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4: Deductive Logic II - Sentential Logic

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  • Sentential logic (also called propositional logic) is logic that includes sentence letters (A,B,C) and logical connectives, but not quantifiers. The semantics of sentential logic uses truth assignments to the letters to determine whether a compound propositional sentence is true.

    • 4.1: Why Another Deductive Logic?
      In his own time, in ancient Greece, Aristotle’s system had a rival—the logic of the Stoic school, culminating in the work of Chrysippus. Recall, for Aristotle, the fundamental logical unit was the class; and since terms pick out classes, his logic is often referred to as a “term logic”. For the Stoics, the fundamental logical unit was the proposition; since sentences pick out propositions, we could call this a “sentential logic”.
    • 4.2: Syntax of Sentential Logic
      First, we cover syntax. This discussion will give us some clues as to the relationship between Sentential Logic and English, but a full accounting of that relationship will have to wait, as we said, for the discussion of semantics.
    • 4.3: Semantics of Sentential Logic
      While the semantics for a natural language like English is complicated (What is the meaning of a sentence? Its truth-conditions? The proposition expressed? Are those two things the same? Is it something else entirely? Ugh.), the semantics for SL sentences is simple: all we care about is truth-value. A sentence in SL can have one of two semantic values: true or false. That’s it.
    • 4.4: Translating from English to Sentential Logic
      In real life, though, we’re not interested in evaluating arguments in some artificial language; we’re interested in evaluating arguments presented in natural languages like English. So in order for our evaluative procedure of SL argument to have any real-world significance, we need to show how SL arguments can be fair representations of natural-language counterparts. We need to show how to translate sentences in English into Sentential Logic.
    • 4.5: Testing the Validity of Sentential Logic
      Having dealt with the task of taming natural language, we are finally in a position to complete the second and third steps of building a logic: defining logical form and developing a test for validity. The test will involve applying skills that we’ve already learned: setting up truth tables and computing the truth-values of compounds. First, we must define logical form in SL.

    Thumbnail: Chrysippus was a member of the Stoic school of philosophy and believed the fundamental logical unit was the propositio, which is the basis of “sentential logic”. Bust of Chrysippus, Uffizi Gallery, Florence (CC BY-SA 4.0; Livioandronico2013 via Wikipedia).

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