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11.4: Sentential Logic

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    36229
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    When we create logical forms for arguments we sometime abbreviate clauses or simple sentences with words or just capital letters. If we were always to use capital letters and always to use the following special symbols for the connective phrases, such as "or" and "and," then we’d be expressing logical forms in the language of Sentential Logic. Sentential Logic is also called Statement Logic and Propositional Logic. The connective symbol ‘v’ abbreviates the English connective word ‘or’ that is used to link together two sentences in order to build a longer sentence. Similarly, the symbol ‘&’ replaces the English conjunction word "and" that also builds bigger sentences from smaller sentences. The ‘~’ symbol represents negation or the phrase "It is not true that..." which can be added to the front of a sentence to produce its negation. The arrow ‘→’ represents ‘if-then.’

    Here is a list of the connective symbols of Sentential Logic with the English phrases they replace. We will use the symbols with two simple sentences A and B:

    ~A not-A (it’s not true that A)
    A v B A or B (either A or B or both)
    A & B A and B (A but B)
    A → B if A then B (B if A) (A only if B)
    A ↔ B A if and only if B (A just in case B)

    There are grammar rules for forming well-formed formal sentences of greater and greater complexity. For our vocabulary of basic formal sentences we use the capital letters from A to O in the alphabet (and perhaps these capitals with numerical subscripts if we need more basic sentences). Then the complex sentences are built from these sentences by applying the connectives, according to the following rules of grammar:

    If P and Q are variables standing for any symbolic sentences, no matter how complex, then ~P is a well-formed sentence, and so is (P v Q), and (P & Q) and (P → Q) and (P ↔ Q).

    ‘P’ and ‘Q’ might represent or abbreviate any declarative sentence such as A, or A & (B → A), or whatever. Formal sentences can be as complicated as we want, but their length must be finite.

    These grammar rules imply that the complex sequence of symbols ((A & B) → C) is well-formed, but (A & B → C) is not. By the way, we customarily drop the outermost pair of parentheses (but only those) when writing well-formed sentences. For example, ((A & B) → C) is the same sentence as (A & B) → C.

    Exercise \(\PageIndex{1}\)

    Is this string of symbols grammatically well-formed? (E v F & G)

    Answer

    No, it cannot be built by applying the grammar rules to basic sentences. However, the following two sentences are well-formed: E v (F & G), and (E v G) & G


    This page titled 11.4: Sentential Logic is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Bradley H. Dowden.

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