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1.3: Using Letters to Symbolize Assertions

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    23873
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    In the remainder of this , we will discuss a logical language called . It provides a convenient way to describe the logical relationship between two (or more) assertions, by using capital letters to represent assertions. Considered only as a symbol of , the letter \(A\) could mean any assertion. So, when translating from English into , it is important to provide a symbolization key that specifies what assertion is represented by each letter.

    For example, consider this deduction:

    Hypothesis:

    There is an apple on the desk.
    If there is an apple on the desk, then Jenny made it to class. Jenny made it to class.

    Conclusion:
    Jenny made it to class.

    This is obviously a valid deduction in English.

    What happens if we replace each assertion with a letter? Our symbolization key would look like this:

    \(A\):There is an apple on the desk.
    \(B\): If there is an apple on the desk, then Jenny made it to class.
    \(C\): Jenny made it to class.

    We would then symbolize the deduction in this way:

    Hypothesis:

    \(A\)
    \(B\)

    Conclusion: \(C\)

    Unfortunately, there is no necessary connection between two assertions \(A\) and \(B\), which could be any assertions, and a third assertion \(C\), which could be any assertion, so this is not a valid deduction. Thus, the validity of the original deduction has been lost in this translation; we need to do something else in order to preserve the logical structure of the original deduction and obtain a valid translation.

    The important thing about the original deduction is that the second hypothesis is not merely any assertion, logically divorced from the other assertions in the deduction. Instead, the second hypothesis contains the first hypothesis and the conclusion as parts. Our symbolization key for the deduction only needs to include meanings for \(A\) and \(C\), and we can build the second hypothesis from those pieces. So we could symbolize the deduction this way:

    Hypothesis:

    \(A\)
    If \(A\), then \(C\).

    Conclusion: \(C\)

    This preserves the structure of the deduction that makes it valid, but it still makes use of the English expression “If\(\ldots\) then\(\ldots\).” Eventually, we will replace all of the English expressions with mathematical notation, but this is a good start.

    The assertions that are symbolized with a single letter are called atomic assertions, because they are the basic building blocks out of which more complex assertions are built. Whatever logical structure an assertion might have is lost when it is translated as an atomic assertion. From the point of view of Propositional Logic, the assertion is just a letter. It can be used to build more complex assertions, but it cannot be taken apart.

    Notation \(1.3.1\).

    The symbol “\(\therefore\)” means “therefore,” and we will often use \[A, B, C, \ldots, \therefore Z\]
    as an abbreviation for the deduction

    Hypotheses:

    \(\begin{aligned}
    &A \\
    &B \\
    &C \\
    &\vdots
    \end{aligned}\)

    Conclusion: \(Z\).


    This page titled 1.3: Using Letters to Symbolize Assertions is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Dave Witte Morris & Joy Morris.

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