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What is the difference between “logical equivalence” and “material equivalence”?

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2

Wikipedia offers this as the difference between "logical equivalence" and "material equivalence":

Logical equivalence is different from material equivalence. Formulas p and q are logically equivalent if and only if the statement of their material equivalence (P ⟺ Q) is a tautology.

Material equivalence is associated with the biconditional. However, I am still unclear what the difference is between the two.

I want to make sure I am using the terms correctly. Recently to avoid confusion I dropped the adjective "logically" in front of the following use of "equivalent":

Using De Morgan's laws, ¬(A ∧ ¬B) is equivalent to ¬A ∨ B.

If there is any difference between the two terms, what is it? Perhaps an example of the correct use of each would help clarify the difference.

If there isn't any difference I probably shouldn't use either one.

---

Wikipedia contributors. (2019, February 13). Logical equivalence. In Wikipedia, The Free Encyclopedia. Retrieved 11:22, August 3, 2019, from https://en.wikipedia.org/w/index.php?title=Logical_equivalence&oldid=883191333

logic

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asked 10 hours ago

Frank HubenyFrank Hubeny

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2

Wikipedia offers this as the difference between "logical equivalence" and "material equivalence":

Logical equivalence is different from material equivalence. Formulas p and q are logically equivalent if and only if the statement of their material equivalence (P ⟺ Q) is a tautology.

Material equivalence is associated with the biconditional. However, I am still unclear what the difference is between the two.

I want to make sure I am using the terms correctly. Recently to avoid confusion I dropped the adjective "logically" in front of the following use of "equivalent":

Using De Morgan's laws, ¬(A ∧ ¬B) is equivalent to ¬A ∨ B.

If there is any difference between the two terms, what is it? Perhaps an example of the correct use of each would help clarify the difference.

If there isn't any difference I probably shouldn't use either one.

---

Wikipedia contributors. (2019, February 13). Logical equivalence. In Wikipedia, The Free Encyclopedia. Retrieved 11:22, August 3, 2019, from https://en.wikipedia.org/w/index.php?title=Logical_equivalence&oldid=883191333

logic

share|improve this question

asked 10 hours ago

Frank HubenyFrank Hubeny

13.7k55 gold badges1717 silver badges6767 bronze badges

add a comment | 

Wikipedia offers this as the difference between "logical equivalence" and "material equivalence":

Logical equivalence is different from material equivalence. Formulas p and q are logically equivalent if and only if the statement of their material equivalence (P ⟺ Q) is a tautology.

Material equivalence is associated with the biconditional. However, I am still unclear what the difference is between the two.

I want to make sure I am using the terms correctly. Recently to avoid confusion I dropped the adjective "logically" in front of the following use of "equivalent":

Using De Morgan's laws, ¬(A ∧ ¬B) is equivalent to ¬A ∨ B.

If there is any difference between the two terms, what is it? Perhaps an example of the correct use of each would help clarify the difference.

If there isn't any difference I probably shouldn't use either one.

---

Wikipedia contributors. (2019, February 13). Logical equivalence. In Wikipedia, The Free Encyclopedia. Retrieved 11:22, August 3, 2019, from https://en.wikipedia.org/w/index.php?title=Logical_equivalence&oldid=883191333

logic

share|improve this question

asked 10 hours ago

Frank HubenyFrank Hubeny

13.7k55 gold badges1717 silver badges6767 bronze badges

share|improve this question

asked 10 hours ago

Frank HubenyFrank Hubeny

13.7k55 gold badges1717 silver badges6767 bronze badges

share|improve this question

asked 10 hours ago

Frank HubenyFrank Hubeny

13.7k55 gold badges1717 silver badges6767 bronze badges

asked 10 hours ago

Frank HubenyFrank Hubeny

13.7k55 gold badges1717 silver badges6767 bronze badges

asked 10 hours ago

Frank HubenyFrank Hubeny

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The answer is suggested by the quote that you provided:

Logical equivalence is different from material equivalence. Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology.

There is a difference between being true and being a tautology. Once you see this you can see the difference between material and logical equivalence. So, for instance, if P = 'today is Saturday' and Q = 'the year is 2019' then (P ↔ Q) is true, because both P and Q are true (at the time of writing), but (P ↔ Q) is not a tautology, because tomorrow P will be false and Q will remain true.

A similar difference exists between material and logical implication, which is also important to remember. With the same example, (P → Q) is true but not a tautology, so P does not logically imply Q. This similarity is to be expected as material equivalence is simply bi-directional material implication, and logical equivalence is bi-directional logical implication.

This is why it can be misleading to write 'P → Q' without specifying if one means logical implication, material implication, or some other kind of conditional relation.

share|improve this answer

answered 8 hours ago

EliranEliran

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3

The answer is suggested by the quote that you provided:

Logical equivalence is different from material equivalence. Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology.

There is a difference between being true and being a tautology. Once you see this you can see the difference between material and logical equivalence. So, for instance, if P = 'today is Saturday' and Q = 'the year is 2019' then (P ↔ Q) is true, because both P and Q are true (at the time of writing), but (P ↔ Q) is not a tautology, because tomorrow P will be false and Q will remain true.

A similar difference exists between material and logical implication, which is also important to remember. With the same example, (P → Q) is true but not a tautology, so P does not logically imply Q. This similarity is to be expected as material equivalence is simply bi-directional material implication, and logical equivalence is bi-directional logical implication.

This is why it can be misleading to write 'P → Q' without specifying if one means logical implication, material implication, or some other kind of conditional relation.

share|improve this answer

answered 8 hours ago

EliranEliran

4,85844 gold badges1515 silver badges3535 bronze badges

add a comment | 

3

The answer is suggested by the quote that you provided:

Logical equivalence is different from material equivalence. Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology.

There is a difference between being true and being a tautology. Once you see this you can see the difference between material and logical equivalence. So, for instance, if P = 'today is Saturday' and Q = 'the year is 2019' then (P ↔ Q) is true, because both P and Q are true (at the time of writing), but (P ↔ Q) is not a tautology, because tomorrow P will be false and Q will remain true.

A similar difference exists between material and logical implication, which is also important to remember. With the same example, (P → Q) is true but not a tautology, so P does not logically imply Q. This similarity is to be expected as material equivalence is simply bi-directional material implication, and logical equivalence is bi-directional logical implication.

This is why it can be misleading to write 'P → Q' without specifying if one means logical implication, material implication, or some other kind of conditional relation.

share|improve this answer

answered 8 hours ago

EliranEliran

4,85844 gold badges1515 silver badges3535 bronze badges

add a comment | 

The answer is suggested by the quote that you provided:

Logical equivalence is different from material equivalence. Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology.

There is a difference between being true and being a tautology. Once you see this you can see the difference between material and logical equivalence. So, for instance, if P = 'today is Saturday' and Q = 'the year is 2019' then (P ↔ Q) is true, because both P and Q are true (at the time of writing), but (P ↔ Q) is not a tautology, because tomorrow P will be false and Q will remain true.

A similar difference exists between material and logical implication, which is also important to remember. With the same example, (P → Q) is true but not a tautology, so P does not logically imply Q. This similarity is to be expected as material equivalence is simply bi-directional material implication, and logical equivalence is bi-directional logical implication.

This is why it can be misleading to write 'P → Q' without specifying if one means logical implication, material implication, or some other kind of conditional relation.

share|improve this answer

answered 8 hours ago

EliranEliran

4,85844 gold badges1515 silver badges3535 bronze badges

share|improve this answer

answered 8 hours ago

EliranEliran

4,85844 gold badges1515 silver badges3535 bronze badges

answered 8 hours ago

EliranEliran

4,85844 gold badges1515 silver badges3535 bronze badges

answered 8 hours ago

EliranEliran

4,85844 gold badges1515 silver badges3535 bronze badges