What is the difference between logical consistency and logical entailment in deductive logic?Difference between implication/conditional and logical entailment?How do disjunctive antecedents work in Marc Lange's stability concept of laws of nature?What is the difference between intensional and extensional logic?What is the difference between deductive and inductive reasoning?Is the use of inconsistent definitions a logical fallacy?What is the difference between relational logic and predicate logic?What is the difference between logic and reasoning?Teaching my son the difference between inductive and deductive reasoningRelationship between entailment and equivalence with consistencyIncorrect statement in Suppes' Introduction to Logic
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What is the difference between logical consistency and logical entailment in deductive logic?
Difference between implication/conditional and logical entailment?How do disjunctive antecedents work in Marc Lange's stability concept of laws of nature?What is the difference between intensional and extensional logic?What is the difference between deductive and inductive reasoning?Is the use of inconsistent definitions a logical fallacy?What is the difference between relational logic and predicate logic?What is the difference between logic and reasoning?Teaching my son the difference between inductive and deductive reasoningRelationship between entailment and equivalence with consistencyIncorrect statement in Suppes' Introduction to Logic
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I am having a little trouble sorting out two definitions out from the first chapter in my logic textbook. I am under the impression that a set in a sentence can be logically consistent without entailing anything (eg. Sara is right-handed, Mark is right-handed, Jean is right-handed). Are there no actual arguments within logically consistent and logically entailed sentences? What are the differences and relation between the two?
I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument). Please correct me if I'm wrong. It's difficult for me to conceptualize and would really appreciate if someone could help simplify, thank you!
logic symbolic-logic
asked 8 hours ago
ToniToni
63 bronze badges
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Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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show 1 more comment
I am having a little trouble sorting out two definitions out from the first chapter in my logic textbook. I am under the impression that a set in a sentence can be logically consistent without entailing anything (eg. Sara is right-handed, Mark is right-handed, Jean is right-handed). Are there no actual arguments within logically consistent and logically entailed sentences? What are the differences and relation between the two?
I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument). Please correct me if I'm wrong. It's difficult for me to conceptualize and would really appreciate if someone could help simplify, thank you!
logic symbolic-logic
asked 8 hours ago
ToniToni
63 bronze badges
New contributor
Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome!
– Frank Hubeny
7 hours ago
You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false.
– Logikal
6 hours ago
Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency.
– Toni
6 hours ago
Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description.
– Toni
6 hours ago
It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic.
– Logikal
2 hours ago
|
show 1 more comment
I am having a little trouble sorting out two definitions out from the first chapter in my logic textbook. I am under the impression that a set in a sentence can be logically consistent without entailing anything (eg. Sara is right-handed, Mark is right-handed, Jean is right-handed). Are there no actual arguments within logically consistent and logically entailed sentences? What are the differences and relation between the two?
I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument). Please correct me if I'm wrong. It's difficult for me to conceptualize and would really appreciate if someone could help simplify, thank you!
logic symbolic-logic
asked 8 hours ago
ToniToni
63 bronze badges
New contributor
Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I am having a little trouble sorting out two definitions out from the first chapter in my logic textbook. I am under the impression that a set in a sentence can be logically consistent without entailing anything (eg. Sara is right-handed, Mark is right-handed, Jean is right-handed). Are there no actual arguments within logically consistent and logically entailed sentences? What are the differences and relation between the two?
I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument). Please correct me if I'm wrong. It's difficult for me to conceptualize and would really appreciate if someone could help simplify, thank you!
logic symbolic-logic
logic symbolic-logic
asked 8 hours ago
ToniToni
63 bronze badges
New contributor
Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
ToniToni
63 bronze badges
New contributor
Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 6 hours ago
Toni
asked 8 hours ago
ToniToni
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Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 8 hours ago
ToniToni
63 bronze badges
asked 8 hours ago
ToniToni
63 bronze badges
63 bronze badges
New contributor
Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
Toni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome!
– Frank Hubeny
7 hours ago
You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false.
– Logikal
6 hours ago
Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency.
– Toni
6 hours ago
Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description.
– Toni
6 hours ago
It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic.
– Logikal
2 hours ago
|
show 1 more comment
Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome!
– Frank Hubeny
7 hours ago
You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false.
– Logikal
6 hours ago
Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency.
– Toni
6 hours ago
Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description.
– Toni
6 hours ago
It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic.
– Logikal
2 hours ago
Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome!
– Frank Hubeny
7 hours ago
Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome!
– Frank Hubeny
7 hours ago
You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false.
– Logikal
6 hours ago
You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false.
– Logikal
6 hours ago
Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency.
– Toni
6 hours ago
Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency.
– Toni
6 hours ago
Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description.
– Toni
6 hours ago
Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description.
– Toni
6 hours ago
It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic.
– Logikal
2 hours ago
It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic.
– Logikal
2 hours ago
|
show 1 more comment
3 Answers
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Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well).
Logical consistency merely means that there exists at least one truth assignment that satisfies all of the statements. The statements "someone is English or drives on the right" and "someone is not English or drives on the left" are logically consistent with each other, because there is at least one truth condition (e.g. someone is English and drives on the left) which satisfies both.
Keep in mind that entailment is directional — the truth assignments that satisfy one statement are a subset of the assignments that satisfy the other — while consistency is a simple relation. You can think of it (roughly) like the difference between causation and correlation, though please don't extend that analogy too far.
answered 6 hours ago
Ted WrigleyTed Wrigley
913 bronze badges
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Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
add a comment |
Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.).
Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other.
This is extended to any set of more than two sentences as follows: A set of sentences is inconsistent if any two sentences of the set are inconsistent with each other.
A set of sentences may be consistent without any of them implying any of the other. You can also have a set of sentences which is consistent and some or all of the sentences imply some or all of the other sentences.
The relation, therefore, between consistency of a set of sentences and implication (entailment) is only that if any sentence of the set implies the negation of any of the other sentences of the set, then the set is inconsistent (and the transposition of that).
Thus, the fact that any two sentences of any set are inconsistent entails that the set itself is inconsistent.
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
add a comment |
There is a relation :
if a set Γ of formulas and a formula A are not consistent, then Γ logically implies (or entails) ¬ A.
Consistency can be defined either sintactically : a set Γ of formulas is inconsistent iff we can derive a contradiction from it, or semantically : a set Γ is inconsistent iff it is unsatisfiable, i.e. we cannot find an interpretation that satisfies all formulas in it.
Logical consequence (or logical entailment) is a relation defined between a set Γ of formulas and a formula A when there is no interpretation that satisfies all formulas in Γ and falsify A.
The proof is quite simple : if Γ, A is inconsistent, it is unsatisfiable, i.e. there is no interpretation that satisfies simultaneously Γ and A.
But this means that, every interpretation that satisfies Γ will satisfies the negation of A.
Example : a very simple example is the following : P, P → ¬Q as Γ and Q as A.
We have that P, P → ¬Q and Q is an inconsistent set of formulas and thus P, P → ¬Q entails ¬Q.
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
add a comment |
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Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well).
Logical consistency merely means that there exists at least one truth assignment that satisfies all of the statements. The statements "someone is English or drives on the right" and "someone is not English or drives on the left" are logically consistent with each other, because there is at least one truth condition (e.g. someone is English and drives on the left) which satisfies both.
Keep in mind that entailment is directional — the truth assignments that satisfy one statement are a subset of the assignments that satisfy the other — while consistency is a simple relation. You can think of it (roughly) like the difference between causation and correlation, though please don't extend that analogy too far.
answered 6 hours ago
Ted WrigleyTed Wrigley
913 bronze badges
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Ted Wrigley is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
add a comment |
Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well).
Logical consistency merely means that there exists at least one truth assignment that satisfies all of the statements. The statements "someone is English or drives on the right" and "someone is not English or drives on the left" are logically consistent with each other, because there is at least one truth condition (e.g. someone is English and drives on the left) which satisfies both.
Keep in mind that entailment is directional — the truth assignments that satisfy one statement are a subset of the assignments that satisfy the other — while consistency is a simple relation. You can think of it (roughly) like the difference between causation and correlation, though please don't extend that analogy too far.
answered 6 hours ago
Ted WrigleyTed Wrigley
913 bronze badges
New contributor
Ted Wrigley is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
add a comment |
Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well).
Logical consistency merely means that there exists at least one truth assignment that satisfies all of the statements. The statements "someone is English or drives on the right" and "someone is not English or drives on the left" are logically consistent with each other, because there is at least one truth condition (e.g. someone is English and drives on the left) which satisfies both.
Keep in mind that entailment is directional — the truth assignments that satisfy one statement are a subset of the assignments that satisfy the other — while consistency is a simple relation. You can think of it (roughly) like the difference between causation and correlation, though please don't extend that analogy too far.
answered 6 hours ago
Ted WrigleyTed Wrigley
913 bronze badges
New contributor
Ted Wrigley is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well).
Logical consistency merely means that there exists at least one truth assignment that satisfies all of the statements. The statements "someone is English or drives on the right" and "someone is not English or drives on the left" are logically consistent with each other, because there is at least one truth condition (e.g. someone is English and drives on the left) which satisfies both.
Keep in mind that entailment is directional — the truth assignments that satisfy one statement are a subset of the assignments that satisfy the other — while consistency is a simple relation. You can think of it (roughly) like the difference between causation and correlation, though please don't extend that analogy too far.
answered 6 hours ago
Ted WrigleyTed Wrigley
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answered 6 hours ago
Ted WrigleyTed Wrigley
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answered 6 hours ago
Ted WrigleyTed Wrigley
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answered 6 hours ago
Ted WrigleyTed Wrigley
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Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
add a comment |
Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences).
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this.
– Toni
6 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right.
– Ted Wrigley
5 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well.
– Toni
4 hours ago
add a comment |
Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.).
Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other.
This is extended to any set of more than two sentences as follows: A set of sentences is inconsistent if any two sentences of the set are inconsistent with each other.
A set of sentences may be consistent without any of them implying any of the other. You can also have a set of sentences which is consistent and some or all of the sentences imply some or all of the other sentences.
The relation, therefore, between consistency of a set of sentences and implication (entailment) is only that if any sentence of the set implies the negation of any of the other sentences of the set, then the set is inconsistent (and the transposition of that).
Thus, the fact that any two sentences of any set are inconsistent entails that the set itself is inconsistent.
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
add a comment |
Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.).
Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other.
This is extended to any set of more than two sentences as follows: A set of sentences is inconsistent if any two sentences of the set are inconsistent with each other.
A set of sentences may be consistent without any of them implying any of the other. You can also have a set of sentences which is consistent and some or all of the sentences imply some or all of the other sentences.
The relation, therefore, between consistency of a set of sentences and implication (entailment) is only that if any sentence of the set implies the negation of any of the other sentences of the set, then the set is inconsistent (and the transposition of that).
Thus, the fact that any two sentences of any set are inconsistent entails that the set itself is inconsistent.
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
add a comment |
Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.).
Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other.
This is extended to any set of more than two sentences as follows: A set of sentences is inconsistent if any two sentences of the set are inconsistent with each other.
A set of sentences may be consistent without any of them implying any of the other. You can also have a set of sentences which is consistent and some or all of the sentences imply some or all of the other sentences.
The relation, therefore, between consistency of a set of sentences and implication (entailment) is only that if any sentence of the set implies the negation of any of the other sentences of the set, then the set is inconsistent (and the transposition of that).
Thus, the fact that any two sentences of any set are inconsistent entails that the set itself is inconsistent.
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.).
Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other.
This is extended to any set of more than two sentences as follows: A set of sentences is inconsistent if any two sentences of the set are inconsistent with each other.
A set of sentences may be consistent without any of them implying any of the other. You can also have a set of sentences which is consistent and some or all of the sentences imply some or all of the other sentences.
The relation, therefore, between consistency of a set of sentences and implication (entailment) is only that if any sentence of the set implies the negation of any of the other sentences of the set, then the set is inconsistent (and the transposition of that).
Thus, the fact that any two sentences of any set are inconsistent entails that the set itself is inconsistent.
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
answered 6 hours ago
SpeakpigeonSpeakpigeon
4531 silver badge10 bronze badges
4531 silver badge10 bronze badges
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
add a comment |
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it.
– Toni
5 hours ago
add a comment |
There is a relation :
if a set Γ of formulas and a formula A are not consistent, then Γ logically implies (or entails) ¬ A.
Consistency can be defined either sintactically : a set Γ of formulas is inconsistent iff we can derive a contradiction from it, or semantically : a set Γ is inconsistent iff it is unsatisfiable, i.e. we cannot find an interpretation that satisfies all formulas in it.
Logical consequence (or logical entailment) is a relation defined between a set Γ of formulas and a formula A when there is no interpretation that satisfies all formulas in Γ and falsify A.
The proof is quite simple : if Γ, A is inconsistent, it is unsatisfiable, i.e. there is no interpretation that satisfies simultaneously Γ and A.
But this means that, every interpretation that satisfies Γ will satisfies the negation of A.
Example : a very simple example is the following : P, P → ¬Q as Γ and Q as A.
We have that P, P → ¬Q and Q is an inconsistent set of formulas and thus P, P → ¬Q entails ¬Q.
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
add a comment |
There is a relation :
if a set Γ of formulas and a formula A are not consistent, then Γ logically implies (or entails) ¬ A.
Consistency can be defined either sintactically : a set Γ of formulas is inconsistent iff we can derive a contradiction from it, or semantically : a set Γ is inconsistent iff it is unsatisfiable, i.e. we cannot find an interpretation that satisfies all formulas in it.
Logical consequence (or logical entailment) is a relation defined between a set Γ of formulas and a formula A when there is no interpretation that satisfies all formulas in Γ and falsify A.
The proof is quite simple : if Γ, A is inconsistent, it is unsatisfiable, i.e. there is no interpretation that satisfies simultaneously Γ and A.
But this means that, every interpretation that satisfies Γ will satisfies the negation of A.
Example : a very simple example is the following : P, P → ¬Q as Γ and Q as A.
We have that P, P → ¬Q and Q is an inconsistent set of formulas and thus P, P → ¬Q entails ¬Q.
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
add a comment |
There is a relation :
if a set Γ of formulas and a formula A are not consistent, then Γ logically implies (or entails) ¬ A.
Consistency can be defined either sintactically : a set Γ of formulas is inconsistent iff we can derive a contradiction from it, or semantically : a set Γ is inconsistent iff it is unsatisfiable, i.e. we cannot find an interpretation that satisfies all formulas in it.
Logical consequence (or logical entailment) is a relation defined between a set Γ of formulas and a formula A when there is no interpretation that satisfies all formulas in Γ and falsify A.
The proof is quite simple : if Γ, A is inconsistent, it is unsatisfiable, i.e. there is no interpretation that satisfies simultaneously Γ and A.
But this means that, every interpretation that satisfies Γ will satisfies the negation of A.
Example : a very simple example is the following : P, P → ¬Q as Γ and Q as A.
We have that P, P → ¬Q and Q is an inconsistent set of formulas and thus P, P → ¬Q entails ¬Q.
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
There is a relation :
if a set Γ of formulas and a formula A are not consistent, then Γ logically implies (or entails) ¬ A.
Consistency can be defined either sintactically : a set Γ of formulas is inconsistent iff we can derive a contradiction from it, or semantically : a set Γ is inconsistent iff it is unsatisfiable, i.e. we cannot find an interpretation that satisfies all formulas in it.
Logical consequence (or logical entailment) is a relation defined between a set Γ of formulas and a formula A when there is no interpretation that satisfies all formulas in Γ and falsify A.
The proof is quite simple : if Γ, A is inconsistent, it is unsatisfiable, i.e. there is no interpretation that satisfies simultaneously Γ and A.
But this means that, every interpretation that satisfies Γ will satisfies the negation of A.
Example : a very simple example is the following : P, P → ¬Q as Γ and Q as A.
We have that P, P → ¬Q and Q is an inconsistent set of formulas and thus P, P → ¬Q entails ¬Q.
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
edited 4 hours ago
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
answered 7 hours ago
Mauro ALLEGRANZAMauro ALLEGRANZA
28.7k2 gold badges20 silver badges68 bronze badges
28.7k2 gold badges20 silver badges68 bronze badges
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
add a comment |
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well.
– Toni
6 hours ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on.
– Toni
1 hour ago
add a comment |
Toni is a new contributor. Be nice, and check out our Code of Conduct.
Toni is a new contributor. Be nice, and check out our Code of Conduct.
Toni is a new contributor. Be nice, and check out our Code of Conduct.
Toni is a new contributor. Be nice, and check out our Code of Conduct.
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Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome!
– Frank Hubeny
7 hours ago
You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false.
– Logikal
6 hours ago
Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency.
– Toni
6 hours ago
Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description.
– Toni
6 hours ago
It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic.
– Logikal
2 hours ago