Truth tables from word problem Sentential Logic
I am reading the book "how to prove it" and the answers says that this argument is valid and that I'm to construct a truth table to verify it but I just can't see how this argument is valid and I'm not sure how to construct the truth table to prove it. Here is the argument:
Either sales or expenses will go up. If sales go up, then the boss will
be happy. If expenses go up, then the boss will be unhappy. Therefore,
sales and expenses will not both go up.
I understand that the boss can't be both happy and unhappy at the same time, but as I see it sales and expenses can both go up at the same time and I have no idea why the bosses mood proves that this can't happen. Am I just understanding this completely wrong?
I tried constructing a truth table with the values for sales going up (S), expenses going up (E), boss being happy (H) and boss being unhappy (U) and then looking at
S->H and E->U
But I just end up with a truth table with the size of 2^4 but it just ends up being a big mess that I have no idea how to read. Can anybody help me here?
logic
asked Nov 26 at 16:03
VictorVH
1157
add a comment |
I am reading the book "how to prove it" and the answers says that this argument is valid and that I'm to construct a truth table to verify it but I just can't see how this argument is valid and I'm not sure how to construct the truth table to prove it. Here is the argument:
Either sales or expenses will go up. If sales go up, then the boss will
be happy. If expenses go up, then the boss will be unhappy. Therefore,
sales and expenses will not both go up.
I understand that the boss can't be both happy and unhappy at the same time, but as I see it sales and expenses can both go up at the same time and I have no idea why the bosses mood proves that this can't happen. Am I just understanding this completely wrong?
I tried constructing a truth table with the values for sales going up (S), expenses going up (E), boss being happy (H) and boss being unhappy (U) and then looking at
S->H and E->U
But I just end up with a truth table with the size of 2^4 but it just ends up being a big mess that I have no idea how to read. Can anybody help me here?
logic
asked Nov 26 at 16:03
VictorVH
1157
add a comment |
I am reading the book "how to prove it" and the answers says that this argument is valid and that I'm to construct a truth table to verify it but I just can't see how this argument is valid and I'm not sure how to construct the truth table to prove it. Here is the argument:
Either sales or expenses will go up. If sales go up, then the boss will
be happy. If expenses go up, then the boss will be unhappy. Therefore,
sales and expenses will not both go up.
I understand that the boss can't be both happy and unhappy at the same time, but as I see it sales and expenses can both go up at the same time and I have no idea why the bosses mood proves that this can't happen. Am I just understanding this completely wrong?
I tried constructing a truth table with the values for sales going up (S), expenses going up (E), boss being happy (H) and boss being unhappy (U) and then looking at
S->H and E->U
But I just end up with a truth table with the size of 2^4 but it just ends up being a big mess that I have no idea how to read. Can anybody help me here?
logic
asked Nov 26 at 16:03
VictorVH
1157
I am reading the book "how to prove it" and the answers says that this argument is valid and that I'm to construct a truth table to verify it but I just can't see how this argument is valid and I'm not sure how to construct the truth table to prove it. Here is the argument:
Either sales or expenses will go up. If sales go up, then the boss will
be happy. If expenses go up, then the boss will be unhappy. Therefore,
sales and expenses will not both go up.
I understand that the boss can't be both happy and unhappy at the same time, but as I see it sales and expenses can both go up at the same time and I have no idea why the bosses mood proves that this can't happen. Am I just understanding this completely wrong?
I tried constructing a truth table with the values for sales going up (S), expenses going up (E), boss being happy (H) and boss being unhappy (U) and then looking at
S->H and E->U
But I just end up with a truth table with the size of 2^4 but it just ends up being a big mess that I have no idea how to read. Can anybody help me here?
logic
logic
asked Nov 26 at 16:03
VictorVH
1157
asked Nov 26 at 16:03
VictorVH
1157
asked Nov 26 at 16:03
VictorVH
1157
asked Nov 26 at 16:03
VictorVH
1157
asked Nov 26 at 16:03
VictorVH
1157
1157
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
Hint
1) Either sales or expenses will go up.
2)If sales go up, then the boss will be happy.
3) If expenses go up, then the boss will be unhappy.
4) Therefore, sales and expenses will not both go up.
In symbols :
1) $S lor E$
2) $S to H$
3) $E to lnot H$
4) $lnot (S land E)$
Having said that, you have to build uo a truth table with the three propositional letters : $S, E$ and $H$, that means $2^3=8$ rows and one column for each premise and the last column for the conclusion : seven columns in total.
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
add a comment |
Instead of using $H$ for 'The boss is Happy' and $U$ for 'The boss is Unhappy', use $H$ for 'The boss is Happy' and $neg H$ for 'The boss is Unhappy'.
Since you cannot have both $H$ and $neg H$, this will work out.
It does not work with your $H$ and $U$, since truth-functional logic does not recognize that $H$ and $U$ cannot both be true. ... indeed, now you get several rows where both $H$ and $U$ are true.
answered Nov 26 at 16:08
Bram28
60.1k44589
add a comment |
The argument claims that the boss cannot be both happy and unhappy at the same time. I think that is the intended reading of the problem, but it is not in line with normal experience or English usage. The boss can be happy about sales rising, unhappy about expenses rising, and on balance somewhat happy or unhappy depending on what is happening to the net.
answered Nov 26 at 16:11
Ross Millikan
291k23196370
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint
1) Either sales or expenses will go up.
2)If sales go up, then the boss will be happy.
3) If expenses go up, then the boss will be unhappy.
4) Therefore, sales and expenses will not both go up.
In symbols :
1) $S lor E$
2) $S to H$
3) $E to lnot H$
4) $lnot (S land E)$
Having said that, you have to build uo a truth table with the three propositional letters : $S, E$ and $H$, that means $2^3=8$ rows and one column for each premise and the last column for the conclusion : seven columns in total.
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
add a comment |
Hint
1) Either sales or expenses will go up.
2)If sales go up, then the boss will be happy.
3) If expenses go up, then the boss will be unhappy.
4) Therefore, sales and expenses will not both go up.
In symbols :
1) $S lor E$
2) $S to H$
3) $E to lnot H$
4) $lnot (S land E)$
Having said that, you have to build uo a truth table with the three propositional letters : $S, E$ and $H$, that means $2^3=8$ rows and one column for each premise and the last column for the conclusion : seven columns in total.
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
add a comment |
Hint
1) Either sales or expenses will go up.
2)If sales go up, then the boss will be happy.
3) If expenses go up, then the boss will be unhappy.
4) Therefore, sales and expenses will not both go up.
In symbols :
1) $S lor E$
2) $S to H$
3) $E to lnot H$
4) $lnot (S land E)$
Having said that, you have to build uo a truth table with the three propositional letters : $S, E$ and $H$, that means $2^3=8$ rows and one column for each premise and the last column for the conclusion : seven columns in total.
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
Hint
1) Either sales or expenses will go up.
2)If sales go up, then the boss will be happy.
3) If expenses go up, then the boss will be unhappy.
4) Therefore, sales and expenses will not both go up.
In symbols :
1) $S lor E$
2) $S to H$
3) $E to lnot H$
4) $lnot (S land E)$
Having said that, you have to build uo a truth table with the three propositional letters : $S, E$ and $H$, that means $2^3=8$ rows and one column for each premise and the last column for the conclusion : seven columns in total.
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
edited Nov 26 at 16:14
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
answered Nov 26 at 16:10
Mauro ALLEGRANZA
64.2k448111
64.2k448111
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
add a comment |
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
You don't need the first line and I don't think it is justified.
– Ross Millikan
Nov 26 at 16:12
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
@RossMillikan - the original problem is written that way ...
– Mauro ALLEGRANZA
Nov 26 at 16:15
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
I don't know why I didn't think of H and -H instead of H and U but this fixed it. Thanks.
– VictorVH
Nov 26 at 16:20
add a comment |
Instead of using $H$ for 'The boss is Happy' and $U$ for 'The boss is Unhappy', use $H$ for 'The boss is Happy' and $neg H$ for 'The boss is Unhappy'.
Since you cannot have both $H$ and $neg H$, this will work out.
It does not work with your $H$ and $U$, since truth-functional logic does not recognize that $H$ and $U$ cannot both be true. ... indeed, now you get several rows where both $H$ and $U$ are true.
answered Nov 26 at 16:08
Bram28
60.1k44589
add a comment |
Instead of using $H$ for 'The boss is Happy' and $U$ for 'The boss is Unhappy', use $H$ for 'The boss is Happy' and $neg H$ for 'The boss is Unhappy'.
Since you cannot have both $H$ and $neg H$, this will work out.
It does not work with your $H$ and $U$, since truth-functional logic does not recognize that $H$ and $U$ cannot both be true. ... indeed, now you get several rows where both $H$ and $U$ are true.
answered Nov 26 at 16:08
Bram28
60.1k44589
add a comment |
Instead of using $H$ for 'The boss is Happy' and $U$ for 'The boss is Unhappy', use $H$ for 'The boss is Happy' and $neg H$ for 'The boss is Unhappy'.
Since you cannot have both $H$ and $neg H$, this will work out.
It does not work with your $H$ and $U$, since truth-functional logic does not recognize that $H$ and $U$ cannot both be true. ... indeed, now you get several rows where both $H$ and $U$ are true.
answered Nov 26 at 16:08
Bram28
60.1k44589
Instead of using $H$ for 'The boss is Happy' and $U$ for 'The boss is Unhappy', use $H$ for 'The boss is Happy' and $neg H$ for 'The boss is Unhappy'.
Since you cannot have both $H$ and $neg H$, this will work out.
It does not work with your $H$ and $U$, since truth-functional logic does not recognize that $H$ and $U$ cannot both be true. ... indeed, now you get several rows where both $H$ and $U$ are true.
answered Nov 26 at 16:08
Bram28
60.1k44589
answered Nov 26 at 16:08
Bram28
60.1k44589
answered Nov 26 at 16:08
Bram28
60.1k44589
answered Nov 26 at 16:08
Bram28
60.1k44589
60.1k44589
add a comment |
add a comment |
The argument claims that the boss cannot be both happy and unhappy at the same time. I think that is the intended reading of the problem, but it is not in line with normal experience or English usage. The boss can be happy about sales rising, unhappy about expenses rising, and on balance somewhat happy or unhappy depending on what is happening to the net.
answered Nov 26 at 16:11
Ross Millikan
291k23196370
add a comment |
The argument claims that the boss cannot be both happy and unhappy at the same time. I think that is the intended reading of the problem, but it is not in line with normal experience or English usage. The boss can be happy about sales rising, unhappy about expenses rising, and on balance somewhat happy or unhappy depending on what is happening to the net.
answered Nov 26 at 16:11
Ross Millikan
291k23196370
add a comment |
The argument claims that the boss cannot be both happy and unhappy at the same time. I think that is the intended reading of the problem, but it is not in line with normal experience or English usage. The boss can be happy about sales rising, unhappy about expenses rising, and on balance somewhat happy or unhappy depending on what is happening to the net.
answered Nov 26 at 16:11
Ross Millikan
291k23196370
The argument claims that the boss cannot be both happy and unhappy at the same time. I think that is the intended reading of the problem, but it is not in line with normal experience or English usage. The boss can be happy about sales rising, unhappy about expenses rising, and on balance somewhat happy or unhappy depending on what is happening to the net.
answered Nov 26 at 16:11
Ross Millikan
291k23196370
answered Nov 26 at 16:11
Ross Millikan
291k23196370
answered Nov 26 at 16:11
Ross Millikan
291k23196370
answered Nov 26 at 16:11
Ross Millikan
291k23196370
291k23196370
add a comment |
add a comment |
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