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11.2: Logical Equivalence

  • Page ID
    22021
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    If you were told, "John stepped on the camera by accident," you wouldn't have learned anything different from having been told "The camera was accidentally stepped on by John." These sentences say the same thing—they make the same statements—even though they are grammatically different. Because the two say the same thing logically, they are said to be equivalent, or, more technically, logically equivalent. Logical equivalence is somewhat like synonymy except that it is for sentences, not words. That phrase say the same thing is a bit imprecise, so here is a definition using more precise terminology:

    Definition

    Statement P is logically equivalent to statement Q provided P follows from Q with certainty and Q also follows from P with certainty

    The certainty mentioned here is not a psychological notion; it is a logical notion. That is, the certainty is not about feeling sure but instead about the solidity of the logical relationship of support among statements. Logically equivalent statements say the same thing as far as logic is concerned.

    alternative Definition

    Statement P is logically equivalent to statement Q provided P logically implies Q, and also Q logically implies P.

    Here is a pair of logically equivalent statements:

    Tiffany is so sincere that you can't doubt her.
    The sincerity of Tiffany cannot be doubted.

    Yet these two are not logically equivalent:

    Tiffany got married and got pregnant.
    Tiffany got pregnant and got married.

    Time order is the problem.

    Here is a much less obvious example of logical equivalence. Suppose P is the sentence "Not all mammals are land dwellers," and Q is the sentence "Some mammals are not creatures that live on land." Does Q follow from P with certainty? Yes. How about vice versa? Yes. So, P and Q are logically equivalent. This relationship between the two sentences would hold even if the word mammal were replaced by the phrase fish in the Indian Ocean. Consequently, logical equivalence between two sentences can be a matter of the form of the two sentences, not just what they are about.

    Exercise \(\PageIndex{1}\)

    Does the definition of logical equivalence permit a true sentence and a false sentence to be logically equivalent to each other?

    Answer

    Do not rush to look at the answer before thinking seriously about the question. The answer is in the next footnote

    Deciding whether two phrases are logically equivalent1 can be critical in assessing the quality of an argument. Here is an example involving an argument in which the conclusion follows from the premises with certainty:

    1. If the attraction that baseball has will persist in America over the next decade, then our income from concessions will also remain steady over the next decade.

    2. I know the attraction that baseball has will in fact persist in America over the next decade.

    3. So, our income from concessions will remain steady over the next decade.

    Would the conclusion still follow if premise 2 were replaced with the following statement?

    2'. Baseball will continue to flourish in America over the next ten years.

    It depends. If statement 2' is logically equivalent to statement 2, then the conclusion would still follow with certainty. However, if you cannot be sure they are equivalent, you cannot be sure the conclusion of the argument with 2' follows with certainty. To decide whether 2 and 2' are equivalent, you should be sensitive to context and use the principle of charity. If, after doing all this, you still cannot tell whether 2 and 2' are equivalent, and if you need to be sure, you will have to ask the speaker or author to be clearer.

    Exercise \(\PageIndex{1}\)

    Does the conclusion follow with certainty from the premises in this argument? Explain why a simple "yes" or "no" answer is unacceptable because of logical equivalence.

    If the latest version of the word processing program Word is warmly received on presentation, its owners and programmers are going to be happy about what they created. All reports indicate Word did hit the market with a splash and got many good reviews. So, we can conclude that WordPerfect's creators felt a sense of accomplishment.

    Answer

    The argument's conclusion follows with certainty from its premises if the principle of charity permits us to say that "warmly received on presentation" means the same as "hit the market with a splash," and if it also permits us to say "happiness" here is the same as "feeling a sense of accomplishment" and if the "owners and programmers" include the "creators." It is likely that these equivalences hold, but you cannot be sure and therefore cannot definitely say, "Yes, the conclusion follows with certainty." If you needed to be sure, you should ask the author to be clearer about all this.

    The concept of logical equivalence is useful in other ways. This usefulness arises from the fact that the deductive validity or invalidity of an argument usually depends on the logical forms of its sentences, as we will see later in this chapter. In turn, the ability to identify the logical forms of sentences requires the ability to translate the sentence into a logically equivalent one.


    1 No, if one is true and the other is false in the same circumstances, they must be saying something different from each other and thus cannot be logically equivalent.


    This page titled 11.2: Logical Equivalence is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Bradley H. Dowden.

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