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Why is it that the natural deduction method can't test for invalidity?
Does transfinite induction indicates limitations of Agrippa’s Trilemma?Shouldn't statements be considered equivalent based on their meaning rather than truth tables?Does predicate logic have truth tables?Why is propositional logic decidable?Why is the biconditional true if both components are false?What are factual propositions?Can Deduction for a Valid Argument produce the wrong conclusion?Without computers, how can you conjecture the (in)validity of a long convoluted argument in Predicate Logic?Why did we define vacuous statements as true rather than false?Are there “partially explosive” logics?
I just got the hang of using truth tables as a method to test for validity and invalidity. Now I'm learning the natural deduction method, and been told that it can test for validity, but not invalidity as the truth table method can. Why is this the case?
As a side note, it's hard to wrap my mind around how something can test for something but not its "opposite." Perhaps validity and invalidity are not contradictions but more like contraries? I don't know.
logic
asked 3 hours ago
csp2018csp2018
384
add a comment |
I just got the hang of using truth tables as a method to test for validity and invalidity. Now I'm learning the natural deduction method, and been told that it can test for validity, but not invalidity as the truth table method can. Why is this the case?
As a side note, it's hard to wrap my mind around how something can test for something but not its "opposite." Perhaps validity and invalidity are not contradictions but more like contraries? I don't know.
logic
asked 3 hours ago
csp2018csp2018
384
add a comment |
I just got the hang of using truth tables as a method to test for validity and invalidity. Now I'm learning the natural deduction method, and been told that it can test for validity, but not invalidity as the truth table method can. Why is this the case?
As a side note, it's hard to wrap my mind around how something can test for something but not its "opposite." Perhaps validity and invalidity are not contradictions but more like contraries? I don't know.
logic
asked 3 hours ago
csp2018csp2018
384
I just got the hang of using truth tables as a method to test for validity and invalidity. Now I'm learning the natural deduction method, and been told that it can test for validity, but not invalidity as the truth table method can. Why is this the case?
As a side note, it's hard to wrap my mind around how something can test for something but not its "opposite." Perhaps validity and invalidity are not contradictions but more like contraries? I don't know.
logic
logic
asked 3 hours ago
csp2018csp2018
384
asked 3 hours ago
csp2018csp2018
384
asked 3 hours ago
csp2018csp2018
384
asked 3 hours ago
csp2018csp2018
384
asked 3 hours ago
csp2018csp2018
384
384
add a comment |
add a comment |
2 Answers
2
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oldest
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Natural Deduction is a proof system that is sound and complete for e.g. classical propositional calculus.
Sound means that if a formula is provable with ND, it is valid.
In general, a proof system is not an algorithm, i.e. if we start from a formula and we do not know if it is valid or not, the simple fact that we are not able to find a proof does not mean that the proof does not exist: maybe, we are simply not clever enough to find it.
Truth table, instead, is an algorithm to test validity for propositional calculus; this means that, applying it to a formula whatever, it always comes to an end with a result: if all rows have TRUE, the formula is valid; if there is some FALSE value, the formula is not valid.
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
add a comment |
Truth tables can become large if there are many sentence letters That is when natural deduction might find a solution in a more economical manner. That assumes one can derive a line in a natural deduction proof that corresponds to the desired conclusion. If not, one has to keep looking. A truth table, although potentially large, would let one know one does not have to continue.
Here is how the authors of forallx describe the situation in Chapter 20: Soundness and Completeness, page 149:
Now that we know that the truth table method is interchangeable with the method of derivations, you can chose which method
you want to use for any given problem. Students often prefer to
use truth tables, because they can be produced purely mechanically, and that seems ‘easier’. However, we have already seen that
truth tables become impossibly large after just a few sentence letters. On the other hand, there are a couple situations where using
proofs simply isn’t possible. We syntactically defined a contingent
sentence as a sentence that couldn’t be proven to be a tautology
or a contradiction. There is no practical way to prove this kind of
negative statement. We will never know if there isn’t some proof
out there that a statement is a contradiction and we just haven’t
found it yet. We have nothing to do in this situation but resort
to truth tables. Similarly, we can use derivations to prove two
sentences equivalent, but what if we want to prove that they are
not equivalent? We have no way of proving that we will never find
the relevant proof. So we have to fall back on truth tables again.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
add a comment |
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2 Answers
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active
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2 Answers
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Natural Deduction is a proof system that is sound and complete for e.g. classical propositional calculus.
Sound means that if a formula is provable with ND, it is valid.
In general, a proof system is not an algorithm, i.e. if we start from a formula and we do not know if it is valid or not, the simple fact that we are not able to find a proof does not mean that the proof does not exist: maybe, we are simply not clever enough to find it.
Truth table, instead, is an algorithm to test validity for propositional calculus; this means that, applying it to a formula whatever, it always comes to an end with a result: if all rows have TRUE, the formula is valid; if there is some FALSE value, the formula is not valid.
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
add a comment |
Natural Deduction is a proof system that is sound and complete for e.g. classical propositional calculus.
Sound means that if a formula is provable with ND, it is valid.
In general, a proof system is not an algorithm, i.e. if we start from a formula and we do not know if it is valid or not, the simple fact that we are not able to find a proof does not mean that the proof does not exist: maybe, we are simply not clever enough to find it.
Truth table, instead, is an algorithm to test validity for propositional calculus; this means that, applying it to a formula whatever, it always comes to an end with a result: if all rows have TRUE, the formula is valid; if there is some FALSE value, the formula is not valid.
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
add a comment |
Natural Deduction is a proof system that is sound and complete for e.g. classical propositional calculus.
Sound means that if a formula is provable with ND, it is valid.
In general, a proof system is not an algorithm, i.e. if we start from a formula and we do not know if it is valid or not, the simple fact that we are not able to find a proof does not mean that the proof does not exist: maybe, we are simply not clever enough to find it.
Truth table, instead, is an algorithm to test validity for propositional calculus; this means that, applying it to a formula whatever, it always comes to an end with a result: if all rows have TRUE, the formula is valid; if there is some FALSE value, the formula is not valid.
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
Natural Deduction is a proof system that is sound and complete for e.g. classical propositional calculus.
Sound means that if a formula is provable with ND, it is valid.
In general, a proof system is not an algorithm, i.e. if we start from a formula and we do not know if it is valid or not, the simple fact that we are not able to find a proof does not mean that the proof does not exist: maybe, we are simply not clever enough to find it.
Truth table, instead, is an algorithm to test validity for propositional calculus; this means that, applying it to a formula whatever, it always comes to an end with a result: if all rows have TRUE, the formula is valid; if there is some FALSE value, the formula is not valid.
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
edited 1 hour ago
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
answered 1 hour ago
Mauro ALLEGRANZAMauro ALLEGRANZA
30.2k22067
30.2k22067
add a comment |
add a comment |
Truth tables can become large if there are many sentence letters That is when natural deduction might find a solution in a more economical manner. That assumes one can derive a line in a natural deduction proof that corresponds to the desired conclusion. If not, one has to keep looking. A truth table, although potentially large, would let one know one does not have to continue.
Here is how the authors of forallx describe the situation in Chapter 20: Soundness and Completeness, page 149:
Now that we know that the truth table method is interchangeable with the method of derivations, you can chose which method
you want to use for any given problem. Students often prefer to
use truth tables, because they can be produced purely mechanically, and that seems ‘easier’. However, we have already seen that
truth tables become impossibly large after just a few sentence letters. On the other hand, there are a couple situations where using
proofs simply isn’t possible. We syntactically defined a contingent
sentence as a sentence that couldn’t be proven to be a tautology
or a contradiction. There is no practical way to prove this kind of
negative statement. We will never know if there isn’t some proof
out there that a statement is a contradiction and we just haven’t
found it yet. We have nothing to do in this situation but resort
to truth tables. Similarly, we can use derivations to prove two
sentences equivalent, but what if we want to prove that they are
not equivalent? We have no way of proving that we will never find
the relevant proof. So we have to fall back on truth tables again.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
add a comment |
Truth tables can become large if there are many sentence letters That is when natural deduction might find a solution in a more economical manner. That assumes one can derive a line in a natural deduction proof that corresponds to the desired conclusion. If not, one has to keep looking. A truth table, although potentially large, would let one know one does not have to continue.
Here is how the authors of forallx describe the situation in Chapter 20: Soundness and Completeness, page 149:
Now that we know that the truth table method is interchangeable with the method of derivations, you can chose which method
you want to use for any given problem. Students often prefer to
use truth tables, because they can be produced purely mechanically, and that seems ‘easier’. However, we have already seen that
truth tables become impossibly large after just a few sentence letters. On the other hand, there are a couple situations where using
proofs simply isn’t possible. We syntactically defined a contingent
sentence as a sentence that couldn’t be proven to be a tautology
or a contradiction. There is no practical way to prove this kind of
negative statement. We will never know if there isn’t some proof
out there that a statement is a contradiction and we just haven’t
found it yet. We have nothing to do in this situation but resort
to truth tables. Similarly, we can use derivations to prove two
sentences equivalent, but what if we want to prove that they are
not equivalent? We have no way of proving that we will never find
the relevant proof. So we have to fall back on truth tables again.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
add a comment |
Truth tables can become large if there are many sentence letters That is when natural deduction might find a solution in a more economical manner. That assumes one can derive a line in a natural deduction proof that corresponds to the desired conclusion. If not, one has to keep looking. A truth table, although potentially large, would let one know one does not have to continue.
Here is how the authors of forallx describe the situation in Chapter 20: Soundness and Completeness, page 149:
Now that we know that the truth table method is interchangeable with the method of derivations, you can chose which method
you want to use for any given problem. Students often prefer to
use truth tables, because they can be produced purely mechanically, and that seems ‘easier’. However, we have already seen that
truth tables become impossibly large after just a few sentence letters. On the other hand, there are a couple situations where using
proofs simply isn’t possible. We syntactically defined a contingent
sentence as a sentence that couldn’t be proven to be a tautology
or a contradiction. There is no practical way to prove this kind of
negative statement. We will never know if there isn’t some proof
out there that a statement is a contradiction and we just haven’t
found it yet. We have nothing to do in this situation but resort
to truth tables. Similarly, we can use derivations to prove two
sentences equivalent, but what if we want to prove that they are
not equivalent? We have no way of proving that we will never find
the relevant proof. So we have to fall back on truth tables again.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
Truth tables can become large if there are many sentence letters That is when natural deduction might find a solution in a more economical manner. That assumes one can derive a line in a natural deduction proof that corresponds to the desired conclusion. If not, one has to keep looking. A truth table, although potentially large, would let one know one does not have to continue.
Here is how the authors of forallx describe the situation in Chapter 20: Soundness and Completeness, page 149:
Now that we know that the truth table method is interchangeable with the method of derivations, you can chose which method
you want to use for any given problem. Students often prefer to
use truth tables, because they can be produced purely mechanically, and that seems ‘easier’. However, we have already seen that
truth tables become impossibly large after just a few sentence letters. On the other hand, there are a couple situations where using
proofs simply isn’t possible. We syntactically defined a contingent
sentence as a sentence that couldn’t be proven to be a tautology
or a contradiction. There is no practical way to prove this kind of
negative statement. We will never know if there isn’t some proof
out there that a statement is a contradiction and we just haven’t
found it yet. We have nothing to do in this situation but resort
to truth tables. Similarly, we can use derivations to prove two
sentences equivalent, but what if we want to prove that they are
not equivalent? We have no way of proving that we will never find
the relevant proof. So we have to fall back on truth tables again.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
answered 1 hour ago
Frank HubenyFrank Hubeny
10.8k51559
10.8k51559
add a comment |
add a comment |
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