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1 - Sentential Logic

Daniel J. Velleman
Daniel J. Velleman
Affiliation:
Amherst College, Massachusetts
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Summary

Deductive Reasoning and Logical Connectives

As we saw in the introduction, proofs play a central role in mathematics, and deductive reasoning is the foundation on which proofs are based. Therefore, we begin our study of mathematical reasoning and proofs by examining how deductive reasoning works.

Example 1.1.1. Here are three examples of deductive reasoning:

  • 1. It will either rain or snow tomorrow.

  • It's too warm for snow.

  • Therefore, it will rain.

  • 2. If today is Sunday, then I don't have to go to work today.

  • Today is Sunday.

  • Therefore, I don't have to go to work today.

  • 3. I will go to work either tomorrow or today.

  • I'm going to stay home today.

  • Therefore, I will go to work tomorrow.

    • In each case, we have arrived at a conclusion from the assumption that some other statements, called premises, are true. For example, the premises in argument 3 are the statements “I will go to work either tomorrow or today” and “I'm going to stay home today.” The conclusion is “I will go to work tomorrow,” and it seems to be forced on us somehow by the premises.

    But is this conclusion really correct? After all, isn't it possible that I'll stay home today, and then wake up sick tomorrow and end up staying home again?

    Type
    Chapter
    Information
    How to Prove It
    A Structured Approach
     , pp. 8 - 54
    Publisher: Cambridge University Press
    Print publication year: 2006

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    • Sentential Logic
    • Daniel J. Velleman, Amherst College, Massachusetts
    • Book: How to Prove It
    • Online publication: 05 September 2012
    • Chapter DOI: https://doi.org/10.1017/CBO9780511808234.003
    Available formats
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    • Sentential Logic
    • Daniel J. Velleman, Amherst College, Massachusetts
    • Book: How to Prove It
    • Online publication: 05 September 2012
    • Chapter DOI: https://doi.org/10.1017/CBO9780511808234.003
    Available formats
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    To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

    • Sentential Logic
    • Daniel J. Velleman, Amherst College, Massachusetts
    • Book: How to Prove It
    • Online publication: 05 September 2012
    • Chapter DOI: https://doi.org/10.1017/CBO9780511808234.003
    Available formats
    ×