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4.1: What logic can do for you

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    138642
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    In Chapter 1 we mentioned that semanticists often use formal logic as a metalanguage for representing the meanings of sentences and other expressions in human languages. For the most part, this book emphasizes prose description more than formalization; we will use the logical notation a fair bit in Unit IV but only sporadically in other sections of the book. Nevertheless, it will be helpful for you to become familiar with this notation, not only for the purposes of this book but also to help you read other books and articles about semantics.

    In this chapter we will introduce some of the basic symbols and rules of inference for standard logic. Before we begin, it will probably be helpful to address a question which many readers may already be asking themselves, and which others are likely to ask before we get too far into the discussion: why are we doing this? How does translating English (or Samoan or Marathi) sentences into logical formulae help us to understand their meaning?

    Representing the complexities of natural language using formal logic is no trivial task, but here are some of the reasons why many scholars have found the effort required in adopting this approach worthwhile. First, every human language is characterized by ambiguity, vagueness, figures of speech, etc. These features can actually be an advantage for communicative purposes, but they make it difficult to provide precise and unambiguous descriptions of word and sentence meanings in English (or Samoan or Marathi). Using formal logic as a metalanguage avoids most of these problems.

    Second, we stated in Chapter 3 that one way of measuring the success or adequacy of a semantic analysis is to see whether it can explain or predict various meaning relations between sentences, such as entailment, paraphrase, or incompatibility. Logic is the science of inference. If the meanings of two sentences can be stated as logical formulae, logic provides very precise rules and methods for determining whether one follows as a logical consequence of the other (entailment), whether each follows as a logical consequence of the other (paraphrase), or whether the two are logically inconsistent, i.e. they cannot both be true (incompatibility).

    Third, it is often useful to test a hypothesis about the meaning of a sentence by expressing it in logical form, and then using the rules of logical inference to see what the implications would be. For example, suppose our analysis predicts that a certain sentence should mean p, and suppose we can show that if a person believes p, he is logically committed to believing q. Now suppose that native speakers of the language feel that there would be no inconsistency in asserting the sentence in question but denying q. This mismatch between logical inference and speaker intuition may give us reason to think that p is not the correct meaning of the sentence after all. We will see examples of this kind of reasoning in future chapters.

    Fourth, formal logic has proven to be a very powerful tool for modeling compositionality, i.e., for explaining how the meanings of sentences can be predicted from the meanings of the words they contain and the syntactic structure used to combine those words. As we noted in Chapter 1, this is one of the fundamental goals of semantic analysis. We will get a glimpse of how this can be done in Unit IV.

    Finally, formal logic is a recursive system. This means that a relatively small number of symbols and rules can be used to form an unlimited number of different formulae. Any adequate metalanguage for describing the meanings of sentences in a human language must have this property, because (as we noted in Chapter 1) there is in principle no limit to the number of distinct meaningful sentences that can be produced in any human language.

    To illustrate the recursive nature of the system, let us introduce the logical negation operator ¬ ‘not’. The negation operator combines with a single proposition to form a new proposition. So, for example, if we let p represent the proposition ‘It is raining,’ then ¬p (read ‘not p’) would represent the proposition ‘It is not raining.’ This proposition in turn can again combine with the negation operator to form a new proposition ¬(¬p) ‘It is not the case that it is not raining.’ There is in principle no limit to the number of formulae that can be produced in this way, though in practice sheer boredom would probably be a limiting factor.

    We begin in §4.2 with a brief discussion of inference and some of the ways in which logic can help us distinguish valid from invalid patterns of inference. §4.3 deals with propositional logic, which specifies ways of combining simple propositions to form complex propositions. An important fact about this part of the logical system is that the inferences of propositional logic depend only on the truth values of the propositions involved, and not on their meanings. §4.4 deals with predicate logic, which provides a way to take into account the meanings of individual content words and to state inferences which arise due to the meanings of quantifier words such as all, some, none, etc.


    This page titled 4.1: What logic can do for you is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Paul Kroeger (Language Library Press) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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