Clarity on implication when p is not true. [duplicate]
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This question already has an answer here:
Implication in mathematics - How can A imply B when A is False? [duplicate]
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"Again, we may put all this together to conclude that our program outputs a boolean value if supplied with an integer input. However, it is important to realise that the presence of $p$ is absolutely essential for the inference to happen. For example, our program might well satisfy $p → q$, but if it doesn’t satisfy $p$ – e.g. if its input is a surname – then we will not be able to derive $q$."
This is text from the book Logic in Computer Science by M. Huth and M. Ryan (page 9).
The second line in the paragraph necessiates the need for p being true, however, the truth table of Implication mentions one case where p is false, but q is true, and the resultant implication is true. How could that be when p is false?
logic propositional-calculus
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
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asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
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Nov 26 '18 at 15:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Implication in mathematics - How can A imply B when A is False? [duplicate]
9 answers
"Again, we may put all this together to conclude that our program outputs a boolean value if supplied with an integer input. However, it is important to realise that the presence of $p$ is absolutely essential for the inference to happen. For example, our program might well satisfy $p → q$, but if it doesn’t satisfy $p$ – e.g. if its input is a surname – then we will not be able to derive $q$."
This is text from the book Logic in Computer Science by M. Huth and M. Ryan (page 9).
The second line in the paragraph necessiates the need for p being true, however, the truth table of Implication mentions one case where p is false, but q is true, and the resultant implication is true. How could that be when p is false?
logic propositional-calculus
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
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Nov 26 '18 at 15:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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It is possible to derive the entire truth table for implication using reasonably self-evident rules of logic. See my blog posting at dcproof.com/IfPigsCanFly.html
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– Dan Christensen
Nov 26 '18 at 18:50
add a comment |
$begingroup$
This question already has an answer here:
Implication in mathematics - How can A imply B when A is False? [duplicate]
9 answers
"Again, we may put all this together to conclude that our program outputs a boolean value if supplied with an integer input. However, it is important to realise that the presence of $p$ is absolutely essential for the inference to happen. For example, our program might well satisfy $p → q$, but if it doesn’t satisfy $p$ – e.g. if its input is a surname – then we will not be able to derive $q$."
This is text from the book Logic in Computer Science by M. Huth and M. Ryan (page 9).
The second line in the paragraph necessiates the need for p being true, however, the truth table of Implication mentions one case where p is false, but q is true, and the resultant implication is true. How could that be when p is false?
logic propositional-calculus
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
$endgroup$
This question already has an answer here:
Implication in mathematics - How can A imply B when A is False? [duplicate]
9 answers
"Again, we may put all this together to conclude that our program outputs a boolean value if supplied with an integer input. However, it is important to realise that the presence of $p$ is absolutely essential for the inference to happen. For example, our program might well satisfy $p → q$, but if it doesn’t satisfy $p$ – e.g. if its input is a surname – then we will not be able to derive $q$."
This is text from the book Logic in Computer Science by M. Huth and M. Ryan (page 9).
The second line in the paragraph necessiates the need for p being true, however, the truth table of Implication mentions one case where p is false, but q is true, and the resultant implication is true. How could that be when p is false?
This question already has an answer here:
Implication in mathematics - How can A imply B when A is False? [duplicate]
9 answers
logic propositional-calculus
logic propositional-calculus
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
edited Nov 26 '18 at 9:18
Mauro ALLEGRANZA
65k448112
65k448112
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
asked Nov 26 '18 at 9:15
haris.a.aminharis.a.amin
11
11
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Nov 26 '18 at 15:01
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
It is possible to derive the entire truth table for implication using reasonably self-evident rules of logic. See my blog posting at dcproof.com/IfPigsCanFly.html
$endgroup$
– Dan Christensen
Nov 26 '18 at 18:50
add a comment |
$begingroup$
It is possible to derive the entire truth table for implication using reasonably self-evident rules of logic. See my blog posting at dcproof.com/IfPigsCanFly.html
$endgroup$
– Dan Christensen
Nov 26 '18 at 18:50
$begingroup$
It is possible to derive the entire truth table for implication using reasonably self-evident rules of logic. See my blog posting at dcproof.com/IfPigsCanFly.html
$endgroup$
– Dan Christensen
Nov 26 '18 at 18:50
$begingroup$
It is possible to derive the entire truth table for implication using reasonably self-evident rules of logic. See my blog posting at dcproof.com/IfPigsCanFly.html
$endgroup$
– Dan Christensen
Nov 26 '18 at 18:50
add a comment |
1 Answer
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In order to conclude that $q$ is True, we need both that the conditional $p to q$ holds, i.e. it is True, and that the antecedent $p$ of the conditional is True.
As you says, the truth table for $p to q$ - in lines where $p$ is False - has True both for $q$ True and for $q$ False.
Thus, from the simple fact that $p to q$ is True, whe cannot conclude that necessarily $q$ will be True.
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In order to conclude that $q$ is True, we need both that the conditional $p to q$ holds, i.e. it is True, and that the antecedent $p$ of the conditional is True.
As you says, the truth table for $p to q$ - in lines where $p$ is False - has True both for $q$ True and for $q$ False.
Thus, from the simple fact that $p to q$ is True, whe cannot conclude that necessarily $q$ will be True.
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
$endgroup$
add a comment |
$begingroup$
In order to conclude that $q$ is True, we need both that the conditional $p to q$ holds, i.e. it is True, and that the antecedent $p$ of the conditional is True.
As you says, the truth table for $p to q$ - in lines where $p$ is False - has True both for $q$ True and for $q$ False.
Thus, from the simple fact that $p to q$ is True, whe cannot conclude that necessarily $q$ will be True.
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
$endgroup$
add a comment |
$begingroup$
In order to conclude that $q$ is True, we need both that the conditional $p to q$ holds, i.e. it is True, and that the antecedent $p$ of the conditional is True.
As you says, the truth table for $p to q$ - in lines where $p$ is False - has True both for $q$ True and for $q$ False.
Thus, from the simple fact that $p to q$ is True, whe cannot conclude that necessarily $q$ will be True.
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
$endgroup$
In order to conclude that $q$ is True, we need both that the conditional $p to q$ holds, i.e. it is True, and that the antecedent $p$ of the conditional is True.
As you says, the truth table for $p to q$ - in lines where $p$ is False - has True both for $q$ True and for $q$ False.
Thus, from the simple fact that $p to q$ is True, whe cannot conclude that necessarily $q$ will be True.
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
answered Nov 26 '18 at 9:24
Mauro ALLEGRANZAMauro ALLEGRANZA
65k448112
65k448112
add a comment |
add a comment |
$begingroup$
It is possible to derive the entire truth table for implication using reasonably self-evident rules of logic. See my blog posting at dcproof.com/IfPigsCanFly.html
$endgroup$
– Dan Christensen
Nov 26 '18 at 18:50