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How to convince students of the implication truth values?
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?How to teach logical implication?How important is it to show students an application of the topics seen in an undergraduate course?Students strictly follow the steps and notations in sample problems without understanding themIs the current education system as bad as most critics and famous pure mathematicians try to convey?How to teach to draw graphs of quadratic equations without knowing calculus?CoTeaching Elementary Linear AlgebraTeaching math by serving it as games with rules first, not intuition?Students understand during course but can't solve examShould young math students be taught an abstract concept of form?How to motivate students to do proofs?What makes education in Finland so good?
$begingroup$
How can I convince students that "P implies Q" is true when P is false, independent of what truth value Q takes?
Is there any real life or a convincing argument for this?
I have given the analogy of if being elected then you will abolish the death penalty which seems ok but rather crass. Are there any other examples?
teaching
edited Apr 17 at 11:36
Adam
2,363818
asked Apr 17 at 9:52
matqkksmatqkks
569313
$endgroup$
|
show 5 more comments
$begingroup$
How can I convince students that "P implies Q" is true when P is false, independent of what truth value Q takes?
Is there any real life or a convincing argument for this?
I have given the analogy of if being elected then you will abolish the death penalty which seems ok but rather crass. Are there any other examples?
teaching
edited Apr 17 at 11:36
Adam
2,363818
asked Apr 17 at 9:52
matqkksmatqkks
569313
$endgroup$
1
$begingroup$
The problem with "If I am elected, then I will abolish the death penalty" is, if you are not elected, then how can you abolish the death penalty?
$endgroup$
– Joel Reyes Noche
Apr 17 at 10:48
1
$begingroup$
So you begin by giving the truth table?
$endgroup$
– Jasper
Apr 17 at 11:25
3
$begingroup$
I'm sure this had been asked before on math.se. You can use $x>8 Rightarrow x>0$.
$endgroup$
– BPP
Apr 17 at 20:51
1
$begingroup$
The mathematical concepts bear only a superficial resemblance to the real world. Yes, math is inspired by real-world problems but abstracts from them and establishes concepts which try to do without natural language as much as possible, for good reason. So "P implies Q" is a math operation with two boolean arguments resulting in a bool. Just present the truth table. "Any resemblance to existing real-world situations is accidental and unintentional."
$endgroup$
– Peter A. Schneider
Apr 18 at 13:36
3
$begingroup$
Possible duplicate of How to teach logical implication?
$endgroup$
– BPP
Apr 18 at 14:41
|
show 5 more comments
$begingroup$
How can I convince students that "P implies Q" is true when P is false, independent of what truth value Q takes?
Is there any real life or a convincing argument for this?
I have given the analogy of if being elected then you will abolish the death penalty which seems ok but rather crass. Are there any other examples?
teaching
edited Apr 17 at 11:36
Adam
2,363818
asked Apr 17 at 9:52
matqkksmatqkks
569313
$endgroup$
How can I convince students that "P implies Q" is true when P is false, independent of what truth value Q takes?
Is there any real life or a convincing argument for this?
I have given the analogy of if being elected then you will abolish the death penalty which seems ok but rather crass. Are there any other examples?
teaching
teaching
edited Apr 17 at 11:36
Adam
2,363818
asked Apr 17 at 9:52
matqkksmatqkks
569313
edited Apr 17 at 11:36
Adam
2,363818
asked Apr 17 at 9:52
matqkksmatqkks
569313
edited Apr 17 at 11:36
Adam
2,363818
edited Apr 17 at 11:36
Adam
2,363818
edited Apr 17 at 11:36
Adam
2,363818
2,363818
asked Apr 17 at 9:52
matqkksmatqkks
569313
asked Apr 17 at 9:52
matqkksmatqkks
569313
asked Apr 17 at 9:52
matqkksmatqkks
569313
569313
1
$begingroup$
The problem with "If I am elected, then I will abolish the death penalty" is, if you are not elected, then how can you abolish the death penalty?
$endgroup$
– Joel Reyes Noche
Apr 17 at 10:48
1
$begingroup$
So you begin by giving the truth table?
$endgroup$
– Jasper
Apr 17 at 11:25
3
$begingroup$
I'm sure this had been asked before on math.se. You can use $x>8 Rightarrow x>0$.
$endgroup$
– BPP
Apr 17 at 20:51
1
$begingroup$
The mathematical concepts bear only a superficial resemblance to the real world. Yes, math is inspired by real-world problems but abstracts from them and establishes concepts which try to do without natural language as much as possible, for good reason. So "P implies Q" is a math operation with two boolean arguments resulting in a bool. Just present the truth table. "Any resemblance to existing real-world situations is accidental and unintentional."
$endgroup$
– Peter A. Schneider
Apr 18 at 13:36
3
$begingroup$
Possible duplicate of How to teach logical implication?
$endgroup$
– BPP
Apr 18 at 14:41
|
show 5 more comments
1
$begingroup$
The problem with "If I am elected, then I will abolish the death penalty" is, if you are not elected, then how can you abolish the death penalty?
$endgroup$
– Joel Reyes Noche
Apr 17 at 10:48
1
$begingroup$
So you begin by giving the truth table?
$endgroup$
– Jasper
Apr 17 at 11:25
3
$begingroup$
I'm sure this had been asked before on math.se. You can use $x>8 Rightarrow x>0$.
$endgroup$
– BPP
Apr 17 at 20:51
1
$begingroup$
The mathematical concepts bear only a superficial resemblance to the real world. Yes, math is inspired by real-world problems but abstracts from them and establishes concepts which try to do without natural language as much as possible, for good reason. So "P implies Q" is a math operation with two boolean arguments resulting in a bool. Just present the truth table. "Any resemblance to existing real-world situations is accidental and unintentional."
$endgroup$
– Peter A. Schneider
Apr 18 at 13:36
3
$begingroup$
Possible duplicate of How to teach logical implication?
$endgroup$
– BPP
Apr 18 at 14:41
1
1
$begingroup$
The problem with "If I am elected, then I will abolish the death penalty" is, if you are not elected, then how can you abolish the death penalty?
$endgroup$
– Joel Reyes Noche
Apr 17 at 10:48
$begingroup$
The problem with "If I am elected, then I will abolish the death penalty" is, if you are not elected, then how can you abolish the death penalty?
$endgroup$
– Joel Reyes Noche
Apr 17 at 10:48
1
1
$begingroup$
So you begin by giving the truth table?
$endgroup$
– Jasper
Apr 17 at 11:25
$begingroup$
So you begin by giving the truth table?
$endgroup$
– Jasper
Apr 17 at 11:25
3
3
$begingroup$
I'm sure this had been asked before on math.se. You can use $x>8 Rightarrow x>0$.
$endgroup$
– BPP
Apr 17 at 20:51
$begingroup$
I'm sure this had been asked before on math.se. You can use $x>8 Rightarrow x>0$.
$endgroup$
– BPP
Apr 17 at 20:51
1
1
$begingroup$
The mathematical concepts bear only a superficial resemblance to the real world. Yes, math is inspired by real-world problems but abstracts from them and establishes concepts which try to do without natural language as much as possible, for good reason. So "P implies Q" is a math operation with two boolean arguments resulting in a bool. Just present the truth table. "Any resemblance to existing real-world situations is accidental and unintentional."
$endgroup$
– Peter A. Schneider
Apr 18 at 13:36
$begingroup$
The mathematical concepts bear only a superficial resemblance to the real world. Yes, math is inspired by real-world problems but abstracts from them and establishes concepts which try to do without natural language as much as possible, for good reason. So "P implies Q" is a math operation with two boolean arguments resulting in a bool. Just present the truth table. "Any resemblance to existing real-world situations is accidental and unintentional."
$endgroup$
– Peter A. Schneider
Apr 18 at 13:36
3
3
$begingroup$
Possible duplicate of How to teach logical implication?
$endgroup$
– BPP
Apr 18 at 14:41
$begingroup$
Possible duplicate of How to teach logical implication?
$endgroup$
– BPP
Apr 18 at 14:41
|
show 5 more comments
6 Answers
6
active
oldest
votes
$begingroup$
I make the statement "If it is raining, then I have an umbrella." Did I lie?
If it is raining and I do not have an umbrella, then I lied.
If it is raining and I do have an umbrella, then I didn't lie.
If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
$endgroup$
2
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
3
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
3
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
2
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
2
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
|
show 1 more comment
$begingroup$
You could say it means "Whenever P is true, Q is true". So "If it rains, I will bring an umbrella" means "Every time it rains, I bring an umbrella". It's not possible to disprove this statement by looking at what happens when it doesn't rain.
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
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add a comment |
$begingroup$
You can introduce implication and equivalence side by side to make the difference clear.
Implication: If A, then also B. (But if not A, this statement does not tell us anything. See the umbrella example in Joel's answer.)
Equivalence: If and only if A, then B. This includes that not A implies not B.
answered Apr 17 at 13:12
JasperJasper
734412
$endgroup$
2
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
2
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
add a comment |
$begingroup$
If I do the work, I will get payed.
- If I do the work and get payed, all is good.
- If I do the work and not get payed, it's bad!
- If I don't do the work and don't get payed, I cannot complain - it's all good.
- If I don't do the work but still get payed - now this is good!
I know it's not exactly how one should interpret implication, but it worked for me. :)
answered Apr 18 at 8:56
michcio1234michcio1234
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
$begingroup$
My approach in the past has always been to work via negation. However, I've recently read one of Tim Gowers' 'discussions' that showed me a new side to the problem.
Basically he points out that in everyday usage (and in proofs too) we are looking for causal relationships. We want to say that 'if A then B' is true when we can understand A as leading to B. But when dealing with more formal logic, we are actually only concerned with evidence. We say 'if A then B' is true when 'in every case where A is true, B is also true', regardless of whether this is something you would expect or just plain chance.
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
$endgroup$
add a comment |
$begingroup$
It's obvious if you view a proposition as a subset of all values in the proposition domain for which the proposition is true.
For example, a proposition "this umbrella is red" selects only red umbrellas form the set of all umbrellas that exist in the world.
A proposition "I own only red umbrellas" can be viewed as implication - P is "I own this umbrella", Q is "this umbrella is red", and it can be interpreted as "the set of umbrellas that I own is a subset of red umbrellas", that is, implication corresponds to "being a subset" relation in the set interpretation (similarly, conjunction corresponds to set intersection and disjunction corresponds to set union).
False proposition corresponds to an empty set - there are no values in the proposition domain for which it's true. From set theory axioms we know that empty set must be a subset of any set. This immediately gives you
"P implies Q" is true when P is false
It's essentially a formality, but a convenient one - for example if you don't own any umbrellas at all you can safely say "I own only umbrellas hand-made by unicorns from rainbows" and it's assumed to be true because it can't be disproved.
answered Apr 17 at 20:18
artemartem
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
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– Andreas Blass
Apr 18 at 10:49
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I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
add a comment |
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6 Answers
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active
oldest
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6 Answers
6
active
oldest
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$begingroup$
I make the statement "If it is raining, then I have an umbrella." Did I lie?
If it is raining and I do not have an umbrella, then I lied.
If it is raining and I do have an umbrella, then I didn't lie.
If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
$endgroup$
2
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
3
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
3
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
2
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
2
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
|
show 1 more comment
$begingroup$
I make the statement "If it is raining, then I have an umbrella." Did I lie?
If it is raining and I do not have an umbrella, then I lied.
If it is raining and I do have an umbrella, then I didn't lie.
If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
$endgroup$
2
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
3
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
3
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
2
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
2
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
|
show 1 more comment
$begingroup$
I make the statement "If it is raining, then I have an umbrella." Did I lie?
If it is raining and I do not have an umbrella, then I lied.
If it is raining and I do have an umbrella, then I didn't lie.
If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
$endgroup$
I make the statement "If it is raining, then I have an umbrella." Did I lie?
If it is raining and I do not have an umbrella, then I lied.
If it is raining and I do have an umbrella, then I didn't lie.
If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
answered Apr 17 at 10:45
Joel Reyes NocheJoel Reyes Noche
5,71521753
5,71521753
2
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
3
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
3
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
2
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
2
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
|
show 1 more comment
2
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
3
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
3
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
2
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
2
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
2
2
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
$begingroup$
I use exactly this example when I teach this concept. Moreover, you can use it to explain the logical negation: $neg (Pimplies Q) iff (P wedge neg Q)$, because the only situation in which you can call me a liar is if you see me walking around in the rain without an umbrella.
$endgroup$
– Brendan W. Sullivan
Apr 17 at 18:19
3
3
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
$begingroup$
If it is not raining, did you tell the truth?
$endgroup$
– immibis
Apr 18 at 3:24
3
3
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
$begingroup$
@immibis, yes..
$endgroup$
– Joel Reyes Noche
Apr 18 at 5:32
2
2
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
$begingroup$
This answer has the implication that all statements that are not false (not a lie) are true. Probably fine for introducing the concept... but @immibis has a good point.
$endgroup$
– Rick
Apr 18 at 13:12
2
2
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
$begingroup$
@Rick And exactly there is the problem for the casual reader. It's called vacuous truth. A condition which is always wrong yields true for any consequence.
$endgroup$
– Peter A. Schneider
Apr 18 at 13:32
|
show 1 more comment
$begingroup$
You could say it means "Whenever P is true, Q is true". So "If it rains, I will bring an umbrella" means "Every time it rains, I bring an umbrella". It's not possible to disprove this statement by looking at what happens when it doesn't rain.
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
$endgroup$
add a comment |
$begingroup$
You could say it means "Whenever P is true, Q is true". So "If it rains, I will bring an umbrella" means "Every time it rains, I bring an umbrella". It's not possible to disprove this statement by looking at what happens when it doesn't rain.
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
$endgroup$
add a comment |
$begingroup$
You could say it means "Whenever P is true, Q is true". So "If it rains, I will bring an umbrella" means "Every time it rains, I bring an umbrella". It's not possible to disprove this statement by looking at what happens when it doesn't rain.
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
$endgroup$
You could say it means "Whenever P is true, Q is true". So "If it rains, I will bring an umbrella" means "Every time it rains, I bring an umbrella". It's not possible to disprove this statement by looking at what happens when it doesn't rain.
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
answered Apr 17 at 17:31
AcccumulationAcccumulation
36212
36212
add a comment |
add a comment |
$begingroup$
You can introduce implication and equivalence side by side to make the difference clear.
Implication: If A, then also B. (But if not A, this statement does not tell us anything. See the umbrella example in Joel's answer.)
Equivalence: If and only if A, then B. This includes that not A implies not B.
answered Apr 17 at 13:12
JasperJasper
734412
$endgroup$
2
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
2
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
add a comment |
$begingroup$
You can introduce implication and equivalence side by side to make the difference clear.
Implication: If A, then also B. (But if not A, this statement does not tell us anything. See the umbrella example in Joel's answer.)
Equivalence: If and only if A, then B. This includes that not A implies not B.
answered Apr 17 at 13:12
JasperJasper
734412
$endgroup$
2
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
2
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
add a comment |
$begingroup$
You can introduce implication and equivalence side by side to make the difference clear.
Implication: If A, then also B. (But if not A, this statement does not tell us anything. See the umbrella example in Joel's answer.)
Equivalence: If and only if A, then B. This includes that not A implies not B.
answered Apr 17 at 13:12
JasperJasper
734412
$endgroup$
You can introduce implication and equivalence side by side to make the difference clear.
Implication: If A, then also B. (But if not A, this statement does not tell us anything. See the umbrella example in Joel's answer.)
Equivalence: If and only if A, then B. This includes that not A implies not B.
answered Apr 17 at 13:12
JasperJasper
734412
edited Apr 17 at 14:35
answered Apr 17 at 13:12
JasperJasper
734412
answered Apr 17 at 13:12
JasperJasper
734412
answered Apr 17 at 13:12
JasperJasper
734412
734412
2
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
2
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
add a comment |
2
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
2
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
2
2
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
$begingroup$
I've never encountered "If and only if A, then B" in writing or in speech. Is it common in your context (country/text book/field)? Otherwise, I'd consider "A if and only if B" or equivalently (by symmetry) "B if and only if A" far better because yours sounds much more asymmetric to me -- which $Leftrightarrow$ is exactly not.
$endgroup$
– ComFreek
Apr 17 at 13:53
2
2
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
$begingroup$
Hmm. You're right, but I've tried to keep the similarity to the implication. I also think that "B if and only if A" is way more common.
$endgroup$
– Jasper
Apr 17 at 14:36
add a comment |
$begingroup$
If I do the work, I will get payed.
- If I do the work and get payed, all is good.
- If I do the work and not get payed, it's bad!
- If I don't do the work and don't get payed, I cannot complain - it's all good.
- If I don't do the work but still get payed - now this is good!
I know it's not exactly how one should interpret implication, but it worked for me. :)
answered Apr 18 at 8:56
michcio1234michcio1234
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
If I do the work, I will get payed.
- If I do the work and get payed, all is good.
- If I do the work and not get payed, it's bad!
- If I don't do the work and don't get payed, I cannot complain - it's all good.
- If I don't do the work but still get payed - now this is good!
I know it's not exactly how one should interpret implication, but it worked for me. :)
answered Apr 18 at 8:56
michcio1234michcio1234
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
If I do the work, I will get payed.
- If I do the work and get payed, all is good.
- If I do the work and not get payed, it's bad!
- If I don't do the work and don't get payed, I cannot complain - it's all good.
- If I don't do the work but still get payed - now this is good!
I know it's not exactly how one should interpret implication, but it worked for me. :)
answered Apr 18 at 8:56
michcio1234michcio1234
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
If I do the work, I will get payed.
- If I do the work and get payed, all is good.
- If I do the work and not get payed, it's bad!
- If I don't do the work and don't get payed, I cannot complain - it's all good.
- If I don't do the work but still get payed - now this is good!
I know it's not exactly how one should interpret implication, but it worked for me. :)
answered Apr 18 at 8:56
michcio1234michcio1234
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Apr 18 at 8:56
michcio1234michcio1234
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Apr 18 at 8:56
michcio1234michcio1234
1211
answered Apr 18 at 8:56
michcio1234michcio1234
1211
1211
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
michcio1234 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
$begingroup$
My approach in the past has always been to work via negation. However, I've recently read one of Tim Gowers' 'discussions' that showed me a new side to the problem.
Basically he points out that in everyday usage (and in proofs too) we are looking for causal relationships. We want to say that 'if A then B' is true when we can understand A as leading to B. But when dealing with more formal logic, we are actually only concerned with evidence. We say 'if A then B' is true when 'in every case where A is true, B is also true', regardless of whether this is something you would expect or just plain chance.
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
$endgroup$
add a comment |
$begingroup$
My approach in the past has always been to work via negation. However, I've recently read one of Tim Gowers' 'discussions' that showed me a new side to the problem.
Basically he points out that in everyday usage (and in proofs too) we are looking for causal relationships. We want to say that 'if A then B' is true when we can understand A as leading to B. But when dealing with more formal logic, we are actually only concerned with evidence. We say 'if A then B' is true when 'in every case where A is true, B is also true', regardless of whether this is something you would expect or just plain chance.
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
$endgroup$
add a comment |
$begingroup$
My approach in the past has always been to work via negation. However, I've recently read one of Tim Gowers' 'discussions' that showed me a new side to the problem.
Basically he points out that in everyday usage (and in proofs too) we are looking for causal relationships. We want to say that 'if A then B' is true when we can understand A as leading to B. But when dealing with more formal logic, we are actually only concerned with evidence. We say 'if A then B' is true when 'in every case where A is true, B is also true', regardless of whether this is something you would expect or just plain chance.
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
$endgroup$
My approach in the past has always been to work via negation. However, I've recently read one of Tim Gowers' 'discussions' that showed me a new side to the problem.
Basically he points out that in everyday usage (and in proofs too) we are looking for causal relationships. We want to say that 'if A then B' is true when we can understand A as leading to B. But when dealing with more formal logic, we are actually only concerned with evidence. We say 'if A then B' is true when 'in every case where A is true, B is also true', regardless of whether this is something you would expect or just plain chance.
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
answered Apr 17 at 22:18
Jessica BJessica B
4,20211128
4,20211128
add a comment |
add a comment |
$begingroup$
It's obvious if you view a proposition as a subset of all values in the proposition domain for which the proposition is true.
For example, a proposition "this umbrella is red" selects only red umbrellas form the set of all umbrellas that exist in the world.
A proposition "I own only red umbrellas" can be viewed as implication - P is "I own this umbrella", Q is "this umbrella is red", and it can be interpreted as "the set of umbrellas that I own is a subset of red umbrellas", that is, implication corresponds to "being a subset" relation in the set interpretation (similarly, conjunction corresponds to set intersection and disjunction corresponds to set union).
False proposition corresponds to an empty set - there are no values in the proposition domain for which it's true. From set theory axioms we know that empty set must be a subset of any set. This immediately gives you
"P implies Q" is true when P is false
It's essentially a formality, but a convenient one - for example if you don't own any umbrellas at all you can safely say "I own only umbrellas hand-made by unicorns from rainbows" and it's assumed to be true because it can't be disproved.
answered Apr 17 at 20:18
artemartem
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
$endgroup$
– Andreas Blass
Apr 18 at 10:49
$begingroup$
I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
add a comment |
$begingroup$
It's obvious if you view a proposition as a subset of all values in the proposition domain for which the proposition is true.
For example, a proposition "this umbrella is red" selects only red umbrellas form the set of all umbrellas that exist in the world.
A proposition "I own only red umbrellas" can be viewed as implication - P is "I own this umbrella", Q is "this umbrella is red", and it can be interpreted as "the set of umbrellas that I own is a subset of red umbrellas", that is, implication corresponds to "being a subset" relation in the set interpretation (similarly, conjunction corresponds to set intersection and disjunction corresponds to set union).
False proposition corresponds to an empty set - there are no values in the proposition domain for which it's true. From set theory axioms we know that empty set must be a subset of any set. This immediately gives you
"P implies Q" is true when P is false
It's essentially a formality, but a convenient one - for example if you don't own any umbrellas at all you can safely say "I own only umbrellas hand-made by unicorns from rainbows" and it's assumed to be true because it can't be disproved.
answered Apr 17 at 20:18
artemartem
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
$endgroup$
– Andreas Blass
Apr 18 at 10:49
$begingroup$
I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
add a comment |
$begingroup$
It's obvious if you view a proposition as a subset of all values in the proposition domain for which the proposition is true.
For example, a proposition "this umbrella is red" selects only red umbrellas form the set of all umbrellas that exist in the world.
A proposition "I own only red umbrellas" can be viewed as implication - P is "I own this umbrella", Q is "this umbrella is red", and it can be interpreted as "the set of umbrellas that I own is a subset of red umbrellas", that is, implication corresponds to "being a subset" relation in the set interpretation (similarly, conjunction corresponds to set intersection and disjunction corresponds to set union).
False proposition corresponds to an empty set - there are no values in the proposition domain for which it's true. From set theory axioms we know that empty set must be a subset of any set. This immediately gives you
"P implies Q" is true when P is false
It's essentially a formality, but a convenient one - for example if you don't own any umbrellas at all you can safely say "I own only umbrellas hand-made by unicorns from rainbows" and it's assumed to be true because it can't be disproved.
answered Apr 17 at 20:18
artemartem
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
It's obvious if you view a proposition as a subset of all values in the proposition domain for which the proposition is true.
For example, a proposition "this umbrella is red" selects only red umbrellas form the set of all umbrellas that exist in the world.
A proposition "I own only red umbrellas" can be viewed as implication - P is "I own this umbrella", Q is "this umbrella is red", and it can be interpreted as "the set of umbrellas that I own is a subset of red umbrellas", that is, implication corresponds to "being a subset" relation in the set interpretation (similarly, conjunction corresponds to set intersection and disjunction corresponds to set union).
False proposition corresponds to an empty set - there are no values in the proposition domain for which it's true. From set theory axioms we know that empty set must be a subset of any set. This immediately gives you
"P implies Q" is true when P is false
It's essentially a formality, but a convenient one - for example if you don't own any umbrellas at all you can safely say "I own only umbrellas hand-made by unicorns from rainbows" and it's assumed to be true because it can't be disproved.
answered Apr 17 at 20:18
artemartem
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Apr 17 at 20:18
artemartem
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Apr 17 at 20:18
artemartem
101
answered Apr 17 at 20:18
artemartem
101
101
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
artem is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
$endgroup$
– Andreas Blass
Apr 18 at 10:49
$begingroup$
I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
add a comment |
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
$endgroup$
– Andreas Blass
Apr 18 at 10:49
$begingroup$
I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
$endgroup$
– Andreas Blass
Apr 18 at 10:49
$begingroup$
This explains the truth values for implications assuming students already accept that implication (of propositions) should correspond to inclusion (of sets). Is it easy to convince them to accept that? Will it cause a problem when they see $subseteq$ defined (in set theory) using $implies$?
$endgroup$
– Andreas Blass
Apr 18 at 10:49
$begingroup$
I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
$begingroup$
I don't think it will cause any problems, you just have to explain that ⊆ is exactly the same thing as ⟹ when viewed from a certain angle.
$endgroup$
– artem
Apr 18 at 23:43
add a comment |
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$begingroup$
The problem with "If I am elected, then I will abolish the death penalty" is, if you are not elected, then how can you abolish the death penalty?
$endgroup$
– Joel Reyes Noche
Apr 17 at 10:48
1
$begingroup$
So you begin by giving the truth table?
$endgroup$
– Jasper
Apr 17 at 11:25
3
$begingroup$
I'm sure this had been asked before on math.se. You can use $x>8 Rightarrow x>0$.
$endgroup$
– BPP
Apr 17 at 20:51
1
$begingroup$
The mathematical concepts bear only a superficial resemblance to the real world. Yes, math is inspired by real-world problems but abstracts from them and establishes concepts which try to do without natural language as much as possible, for good reason. So "P implies Q" is a math operation with two boolean arguments resulting in a bool. Just present the truth table. "Any resemblance to existing real-world situations is accidental and unintentional."
$endgroup$
– Peter A. Schneider
Apr 18 at 13:36
3
$begingroup$
Possible duplicate of How to teach logical implication?
$endgroup$
– BPP
Apr 18 at 14:41