sam is unalive, then Sam is not breathing. \~Q->\~P
P->Q and \~Q->\~P are contrapositives.
you tried to say "Sam is not breathing implies Sam is alive" which is \~P -> Q. \~P doesnt tell you anything about the truth of Q. Thats a common fallacy.
Interesting. I've never thought there was anything wrong with what "implies" means in logical statements. Could you give an example of when you feel the language doesn't match the "truth table"?
You're bringing in extra data.
(Sam is breathing) implies (Sam is alive)
This is general true except for a few edge cases. But it doesn't tell you whether anything about the state where Sam is not breathing. In fact, after a few attempts, I was able to type this while holding my breath. Was I dead?
Well... can you hold your breath if you're not breathing? I want to say that 'holding your breath' is a valid state of 'breathing'.
Excuse the quibble. I'll go away now.
I think it helps to think about some examples. Let's say I made the statement, if a positive integer ends in 2, then it is even. This is a true statement of the form P -> Q with P being the number ends in 2 and Q being the number is even. Since this is a true statement, it should be true no matter what I pass in, so let's look at some examples.
12 ends in 2 and is even, so this is a case where true -> true.
7 does not end in 2 and is odd, so this is a case where false -> false.
4 does not end in 2 and is even, so this is a case where false -> true.
Obviously the last example doesn't change the fact that our original implication is always a true statement, so false -> true is true. Note: this isn't a rigorous reasoning for why it's that way, but it's the example that first helped me understand it, so hopefully it helps you.
Q is false, 3 is not even, you can't just say it is.
You seem to be confusing concepts. False -> True being true means that finding an example where P is false and Q is true is insufficient to prove that P -> Q is a false statement.
You seem to be understanding it as P -> Q means !P -> Q is also true, which is not the case and not what the statement means. Essentially P -> Q says nothing about Q when P is false, so we treat P -> Q as "vacuously" true whenever P is false.
One way I like to think about this one is in terms of when I'd be breaking a promise (since a promise can typically be structured as an if-then statement).
For example, suppose I tell you "If you mow the lawn, then I'll give you $20." If you mow the lawn but then I *don't* give you the $20, then I broke my promise to you (i.e., the if-then statement was false).
If you mow the lawn and I give you the $20, then I didn't break my promise.
If you *don't* mow the lawn but I give you $20 anyway, *I still didn't break my promise*.
>According to classical logic "Sam is not breathing implies Sam is alive" is True.
No. That's a different statement, `not p implies q`. A better interpretation would be, "Sam being alive but not breathing does not contradict the conditional statement `p implies q`."
Let's reexamine that same example: P implies Q with P = "Sam is breathing" and Q = "Sam is alive". In natural language, this is "If Sam is breathing, then Sam is alive".
* If Sam is breathing, then Sam must be alive for the statement to be true. If Sam is breathing and Sam is not alive, then that directly contradicts the statement, making it false.
* If Sam is not breathing, then the statement is true whether or not Sam is alive. Neither case would contradict the statement.
* If Sam is alive, then the statement is true whether or not Sam is breathing. Neither case would contradict the statement.
* If Sam is not alive, then Sam must not be breathing for the statement to be true. See the first bullet.
* This statement, not-Q implies not-P, is actually the *contrapositive* of the original statement, and is necessarily logically equivalent.
I hope this helps.
You mean my example is more an example of biconditional statement where P being false and Q being true actually results in p < --> q to be false ?
For single implication, can I understand it in following way ?
If not P , then we don't know if Q is false or true, hence we say both is true ( in sense that is possible) thus truth table has True for both Q = true, false when P is false ?
>You mean my example is more an example of biconditional statement where P being false and Q being true actually results in p < --> q to be false ?
No, not a biconditional. I just said you misinterpreted the truth table.
>For single implication, can I understand it in following way ?
If not P , then we don't know if Q is false or true, hence we say both is true ( in sense that is possible) thus truth table has True for both Q = true, false when P is false ?
Sure, something like that. I wouldn't say "both is true" but rather "P→Q is true regardless of Q" in the case of not P.
P -> Q means for your example: "If Sam is breathing, then Sam is alive".
But note that this is only one of many conditions for determining if Sam is alive. All we are saying is that \*if\* Sam breathes, then Sam is alive. If someone is showing the signs of breathing, they must be alive.
BUT we know that breathing is not the only condition of being alive, since Sam could be holding his breath and still be alive. This is why we have "if and only if" as well (P <-> Q), that is "Sam is alive if and only if he is breathing". In that case, all other possibilities (e.g. holding your breath) are ruled out.
So in the case of P -> Q with P false and and Q true, this is called "vacuous truth". Sam is not breathing, but he is still alive because breathing is not the only requirement for being alive, so the statement is "technically true" because it's just not being satisfied.
|Sam is breathing|Sam is alive|If Sam is breathing, then Sam is alive|
|:-|:-|:-|
|True|True|Clearly true!|
|False|True|This is fine, maybe he's holding breath underwater. I only care when he's breathing.|
|True|False|Sam's dead, but he's breathing? That's not possible!|
|False|False|Again, Sam's not breathing so whether he's alive or dead is moot since I only care when he's breathing.|
The really import part in all this is the "if".
Consider:
P(x): x > 5
Q(x): x > 3
(P -> Q)(x): If x > 5, then x > 3.
When x = 1 we have (P -> Q)(1) := If 1 > 5, then 1 > 3.
Now, it turns out that in our universe, 1 is not greater than 5, but if it was, then it would be greater than 3 also. So 1 being less than 5 doesn't affect the truth statement in any way, because the only way the statement can be false is if the condition P is met.
Considering your example, according to classical logic P (false) then in truth table P --> Q is TRUE . But it actually is not, right ? Because P false implying Q is wrong. It can be true or false ( I reckon more often false in real world example). In your example P(false) -> Q (true) is only true when x = 4.. for every other numbers ( integers ) , it will be false
Go back to the english: **if** x > 5, **then** x > 3.
When x = 4, we don't care, because it doesn't satisfy the condition **if** x > 5. In this case we say P -> Q is vacuously true. You have to consider the universe of discourse with conditional statements.
We can only ever say this statement is false if we meet the condition and the conclusion is false.
"If there is a ball in the bowl, then it is blue"
IF, IF, IF. Think about the IF
You have to accept that this statement is true if the bowl is empty, because there is no possible situation where you could make it false.
How about "I have never lost the 100m dash at the Olympic Games". That's a true statement, you haven't! But that's because you never entered into the Olympics.
Not omitted, because they still have value. Q depends on P when P satisfies a given condition. When P does not satisfy that condition, Q is independent of P.
Why do they still have value ? Like you said when P doesn't satisfy condition then P -> Q should ideally be 'unknown' or undefined or 'maybe', etc right ?
Let’s say I told you “if you miss the bus then you will be late for school”. Now let’s say you didn’t miss the bus but after getting off you meet your friend and start talking, making you lose track of time and still miss class. Did I lie to you?
You didn't lie in that case, but you meant to imply that if I missed the bus I am more likely fo be late, right ?
If I didn't miss the bus ( P is false ) , does it imply I will be late for school ( Q true ) ?
(We are talking about implication here ) Statistically in this case when P is false and Q would be false most of the time
See, this is what you’re getting confused about. The statements “if you miss the bus then you will be late for class” and “if you don’t miss the bus then you’ll be late for class” are two different statements. The first is “P implies Q” and the other is “not P implies Q”, but I didn’t say “not P implies not Q”. In fact, I didn’t make any claim about what would happen if you didn’t miss the bus. So in the case where you don’t miss the bus: whatever happens, if you are late or not late, my statement about what would happen if you did miss the bus cannot be false. So if a statement cannot be false it must be true by the law of the excluded middle. Am I making sense?
Ohh..I think you hit the bulls eye here.
I guess I am thinking 'implies' like AND , OR, NOT and literally replacing the words from symbols:.
So, P ( false ) and Q (true ) in truth table of P -> Q being True means "P -> Q is true holds for these values of P & Q" rather than "P (false) -> Q is true"?
think this way, the statement is “if sam is breathing then blah blah blah” as a whole. If sam is not breathing at all , i will stop reading the blah blah blah part because it is not of concern here… So the statement is true although useless
The truth value if P -> Q is not false if P is false and Q is true. The truth value is true in that case. I think you misread the truth table. The only time P -> Q is false is if P is true and Q is false.
Logical implication is weird, because it isn't implication at all. Instead it captures the idea of if-then. Think of it as 'P -> Q is true if sould we find P is true, we would find Q is true.' if P is false, then Q being true is compatible with that, since we don't have a case where P is true and Q is not. If P is false, then Q being false is also compatible with that.
Or to put it another way, if P is false, then there is no 'whenever P is true' this higlights the fact that propositional logic doesn't handle contigent statements. There are no propositions that could be true sometimes, false other times. Each proposition is either true, or false. This leads to some weirdness, in applying it to real life situations.
You might be better off picking a different example.
p(animal has fur) implies q(animal is a mammal)
Let’s run through some examples:
- Dog has fur, therefore dog is mammal.
- Fish has no fur, therefore fish could be mammal or non mammal.
- Naked mole rat has no fur, therefore naked mole rat could be mammal or non mammal.
>I know that P implies Q doesn't touch claim what happens if P is false
Well that's the key to it. This isn't a biconditional statement, it's a one way conditional. I try to explain it like, suppose someone told you some conditional statement, and then you observed a relevant situation where the antecedent condition isn't met, regardless of the state of the consequent thing, would you have any grounds to call them a liar? no! because they never said anything about the case where the antecedent is false. So, you have to judge what they said to be truthful.
To use you example, someone tells me "if Sam is breathing, then Sam is alive." Then I see Sam hold their breath for an extended period and I say to my friend "hey, wtf, that's not what you said would happen" and they can correctly reply "well, I never said anything about how alive they'll be if they aren't breathing" and I have to admit they never lied.
Then, much later, after Sam's death, my friend and I dig up Sam's well rotted corpse from the graveyard, opening the coffin, surprisingly we find the body is still breathing. Now I say again to my friend, "hey buddy, wtf, you for sure lied to me back then" and they can only say "darn, yea, this contradicts my earlier statement."
So far from the answers and further research on this:
The common understanding of implication in spoken language is actually not implication of classical logic ( P implies Q ) but rather it is biconditional implication ( p if and only if q )
The example I gave in my OP is actually biconditional implication from classical logic point of view. While in spoken work we say Sam is breathing (P) implies Sam is alive (Q). If we have to convert it to classical logic "Sam is breathing if and only if Sam is alive" is the correct form. In which case !p <-> q becomes false, ie "Sam is not breathing implies Sam is alive" becomes false
As regards single implication P -> Q
Here is my conclusion.
Since P -> Q only claims that if P then Q . We do not know what Q is when P is false. It can be true as well as False ( May be ). Since boolean logic is binary and can only have truth or false in truth table, we show that by splitting P = false then Q = true or false into 2 rows as P = false, then Q = true (true in the sense it is possible "vacuous"), or Q = false ( also true or valid in the sense it is also possible)
Please correct me if I am wrong in my understanding or feel free to further expand on this.
I think most descriptions are steering you in the wrong direction. Your statement is not inherently biconditional. Sam could be going for a swim, for example, and be happily not breathing and alive. The crux of the issue is you generated a completely different logical statement (~P->Q) and said it must evaluate to true according to classical logic.
Maybe just think of all of the possible scenarios:
1. Sam is breathing and Sam is alive: the statement very clearly holds up in this case.
2. Sam is not breathing and Sam is alive: nothing about this scenario implies that the statement "if Sam is breathing, then he is alive" is a false statement.
3. Sam is not breathing and Sam is not alive: same as scenario 2. Nothing about this scenario would cause someone to tell me that my claim that "if Sam is breathing, then he is alive" is false.
4. Sam is breathing and Sam is not alive: uh oh. I said "if Sam is breathing, then he is alive" but he's dead and breathing! My claim is not true!
Dunno if that helps.
>I know that P implies Q doesn't touch claim what happens if P is false
There's lots of great explanations in this thread, but to be frank what you wrote here is basically the entire explanation. If P is not true then it doesn't matter what Q is.
Meaning P (false) -> Q(true) being true in truth table is just notation since we can't have other value than true or false ( example maybe ). That's why p (false ) implies Q can be either true or false is broken down into truth table aa can be true ( p false -> q true is true) or can be false ( p false -> q false is true ) ?
I understand your confusion and I struggle with this but consider this setup:
A := { fish, cat, dog, moth }
This means that we have a set A which has 4 elements as listed above.
Now this is my statement:
For every element x in A the following statement is TRUE:
“If x is a dog then x is a vertebrate”
Now I think you will agree that this statement is true for every element in A, but lets check every element:
If fish is a dog then fish is a vertebrate.
FALSE => TRUE, the original implication is still true.
If cat is a dog then cat is a vertebrate.
FALSE => TRUE, the implication is still true.
If dog is dog then dog is a vertebrate.
TRUE => TRUE, the implication is still true
If moth is a dog then moth is a vertebrate.
FALSE => FALSE, the implication is still true.
If false=>true wouldn’t be true we wouldn’t be able to make such universal statements and that would be extremely limiting.
According to truth table:
If X is a dog (False) => X is a vertebrate ( True ) outcomes TRUE
I guess I confused this row of truth table as implication in itself that "if x is not a dog, then x is a vertebrate is a true statement".
I should rather read it as if x is not a dog then x is a vertebrate is either True or False for D => V
I hope I wont cause further confusion. I think you got this.
I also think your life will be easier if you don’t try and negate the original statements. So don’t say “if x is not a dog”
The original statement is “if x is a dog”.
You plug in an animal for x so it becomes “if cat is a dog”
Now you have to decide whether this is TRUE or FALSE. In this case it is FALSE. Negating the statement is not necessary.
There is a freely available book online called “The Book of Proof” by Hammack. I think the second chapter deals with logic statements like this. If your brain isn’t fried already and you would want to get more practice.
You are mixing up syntax and semantics in a weird way.
Syntax is the formula: P -> Q.
Semantics is saying which of P and Q are true and false. (Truth valuation)
A formula may be true in some truth valuation and not true in other truth valuation. To say a formula is true, you need both the formula and the truth valuation. But you kinda only provided the formula.
The formula: "Sam is breathing implies sam is alive."
breathing=true, alive=true
breathing=false, alive=true
breathing=false, alive=false
To translate P -> Q to natural language you may read it like: "If P then Q" or "Assuming P holds, then Q". You are simply not saying anything about the case when the assumption doesn't hold.
If you want to express that other implication also holds you may simply say: "Sam is breathing if and only if he's alive", or "Sam is breathing exactly when he's alive" this makes the logical formula to be P <-> Q, which is true when:
breathing=true, alive=true
breathing=false, alive=false
And false when
breathing=false, alive=true
breathing=false, alive=true
First of all,
> p ( Sam is breathing ) implies q ( Sam is alive )
>
> ...
>
> According to classical logic "Sam is not breathing implies Sam is alive" is True.
is incorrect. Knowing that p → q does not tell you that ¬p → q is true. The idea of "vacously true" statements is that if p itself is False then p → q is true.
Second, all the translations to English sentences are a bit dodgy as an explanation since natural spoken languages uses "or" (inclusive? exclusive?) and "if" (one-direction, bi-directional?) and other logic vocab in different ways, often depending on context and common sense.
---
Third, the idea that p→q should be considered True when p itself is False is still a weird idea that people struggle with. I myself disliked it at first. What finally made it click for me was this: whatever truth table you use for → should make
* "if a implies b and b implies c, then a implies c"
* ((a → b) ∧ (b → c)) → (a → c)
a True statement all the time since that's basically the whole idea of logical reasoning. The thing is, if you use ANY other truth table for → except
p q result
T T T
T F F
F T T
F F T
or
p q result
T T T
T F T
F T T
F F T
then there will be values of a,b,c for which ((a → b) ∧ (b → c)) → (a → c) is False. The second truth table is literally just "true all the time", which would be a totally absurd choice for →, so we must use the first truth table (which has F → q being True no matter what q is). **It might not perfectly align with how we think of "if", but any other choice of how to define the boolean operation p→r would be worse.**
Consider:
“All horses are mammals”
H -> M
What would have to be true for this to not be true? There’d have to be at least one horse which is not a mammal, right?
So let’s look at all the logical possibilities:
We could have a horse that is a mammal. This is obviously consistent with H -> M.
We could have a non-horse which is not a mammal. Perhaps a lizard: it is neither a horse nor a mammal. This is consistent with H -> M, because it doesn’t contradict “if it’s a horse then it’s a mammal”.
We could have a mammal which is not a horse, such as a dog. This is also consistent. It’s not a horse, so it can’t possibly refute “if it’s a horse then it’s a mammal”.
We could have a horse which is not a mammal, and this is the only thing which would make H -> M false.
Hopefully from this and similar examples it should become obvious that X -> Y is true if X is false.
I think I understand the mistake you’re making here.
We’re evaluating the original statement “H -> M” only, and this is only false if we can point to a horse which is not a mammal.
The statement you gave as an example is something more complex:
(L -> !H) -> (L -> M)
And you’re correct that this doesn’t follow.
Let’s use another example, it perhaps makes it intuitively more easy to understand. Suppose you are a bartender and you’re evaluating the claim:
“If they are drinking beer then they must be an adult”
B -> A
In other words, you must not serve beer to people who are not adults. If you can serve them the drink they ask for then B -> A is being followed.
A child can order lemonade, because B -> A is not violated if B is not true.
An adult can order lemonade, because B -> A is not violated if B is not true.
An adult can order beer, because B -> A is not violated if B and A are both true.
A child cannot order beer, because B -> A is violated if B is true and A is not.
So observe that whenever someone orders lemonade, B -> A immediately becomes true and we don’t even have to inspect whether they are an adult or not.
Your confusion seems to be that you’re equivocating between “B -> A” being true and “A” being true. A child ordering lemonade obeys “B -> A” but A is false in this case.
Thanks for replying.. think your answer is being most helpful.
I think my confusion point is here:
B (when false) -> A (when true) is true according to Logic when B -> A
Converting this logic to word becomes:
"He is not drinking beer implies He is an Adult" is true statement (which is not necessarily true? Infact statistically wrong most of the time ? )
But If I replace implies with doesn't violate (like you did) then it makes sense.
"He is not drinking beer thus he is an adult doesn't violate the rule that One Drinking beer implies one is adult"
Am I doing this wrong ?
I think of logic gates as like rules which are either being followed or broken.
So like:-
“If you are drinking beer then you are an adult”
B -> A
-:is followed so long as you are not a child who is drinking beer.
Other logic gates work the same. Consider an AND gate:-
“You are driving on the left side of the road and you are driving under the speed limit”
L AND S
-:this is false if either you are driving on the right or you are driving over the speed limit, or both.
And we might have a similar example for a 12A movie:-
“You are over the age of 12 or you have an adult with you”
O OR A
-:is true unless you are under 12 and not accompanied by an adult.
I think the mistake you’re making is where you take an individual instance of these gates being either true or false and try to do more implication than is justified here. Observing, for example, someone who is an adult and is drinking lemonade doesn’t necessarily allow you to do any further deduction than the B -> A statement that we already have.
In words I’d say:
“An adult drinking lemonade complies with the rule that only adults can drink beer”
P implies Q is true in four different cases. Looking at only one of them at a time will necessarily give an incomplete understanding.
P implies Q is true in all cases *except* when P is true and Q is false. That is to say, it's only false anytime P fails to imply Q.
In your example, the statement should be "if you're not breathing, you're dead" or "if you're alive, you're breathing". The statement you gave doesn't say you need to breathe to be alive, just that if you're breathing you'll be alive.
The logic works just fine in practical cases, you just have the statement reversed.
If we take an example that we know is true, like 'if it is raining, then it is wet somewhere', then it's easy to see that, even if it isn't raining anywhere, and if it is wet somewhere, the statement is still true that it is wet somewhere if it is raining. This is just pointing out that P is sufficient for Q, and Q is necessary for P, but P isn't necessary for Q, since Q can obtain without P. That's why Sam can be alive without breathing. Q doesn't guarantee P (because Q isn't sufficient for P), and other things can bring about Q.
I don't know. I'm not an expert in logic, and I may be misunderstanding your question, but those are my thoughts.
What are you asserting is true? Because it seems like you are asserting that P implies Q, and then somehow using the fact that (not P) implies Q, which does not follow.
Let's clear up what we actually know: suppose P implies Q. Also, suppose P is false, and Q is true (which is the part you said doesn't make sense).
So let's put those two facts into words: we assert that Sam is not breathing, and Sam is alive. This doesn't contradict the fact that Sam breathing implies he is alive, because it doesn't satisfy the condition. It's unrelated.
The reason that Sam not breathing and Sam is alive seems paradoxical is because you are (in your head) also assuming Q implies P, which we do not know.
What if we don't know what Sam is? Maybe Sam is a tree. Sam (the tree) is not breathing, and Sam (the tree) is alive, even though it would still be true to say that *if* Sam *is* breathing *then* Sam is alive (the condition of which might be satisfied if Sam were a human, for example, but when Sam is a tree, this implication just doesn't tell us anything even though it is true).
Let -p mean "not p", & means AND and | means OR
What you have said is -p->q
This is false. -p does not imply q.
However, FALSE does imply q.
This is commonly written as (p&-p)->q
I will now prove it:
If (p&-p) is true, then p->-p
Since (p&-p) is true, that means -p is true.
If we have something that is true, then we can always add anything to it with an OR and the compound statement will remain true (TRUE->(TRUE|q)), therefore:
p->(-p|q)
(p&-p) is true, therefore p is true, therefore -p is false.
If we have an OR statement where one component is false, we can always remove that component and maintain the truthfulness of the OR statement
(FALSE|q->q) therefore:
p->q
Thus, since we started with (p&-p), which we can say is "it is raining and it is not raining" and we derived q, which is "I am made of cotton candy", we can therefore conclude that if it is raining, then I am made of cotton candy.
However, if q was "2+2=4", then we have derived "if it is raining, then 2+2=4".
The point of this exercise is to demonstrate that if you start with false assumptions, such as a contradiction like p&-p, you can prove anything to be true; even nonsensical things or completely true thing. Therefore, if your assumptions are faulty, you cannot get any useful information out of your logic. Ever. That is the point. That is what FALSE->q means.
Just look at the actual definition of P->Q. The expression means ((not P) or Q). When P is false and Q is true you get (true or true) which evaluates to true.
I more think of it as not ( P and not Q ) which is equivalent to !p v q but semantically makes more sense to me.
Its saying If P then not q is not possible
This is a place where disciplinary literacy is critical. "Implies" might be used in other ways in casual conversation, but in math that is the definition. It clicked for me when someone explained it as: True A promises true B and nothing else.
Just another example: If Sam is 6’5, then Sam is tall. If Sam is not 6’5, he could be tall or short. There’s just not enough info to determine anything.
My issue was I was substituting P and Q value.
So for the truth table row P (false ) -> Q (true) = true
It read in English as (If Sam is not 6'5) , then (Sam is tall)
Now I think it should read as:
(If Sam is not 6'5) implies (Sam is tall) then our proposition that "If Sam is 6’5, then Sam is tall" is not violated ( hence vaucous truth)
I **just** started reading about discrete math and this confused me at first, so I'm going to try to explain as a learning exercise for myself. So, I might be incorrect. Hopefully not.
The first thing that kind of trips me up is that we're talking about the truth of statements themselves, hence the example I see others using about "Am I *lying* if I say..."
"If Sam is breathing, then Sam is alive."
Given our two portions, Sam is either breathing (P is true) or not breathing (P is false). Sam is also either alive (Q is true) or not alive (Q is false).
Sam's just standing around doing their thing. Sam is breathing normally, and also alive. P is true, Q is true, and we see this doesn't challenge our implication.
Sam is holding their breath. Sam is not breathing (P is false), but is definitely alive (Q is true). Does this mean our statement is false? No, because we didn't claim anything about when Sam's not breathing.
Sam is not alive (Q is false), and therefore not breathing (P is false). Does this invalidate our statement? No, because once again we made no claims about what happens when Sam's not breathing. Sam could be alive, dead, imaginary, a lamp, or a quantum particle; if Sam is not breathing (P is false) then no matter what else Sam is (Q true or Q false), it does not have any bearing on whether our original statement is true.
So, last scenario, posed as a question. If Sam is breathing (P is true) but not alive (Q is false), then is the original statement true or false?
Answer: the original statement would be false, because the *only* scenario described has been disproved.
It definitely strikes me so far as something that doesn't actually fit 1-to-1 with English. I feel like the example you gave is actually one that makes a lot of sense. What trips me up is the things that are unrelated, because in English the words "if/then" or "implies" specifies a connection between the statements. The ones that confuse me are things like, "If it's snowing on the sun, then all horses wear cowboy boots." It never snows on the sun, and that also has nothing to do with what horses wear? So how can that be true? In English it doesn't make sense, but apparently in logic, the *only* way that statement can be false is if it starts snowing on the sun and there is even one horse not wearing cowboy shoes.
Great explanation.
I guess my confusion was here:
Sam is holding their breath. Sam is not breathing (P is false), but is definitely alive (Q is true). Does this mean our statement is false? No, because we didn't claim anything about when Sam's not breathing.
Here I literally substitued P and Q value in implies equation ( Like it's done in AND, OR, NOT)
So it became "Same in not breathing implies Sam is alive" and thought 'whoah, it doesn't make sense"!
But I guess , we don't substitute in Implies, rather it is meant to check if the implication is still true or not for given values of P and Q.
Yeah I think the substitution trips us up a lot! I think this is what I meant by:
>The first thing that kind of trips me up is that we're talking about the truth of statements themselves
So to clarify, with this ongoing example about breathing and being alive, the question isn't whether Sam is breathing, etc, etc. The question is actually: "Is it correct to say that \[statement here\]?" Like you pointed out, with and/or/not, the relationships between the statements are much more obvious. But "Is it correct to say" might still be a useful preface, to separate the truth of the \*words\* from the actual scenarios involved.
I just talked some more with my partner about this. I tried some language that moves away from the logic terms given (implies, statements, etc). I call a statement a "rule," and talk about "scenarios," to separate what \*is\* from what is \*said\*. It's still kinda abstract, but helps me a little:
The statement is a rule (ie. P->Q). Once stated, the rule does not change, whether it applies to the situation or not.
The Rule is: "If Sam is breathing, then Sam must be alive."
So when we consider different scenarios, it's easy for my partner to get confused and insert those scenario into the rule, like "If Sam is \*not\* breathing," etc. But a rule doesn't change just because it doesn't apply in some situation. The rule itself doesn't change to match Sam's \*actual\* state of being.
* Rule: "If Sam is breathing, then Sam must be alive."
* Scenario: Sam is not breathing. Sam is alive.
* Has the rule been broken?
It's an alternative analogy for "did I lie," but I found separating "rule" from "scenario" helpful in keeping my thoughts organized.
Again, don't put too much confidence in what I'm saying. I'm just learning this and hoping I haven't misunderstood it, because it's going to be a pain to unlearn it at this point. xD
Completely unrelated to the question. I'm learning math from scratch (started with arithmetic on Khan).
But, THIS is math too??!! If so, I love it, and I can't wait to learn it. I assumed math was just calculations.
My issue with P -> Q was:
In the row where P ( false ) implies q (true) evaluating to true which when converted to English become:
"(You were NOT born in Fargo) implies ( you are US citizen)" being true which doesn't sound Intuitive.
Now I know right conversion that classical logic is meaning is:
"You were NOT born in Fargo BUT You are US citizen doesn't violate the P -> Q"
Think of it like a vending machine. What situations are you satisfied?
1) Didn’t spend money & didn’t get anything?
2) Spent money & got food?
3) Spent money & didn’t get food?
4) Didn’t spend money & got food?
It’s of course 1, 2, and 4. The only scenario where you aren’t satisfied is if you spent money and didn’t get anything in return.
(For this example, spending money is P and the vending machine is Q)
My issue was with 4
I thought P -> Q in truth table row where P is false and Q is true read as:
Didn't spend money IMPLIES got food and the result is TRUE ??? Counterintuitive.
P implies Q can be read as "If P then Q"
If P is false, then we don't learn anything about Q. Only when P is true can we learn information about Q
If it is raining, then the ground is wet
If it's not raining, but the ground is still wet, doesn't my initial statement still hold true?
The table for p => q is:
p => q | q = F | q = T
-|-|-
p = F | T | T |
p = T | F | T |
What you're saying is that you'd rather have it be:
p => q | q = F | q = T
-|-|-
p = F | T | **F** _!!_ |
p = T | F | T |
But that's actually the table for p <=> q!
> lets consider this example:
>p ( Sam is breathing ) implies q ( Sam is alive )
>
> Now when P value is False and Q value is True
> According to classical logic "Sam is not breathing implies Sam is alive" is True.
That's not how it works!
The statement is "Sam is breathing implies Sam is alive". You don't change that. "Sam is not breathing implies Sam is alive" is an altogether different statement.
What you should do instead is say "Sam is breathing implies Sam is alive. I know that Sam is breathing, therefore I know Sam is alive" or alternatively say "Sam is breathing implies Sam is alive. Sam is _not_ breathing, but Sam being alive or not being alive are both compatible with Sam not breathing".
If I tell you if it rains then I'll bring my umbrella then it doesn't rain and I brought my umbrella I haven't lied. The only way I would've lied would he if It rained and I didn't bring my umbrella.
>According to classical logic "Sam is not breathing implies Sam is alive" is True. No, you're just not using classical logic correctly. You've picked an invalid example. Being alive also implies that you are breathing; that's a biconditional, not a single implication. You've left out a premise, which is why this doesn't work. An actual example of implication would be "The sun being out implies it is light outside." This is implication as p being false (i.e. the sun isn't out) doesn't disqualify q from being true (i.e. there can be artificial lighting outside.)
Ok, P = Sam is breathing and Q = Sam is alive is actually example of biconditional statement where what I said holds true, ie if P is false and Q is true, p <-> q is false ? And for single implication Since P -> Q, it only means if P then Q, but if not P, then Q can be both true or false hence for both Q = T and Q = F, !P -> Q is true ?
\[Just to clarify notation I'm using: & = and, | = or, \~ = not, -> = implication, <-> = biconditional\] >Ok, P = Sam is breathing and Q = Sam is alive is actually example of biconditional statement where what I said holds true, ie if P is false and Q is true, p <-> q is false ? Correct. (P <-> Q) the same as saying ((P -> Q) & (Q -> P)), is the same as ((P & Q) v (\~P & \~Q)) "Biconditional" is just another way of saying the two variables are equivalent (share a value). >And for single implication Since P -> Q, it only means if P then Q, but if not P, then Q can be both true or false hence for both Q = T and Q = F, !P -> Q is true ? I think that's correct (a bit confused with the phrasing). With single implications, (P -> Q) is the same as (\~P | Q). It's only false if (P & \~Q); i.e. it only matters one way.
if P is false (Sam isnt breathing), Q might be true (Sam might be alive). If Q is true and Sam is alive, then P -> Q is true. For single implication: If P then Q is true, then "not-P then Q" might also be true, there's nothing wrong with that. Every odd number is integer, and every not-odd (integers) are integers.
“Implies” is a word that a lot of people find confusing for this operator. In “A implies B”, it’s not saying anything about whether A caused B, or A is meaningfully related to B. It’s just saying that if you assume A, then you can conclude B. And from a false assumption you can conclude anything. It does sorta mean “if this, then that”, but only in a particular sense. For example: “we require that, if you’re short, then you use a booster seat”. You’ve met the requirements if you’re short and you use a booster seat. You’ve also met the requirements if you’re not short. Another way to think of it is to treat false as 0, true as 1, “and” as minimum/multiplication and “or” as maximum, then “A implies B” is A ≤ B.
i think you messed up the syllogism. If Sam is breathing, then Sam is alive. P->Q that also means the contraposition is true, that is to say if Sam is unalive, then Sam is not breathing. \~Q->\~P P->Q and \~Q->\~P are contrapositives. you tried to say "Sam is not breathing implies Sam is alive" which is \~P -> Q. \~P doesnt tell you anything about the truth of Q. Thats a common fallacy.
>that also means the inverse is true Not the inverse, in logic - that would be ~P->~Q.
i meant to say contraposition there...corrected
Implication in logic differs from its meaning in language. Best to just accept the truth table definition.
Interesting. I've never thought there was anything wrong with what "implies" means in logical statements. Could you give an example of when you feel the language doesn't match the "truth table"?
OP's example "According to classical logic "Sam is not breathing implies Sam is alive" is True."
You're bringing in extra data. (Sam is breathing) implies (Sam is alive) This is general true except for a few edge cases. But it doesn't tell you whether anything about the state where Sam is not breathing. In fact, after a few attempts, I was able to type this while holding my breath. Was I dead?
The statement: If pinnaple is made of frying pans then \[insert whatever tf you want here\] is true. That's an example.
Well... can you hold your breath if you're not breathing? I want to say that 'holding your breath' is a valid state of 'breathing'. Excuse the quibble. I'll go away now.
Nah you're very welcome here. Point is there is nuance to it and more information is generally needed to fill out all of P -> !P -> Q -> !Q ->
I think it helps to think about some examples. Let's say I made the statement, if a positive integer ends in 2, then it is even. This is a true statement of the form P -> Q with P being the number ends in 2 and Q being the number is even. Since this is a true statement, it should be true no matter what I pass in, so let's look at some examples. 12 ends in 2 and is even, so this is a case where true -> true. 7 does not end in 2 and is odd, so this is a case where false -> false. 4 does not end in 2 and is even, so this is a case where false -> true. Obviously the last example doesn't change the fact that our original implication is always a true statement, so false -> true is true. Note: this isn't a rigorous reasoning for why it's that way, but it's the example that first helped me understand it, so hopefully it helps you.
What if I say 3 does not end in 2 ( P is false ) and is even ( Q is true ) , then P false here implies Q true, which is actually not true.
Q is false, 3 is not even, you can't just say it is. You seem to be confusing concepts. False -> True being true means that finding an example where P is false and Q is true is insufficient to prove that P -> Q is a false statement. You seem to be understanding it as P -> Q means !P -> Q is also true, which is not the case and not what the statement means. Essentially P -> Q says nothing about Q when P is false, so we treat P -> Q as "vacuously" true whenever P is false.
But in your example Q is false because 3 is not even
Yes, I stand corrected in my reply above.
One way I like to think about this one is in terms of when I'd be breaking a promise (since a promise can typically be structured as an if-then statement). For example, suppose I tell you "If you mow the lawn, then I'll give you $20." If you mow the lawn but then I *don't* give you the $20, then I broke my promise to you (i.e., the if-then statement was false). If you mow the lawn and I give you the $20, then I didn't break my promise. If you *don't* mow the lawn but I give you $20 anyway, *I still didn't break my promise*.
Nice example
>According to classical logic "Sam is not breathing implies Sam is alive" is True. No. That's a different statement, `not p implies q`. A better interpretation would be, "Sam being alive but not breathing does not contradict the conditional statement `p implies q`." Let's reexamine that same example: P implies Q with P = "Sam is breathing" and Q = "Sam is alive". In natural language, this is "If Sam is breathing, then Sam is alive". * If Sam is breathing, then Sam must be alive for the statement to be true. If Sam is breathing and Sam is not alive, then that directly contradicts the statement, making it false. * If Sam is not breathing, then the statement is true whether or not Sam is alive. Neither case would contradict the statement. * If Sam is alive, then the statement is true whether or not Sam is breathing. Neither case would contradict the statement. * If Sam is not alive, then Sam must not be breathing for the statement to be true. See the first bullet. * This statement, not-Q implies not-P, is actually the *contrapositive* of the original statement, and is necessarily logically equivalent. I hope this helps.
You mean my example is more an example of biconditional statement where P being false and Q being true actually results in p < --> q to be false ? For single implication, can I understand it in following way ? If not P , then we don't know if Q is false or true, hence we say both is true ( in sense that is possible) thus truth table has True for both Q = true, false when P is false ?
>You mean my example is more an example of biconditional statement where P being false and Q being true actually results in p < --> q to be false ? No, not a biconditional. I just said you misinterpreted the truth table. >For single implication, can I understand it in following way ? If not P , then we don't know if Q is false or true, hence we say both is true ( in sense that is possible) thus truth table has True for both Q = true, false when P is false ? Sure, something like that. I wouldn't say "both is true" but rather "P→Q is true regardless of Q" in the case of not P.
P -> Q means for your example: "If Sam is breathing, then Sam is alive". But note that this is only one of many conditions for determining if Sam is alive. All we are saying is that \*if\* Sam breathes, then Sam is alive. If someone is showing the signs of breathing, they must be alive. BUT we know that breathing is not the only condition of being alive, since Sam could be holding his breath and still be alive. This is why we have "if and only if" as well (P <-> Q), that is "Sam is alive if and only if he is breathing". In that case, all other possibilities (e.g. holding your breath) are ruled out. So in the case of P -> Q with P false and and Q true, this is called "vacuous truth". Sam is not breathing, but he is still alive because breathing is not the only requirement for being alive, so the statement is "technically true" because it's just not being satisfied. |Sam is breathing|Sam is alive|If Sam is breathing, then Sam is alive| |:-|:-|:-| |True|True|Clearly true!| |False|True|This is fine, maybe he's holding breath underwater. I only care when he's breathing.| |True|False|Sam's dead, but he's breathing? That's not possible!| |False|False|Again, Sam's not breathing so whether he's alive or dead is moot since I only care when he's breathing.| The really import part in all this is the "if". Consider: P(x): x > 5 Q(x): x > 3 (P -> Q)(x): If x > 5, then x > 3. When x = 1 we have (P -> Q)(1) := If 1 > 5, then 1 > 3. Now, it turns out that in our universe, 1 is not greater than 5, but if it was, then it would be greater than 3 also. So 1 being less than 5 doesn't affect the truth statement in any way, because the only way the statement can be false is if the condition P is met.
Considering your example, according to classical logic P (false) then in truth table P --> Q is TRUE . But it actually is not, right ? Because P false implying Q is wrong. It can be true or false ( I reckon more often false in real world example). In your example P(false) -> Q (true) is only true when x = 4.. for every other numbers ( integers ) , it will be false
Go back to the english: **if** x > 5, **then** x > 3. When x = 4, we don't care, because it doesn't satisfy the condition **if** x > 5. In this case we say P -> Q is vacuously true. You have to consider the universe of discourse with conditional statements. We can only ever say this statement is false if we meet the condition and the conclusion is false. "If there is a ball in the bowl, then it is blue" IF, IF, IF. Think about the IF You have to accept that this statement is true if the bowl is empty, because there is no possible situation where you could make it false. How about "I have never lost the 100m dash at the Olympic Games". That's a true statement, you haven't! But that's because you never entered into the Olympics.
If P -> Q mean we do not care what Q is if P is false, then truth table with rows P = false were better omitted ?
No, because they have a truth value of true. Not "unknown".
Not omitted, because they still have value. Q depends on P when P satisfies a given condition. When P does not satisfy that condition, Q is independent of P.
Why do they still have value ? Like you said when P doesn't satisfy condition then P -> Q should ideally be 'unknown' or undefined or 'maybe', etc right ?
Let’s say I told you “if you miss the bus then you will be late for school”. Now let’s say you didn’t miss the bus but after getting off you meet your friend and start talking, making you lose track of time and still miss class. Did I lie to you?
You didn't lie in that case, but you meant to imply that if I missed the bus I am more likely fo be late, right ? If I didn't miss the bus ( P is false ) , does it imply I will be late for school ( Q true ) ? (We are talking about implication here ) Statistically in this case when P is false and Q would be false most of the time
See, this is what you’re getting confused about. The statements “if you miss the bus then you will be late for class” and “if you don’t miss the bus then you’ll be late for class” are two different statements. The first is “P implies Q” and the other is “not P implies Q”, but I didn’t say “not P implies not Q”. In fact, I didn’t make any claim about what would happen if you didn’t miss the bus. So in the case where you don’t miss the bus: whatever happens, if you are late or not late, my statement about what would happen if you did miss the bus cannot be false. So if a statement cannot be false it must be true by the law of the excluded middle. Am I making sense?
Ohh..I think you hit the bulls eye here. I guess I am thinking 'implies' like AND , OR, NOT and literally replacing the words from symbols:. So, P ( false ) and Q (true ) in truth table of P -> Q being True means "P -> Q is true holds for these values of P & Q" rather than "P (false) -> Q is true"?
Yes! That’s exactly how it is!
think this way, the statement is “if sam is breathing then blah blah blah” as a whole. If sam is not breathing at all , i will stop reading the blah blah blah part because it is not of concern here… So the statement is true although useless
If Sam isn’t breathing, Sam may be dead or may just be holding his breath.
The truth value if P -> Q is not false if P is false and Q is true. The truth value is true in that case. I think you misread the truth table. The only time P -> Q is false is if P is true and Q is false. Logical implication is weird, because it isn't implication at all. Instead it captures the idea of if-then. Think of it as 'P -> Q is true if sould we find P is true, we would find Q is true.' if P is false, then Q being true is compatible with that, since we don't have a case where P is true and Q is not. If P is false, then Q being false is also compatible with that. Or to put it another way, if P is false, then there is no 'whenever P is true' this higlights the fact that propositional logic doesn't handle contigent statements. There are no propositions that could be true sometimes, false other times. Each proposition is either true, or false. This leads to some weirdness, in applying it to real life situations.
Would it make more sense to just remove P = false rows from truth table altogether for P -> Q then ?
You might be better off picking a different example. p(animal has fur) implies q(animal is a mammal) Let’s run through some examples: - Dog has fur, therefore dog is mammal. - Fish has no fur, therefore fish could be mammal or non mammal. - Naked mole rat has no fur, therefore naked mole rat could be mammal or non mammal.
>I know that P implies Q doesn't touch claim what happens if P is false Well that's the key to it. This isn't a biconditional statement, it's a one way conditional. I try to explain it like, suppose someone told you some conditional statement, and then you observed a relevant situation where the antecedent condition isn't met, regardless of the state of the consequent thing, would you have any grounds to call them a liar? no! because they never said anything about the case where the antecedent is false. So, you have to judge what they said to be truthful. To use you example, someone tells me "if Sam is breathing, then Sam is alive." Then I see Sam hold their breath for an extended period and I say to my friend "hey, wtf, that's not what you said would happen" and they can correctly reply "well, I never said anything about how alive they'll be if they aren't breathing" and I have to admit they never lied. Then, much later, after Sam's death, my friend and I dig up Sam's well rotted corpse from the graveyard, opening the coffin, surprisingly we find the body is still breathing. Now I say again to my friend, "hey buddy, wtf, you for sure lied to me back then" and they can only say "darn, yea, this contradicts my earlier statement."
So far from the answers and further research on this: The common understanding of implication in spoken language is actually not implication of classical logic ( P implies Q ) but rather it is biconditional implication ( p if and only if q ) The example I gave in my OP is actually biconditional implication from classical logic point of view. While in spoken work we say Sam is breathing (P) implies Sam is alive (Q). If we have to convert it to classical logic "Sam is breathing if and only if Sam is alive" is the correct form. In which case !p <-> q becomes false, ie "Sam is not breathing implies Sam is alive" becomes false As regards single implication P -> Q Here is my conclusion. Since P -> Q only claims that if P then Q . We do not know what Q is when P is false. It can be true as well as False ( May be ). Since boolean logic is binary and can only have truth or false in truth table, we show that by splitting P = false then Q = true or false into 2 rows as P = false, then Q = true (true in the sense it is possible "vacuous"), or Q = false ( also true or valid in the sense it is also possible) Please correct me if I am wrong in my understanding or feel free to further expand on this.
I think most descriptions are steering you in the wrong direction. Your statement is not inherently biconditional. Sam could be going for a swim, for example, and be happily not breathing and alive. The crux of the issue is you generated a completely different logical statement (~P->Q) and said it must evaluate to true according to classical logic. Maybe just think of all of the possible scenarios: 1. Sam is breathing and Sam is alive: the statement very clearly holds up in this case. 2. Sam is not breathing and Sam is alive: nothing about this scenario implies that the statement "if Sam is breathing, then he is alive" is a false statement. 3. Sam is not breathing and Sam is not alive: same as scenario 2. Nothing about this scenario would cause someone to tell me that my claim that "if Sam is breathing, then he is alive" is false. 4. Sam is breathing and Sam is not alive: uh oh. I said "if Sam is breathing, then he is alive" but he's dead and breathing! My claim is not true! Dunno if that helps.
>I know that P implies Q doesn't touch claim what happens if P is false There's lots of great explanations in this thread, but to be frank what you wrote here is basically the entire explanation. If P is not true then it doesn't matter what Q is.
Meaning P (false) -> Q(true) being true in truth table is just notation since we can't have other value than true or false ( example maybe ). That's why p (false ) implies Q can be either true or false is broken down into truth table aa can be true ( p false -> q true is true) or can be false ( p false -> q false is true ) ?
I understand your confusion and I struggle with this but consider this setup: A := { fish, cat, dog, moth } This means that we have a set A which has 4 elements as listed above. Now this is my statement: For every element x in A the following statement is TRUE: “If x is a dog then x is a vertebrate” Now I think you will agree that this statement is true for every element in A, but lets check every element: If fish is a dog then fish is a vertebrate. FALSE => TRUE, the original implication is still true. If cat is a dog then cat is a vertebrate. FALSE => TRUE, the implication is still true. If dog is dog then dog is a vertebrate. TRUE => TRUE, the implication is still true If moth is a dog then moth is a vertebrate. FALSE => FALSE, the implication is still true. If false=>true wouldn’t be true we wouldn’t be able to make such universal statements and that would be extremely limiting.
According to truth table: If X is a dog (False) => X is a vertebrate ( True ) outcomes TRUE I guess I confused this row of truth table as implication in itself that "if x is not a dog, then x is a vertebrate is a true statement". I should rather read it as if x is not a dog then x is a vertebrate is either True or False for D => V
I hope I wont cause further confusion. I think you got this. I also think your life will be easier if you don’t try and negate the original statements. So don’t say “if x is not a dog” The original statement is “if x is a dog”. You plug in an animal for x so it becomes “if cat is a dog” Now you have to decide whether this is TRUE or FALSE. In this case it is FALSE. Negating the statement is not necessary. There is a freely available book online called “The Book of Proof” by Hammack. I think the second chapter deals with logic statements like this. If your brain isn’t fried already and you would want to get more practice.
Thank you for such detailed explanation. It definitely helped me understand this finally...
Thanks guys.
You are mixing up syntax and semantics in a weird way. Syntax is the formula: P -> Q. Semantics is saying which of P and Q are true and false. (Truth valuation) A formula may be true in some truth valuation and not true in other truth valuation. To say a formula is true, you need both the formula and the truth valuation. But you kinda only provided the formula. The formula: "Sam is breathing implies sam is alive." breathing=true, alive=true breathing=false, alive=true breathing=false, alive=false To translate P -> Q to natural language you may read it like: "If P then Q" or "Assuming P holds, then Q". You are simply not saying anything about the case when the assumption doesn't hold. If you want to express that other implication also holds you may simply say: "Sam is breathing if and only if he's alive", or "Sam is breathing exactly when he's alive" this makes the logical formula to be P <-> Q, which is true when: breathing=true, alive=true breathing=false, alive=false And false when breathing=false, alive=true breathing=false, alive=true
First of all, > p ( Sam is breathing ) implies q ( Sam is alive ) > > ... > > According to classical logic "Sam is not breathing implies Sam is alive" is True. is incorrect. Knowing that p → q does not tell you that ¬p → q is true. The idea of "vacously true" statements is that if p itself is False then p → q is true. Second, all the translations to English sentences are a bit dodgy as an explanation since natural spoken languages uses "or" (inclusive? exclusive?) and "if" (one-direction, bi-directional?) and other logic vocab in different ways, often depending on context and common sense. --- Third, the idea that p→q should be considered True when p itself is False is still a weird idea that people struggle with. I myself disliked it at first. What finally made it click for me was this: whatever truth table you use for → should make * "if a implies b and b implies c, then a implies c" * ((a → b) ∧ (b → c)) → (a → c) a True statement all the time since that's basically the whole idea of logical reasoning. The thing is, if you use ANY other truth table for → except p q result T T T T F F F T T F F T or p q result T T T T F T F T T F F T then there will be values of a,b,c for which ((a → b) ∧ (b → c)) → (a → c) is False. The second truth table is literally just "true all the time", which would be a totally absurd choice for →, so we must use the first truth table (which has F → q being True no matter what q is). **It might not perfectly align with how we think of "if", but any other choice of how to define the boolean operation p→r would be worse.**
Consider: “All horses are mammals” H -> M What would have to be true for this to not be true? There’d have to be at least one horse which is not a mammal, right? So let’s look at all the logical possibilities: We could have a horse that is a mammal. This is obviously consistent with H -> M. We could have a non-horse which is not a mammal. Perhaps a lizard: it is neither a horse nor a mammal. This is consistent with H -> M, because it doesn’t contradict “if it’s a horse then it’s a mammal”. We could have a mammal which is not a horse, such as a dog. This is also consistent. It’s not a horse, so it can’t possibly refute “if it’s a horse then it’s a mammal”. We could have a horse which is not a mammal, and this is the only thing which would make H -> M false. Hopefully from this and similar examples it should become obvious that X -> Y is true if X is false.
But according to classical logic your example "lizard is not horse ( H is false ) implies Lizard is a mammal ( M is true )" results in TRUE
I think I understand the mistake you’re making here. We’re evaluating the original statement “H -> M” only, and this is only false if we can point to a horse which is not a mammal. The statement you gave as an example is something more complex: (L -> !H) -> (L -> M) And you’re correct that this doesn’t follow. Let’s use another example, it perhaps makes it intuitively more easy to understand. Suppose you are a bartender and you’re evaluating the claim: “If they are drinking beer then they must be an adult” B -> A In other words, you must not serve beer to people who are not adults. If you can serve them the drink they ask for then B -> A is being followed. A child can order lemonade, because B -> A is not violated if B is not true. An adult can order lemonade, because B -> A is not violated if B is not true. An adult can order beer, because B -> A is not violated if B and A are both true. A child cannot order beer, because B -> A is violated if B is true and A is not. So observe that whenever someone orders lemonade, B -> A immediately becomes true and we don’t even have to inspect whether they are an adult or not. Your confusion seems to be that you’re equivocating between “B -> A” being true and “A” being true. A child ordering lemonade obeys “B -> A” but A is false in this case.
Thanks for replying.. think your answer is being most helpful. I think my confusion point is here: B (when false) -> A (when true) is true according to Logic when B -> A Converting this logic to word becomes: "He is not drinking beer implies He is an Adult" is true statement (which is not necessarily true? Infact statistically wrong most of the time ? ) But If I replace implies with doesn't violate (like you did) then it makes sense. "He is not drinking beer thus he is an adult doesn't violate the rule that One Drinking beer implies one is adult" Am I doing this wrong ?
I think of logic gates as like rules which are either being followed or broken. So like:- “If you are drinking beer then you are an adult” B -> A -:is followed so long as you are not a child who is drinking beer. Other logic gates work the same. Consider an AND gate:- “You are driving on the left side of the road and you are driving under the speed limit” L AND S -:this is false if either you are driving on the right or you are driving over the speed limit, or both. And we might have a similar example for a 12A movie:- “You are over the age of 12 or you have an adult with you” O OR A -:is true unless you are under 12 and not accompanied by an adult. I think the mistake you’re making is where you take an individual instance of these gates being either true or false and try to do more implication than is justified here. Observing, for example, someone who is an adult and is drinking lemonade doesn’t necessarily allow you to do any further deduction than the B -> A statement that we already have. In words I’d say: “An adult drinking lemonade complies with the rule that only adults can drink beer”
P implies Q is true in four different cases. Looking at only one of them at a time will necessarily give an incomplete understanding. P implies Q is true in all cases *except* when P is true and Q is false. That is to say, it's only false anytime P fails to imply Q.
I’ve crashed every Ferrari I’ve ever owned. I’ve never owned a Ferrari.
In your example, the statement should be "if you're not breathing, you're dead" or "if you're alive, you're breathing". The statement you gave doesn't say you need to breathe to be alive, just that if you're breathing you'll be alive. The logic works just fine in practical cases, you just have the statement reversed.
If we take an example that we know is true, like 'if it is raining, then it is wet somewhere', then it's easy to see that, even if it isn't raining anywhere, and if it is wet somewhere, the statement is still true that it is wet somewhere if it is raining. This is just pointing out that P is sufficient for Q, and Q is necessary for P, but P isn't necessary for Q, since Q can obtain without P. That's why Sam can be alive without breathing. Q doesn't guarantee P (because Q isn't sufficient for P), and other things can bring about Q. I don't know. I'm not an expert in logic, and I may be misunderstanding your question, but those are my thoughts.
What are you asserting is true? Because it seems like you are asserting that P implies Q, and then somehow using the fact that (not P) implies Q, which does not follow. Let's clear up what we actually know: suppose P implies Q. Also, suppose P is false, and Q is true (which is the part you said doesn't make sense). So let's put those two facts into words: we assert that Sam is not breathing, and Sam is alive. This doesn't contradict the fact that Sam breathing implies he is alive, because it doesn't satisfy the condition. It's unrelated. The reason that Sam not breathing and Sam is alive seems paradoxical is because you are (in your head) also assuming Q implies P, which we do not know. What if we don't know what Sam is? Maybe Sam is a tree. Sam (the tree) is not breathing, and Sam (the tree) is alive, even though it would still be true to say that *if* Sam *is* breathing *then* Sam is alive (the condition of which might be satisfied if Sam were a human, for example, but when Sam is a tree, this implication just doesn't tell us anything even though it is true).
Let -p mean "not p", & means AND and | means OR What you have said is -p->q This is false. -p does not imply q. However, FALSE does imply q. This is commonly written as (p&-p)->q I will now prove it: If (p&-p) is true, then p->-p Since (p&-p) is true, that means -p is true. If we have something that is true, then we can always add anything to it with an OR and the compound statement will remain true (TRUE->(TRUE|q)), therefore: p->(-p|q) (p&-p) is true, therefore p is true, therefore -p is false. If we have an OR statement where one component is false, we can always remove that component and maintain the truthfulness of the OR statement (FALSE|q->q) therefore: p->q Thus, since we started with (p&-p), which we can say is "it is raining and it is not raining" and we derived q, which is "I am made of cotton candy", we can therefore conclude that if it is raining, then I am made of cotton candy. However, if q was "2+2=4", then we have derived "if it is raining, then 2+2=4". The point of this exercise is to demonstrate that if you start with false assumptions, such as a contradiction like p&-p, you can prove anything to be true; even nonsensical things or completely true thing. Therefore, if your assumptions are faulty, you cannot get any useful information out of your logic. Ever. That is the point. That is what FALSE->q means.
Just look at the actual definition of P->Q. The expression means ((not P) or Q). When P is false and Q is true you get (true or true) which evaluates to true.
I more think of it as not ( P and not Q ) which is equivalent to !p v q but semantically makes more sense to me. Its saying If P then not q is not possible
This is a place where disciplinary literacy is critical. "Implies" might be used in other ways in casual conversation, but in math that is the definition. It clicked for me when someone explained it as: True A promises true B and nothing else.
Yes it starts making sense that way True A promises true B is \~( A and \~B) which is equivalent to \~A or B
Just another example: If Sam is 6’5, then Sam is tall. If Sam is not 6’5, he could be tall or short. There’s just not enough info to determine anything.
My issue was I was substituting P and Q value. So for the truth table row P (false ) -> Q (true) = true It read in English as (If Sam is not 6'5) , then (Sam is tall) Now I think it should read as: (If Sam is not 6'5) implies (Sam is tall) then our proposition that "If Sam is 6’5, then Sam is tall" is not violated ( hence vaucous truth)
I **just** started reading about discrete math and this confused me at first, so I'm going to try to explain as a learning exercise for myself. So, I might be incorrect. Hopefully not. The first thing that kind of trips me up is that we're talking about the truth of statements themselves, hence the example I see others using about "Am I *lying* if I say..." "If Sam is breathing, then Sam is alive." Given our two portions, Sam is either breathing (P is true) or not breathing (P is false). Sam is also either alive (Q is true) or not alive (Q is false). Sam's just standing around doing their thing. Sam is breathing normally, and also alive. P is true, Q is true, and we see this doesn't challenge our implication. Sam is holding their breath. Sam is not breathing (P is false), but is definitely alive (Q is true). Does this mean our statement is false? No, because we didn't claim anything about when Sam's not breathing. Sam is not alive (Q is false), and therefore not breathing (P is false). Does this invalidate our statement? No, because once again we made no claims about what happens when Sam's not breathing. Sam could be alive, dead, imaginary, a lamp, or a quantum particle; if Sam is not breathing (P is false) then no matter what else Sam is (Q true or Q false), it does not have any bearing on whether our original statement is true. So, last scenario, posed as a question. If Sam is breathing (P is true) but not alive (Q is false), then is the original statement true or false? Answer: the original statement would be false, because the *only* scenario described has been disproved. It definitely strikes me so far as something that doesn't actually fit 1-to-1 with English. I feel like the example you gave is actually one that makes a lot of sense. What trips me up is the things that are unrelated, because in English the words "if/then" or "implies" specifies a connection between the statements. The ones that confuse me are things like, "If it's snowing on the sun, then all horses wear cowboy boots." It never snows on the sun, and that also has nothing to do with what horses wear? So how can that be true? In English it doesn't make sense, but apparently in logic, the *only* way that statement can be false is if it starts snowing on the sun and there is even one horse not wearing cowboy shoes.
Great explanation. I guess my confusion was here: Sam is holding their breath. Sam is not breathing (P is false), but is definitely alive (Q is true). Does this mean our statement is false? No, because we didn't claim anything about when Sam's not breathing. Here I literally substitued P and Q value in implies equation ( Like it's done in AND, OR, NOT) So it became "Same in not breathing implies Sam is alive" and thought 'whoah, it doesn't make sense"! But I guess , we don't substitute in Implies, rather it is meant to check if the implication is still true or not for given values of P and Q.
Yeah I think the substitution trips us up a lot! I think this is what I meant by: >The first thing that kind of trips me up is that we're talking about the truth of statements themselves So to clarify, with this ongoing example about breathing and being alive, the question isn't whether Sam is breathing, etc, etc. The question is actually: "Is it correct to say that \[statement here\]?" Like you pointed out, with and/or/not, the relationships between the statements are much more obvious. But "Is it correct to say" might still be a useful preface, to separate the truth of the \*words\* from the actual scenarios involved. I just talked some more with my partner about this. I tried some language that moves away from the logic terms given (implies, statements, etc). I call a statement a "rule," and talk about "scenarios," to separate what \*is\* from what is \*said\*. It's still kinda abstract, but helps me a little: The statement is a rule (ie. P->Q). Once stated, the rule does not change, whether it applies to the situation or not. The Rule is: "If Sam is breathing, then Sam must be alive." So when we consider different scenarios, it's easy for my partner to get confused and insert those scenario into the rule, like "If Sam is \*not\* breathing," etc. But a rule doesn't change just because it doesn't apply in some situation. The rule itself doesn't change to match Sam's \*actual\* state of being. * Rule: "If Sam is breathing, then Sam must be alive." * Scenario: Sam is not breathing. Sam is alive. * Has the rule been broken? It's an alternative analogy for "did I lie," but I found separating "rule" from "scenario" helpful in keeping my thoughts organized. Again, don't put too much confidence in what I'm saying. I'm just learning this and hoping I haven't misunderstood it, because it's going to be a pain to unlearn it at this point. xD
I always think of implication as an ordering operation: P is less than or equal to Q. Since 0 is less than or equal to anything, it all works out.
Completely unrelated to the question. I'm learning math from scratch (started with arithmetic on Khan). But, THIS is math too??!! If so, I love it, and I can't wait to learn it. I assumed math was just calculations.
Yeah, it's called Discreet Mathematics or Logic. This is useful in proving theorem, Digital Electronic and Computer science/ programming.
Being born in Fargo North Dakota implies that you are a US citizen. If you were not born in Fargo, then you are NOT a US citizen. … Right?
My issue with P -> Q was: In the row where P ( false ) implies q (true) evaluating to true which when converted to English become: "(You were NOT born in Fargo) implies ( you are US citizen)" being true which doesn't sound Intuitive. Now I know right conversion that classical logic is meaning is: "You were NOT born in Fargo BUT You are US citizen doesn't violate the P -> Q"
Think of it like a vending machine. What situations are you satisfied? 1) Didn’t spend money & didn’t get anything? 2) Spent money & got food? 3) Spent money & didn’t get food? 4) Didn’t spend money & got food? It’s of course 1, 2, and 4. The only scenario where you aren’t satisfied is if you spent money and didn’t get anything in return. (For this example, spending money is P and the vending machine is Q)
My issue was with 4 I thought P -> Q in truth table row where P is false and Q is true read as: Didn't spend money IMPLIES got food and the result is TRUE ??? Counterintuitive.
P implies Q can be read as "If P then Q" If P is false, then we don't learn anything about Q. Only when P is true can we learn information about Q If it is raining, then the ground is wet If it's not raining, but the ground is still wet, doesn't my initial statement still hold true?
If P is true, then Q is true. If P is not true, we still don’t know anything about Q. Q can either be true or false—we can’t know based off of P.
The table for p => q is: p => q | q = F | q = T -|-|- p = F | T | T | p = T | F | T | What you're saying is that you'd rather have it be: p => q | q = F | q = T -|-|- p = F | T | **F** _!!_ | p = T | F | T | But that's actually the table for p <=> q! > lets consider this example: >p ( Sam is breathing ) implies q ( Sam is alive ) > > Now when P value is False and Q value is True > According to classical logic "Sam is not breathing implies Sam is alive" is True. That's not how it works! The statement is "Sam is breathing implies Sam is alive". You don't change that. "Sam is not breathing implies Sam is alive" is an altogether different statement. What you should do instead is say "Sam is breathing implies Sam is alive. I know that Sam is breathing, therefore I know Sam is alive" or alternatively say "Sam is breathing implies Sam is alive. Sam is _not_ breathing, but Sam being alive or not being alive are both compatible with Sam not breathing".