At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
The SYNTAX of a language is given by the rules that determine the set of expressions that are said to belong to the language, or, looking at it the other way around, the rules that tell you how to combine grammatical expressions of the language to obtain new grammatical expressions. You're probably most familiar with the notion of syntax in terms of grammar -grammar and syntax are actually exactly the same thing. Thus, just as the grammar of English tells you how to combine words and phrases of English to make larger expressions, sentences, the formal syntax of sentential logic tells you how to combine expressions of sentential logic to make larger expressions, logical formulae. Looking at it in the other direction, the formal syntax of sentential logic also allows you to determine, of any expression, whether or not that expression is a syntactically well-formed formula of sentential logic. In this chapter, we are going to study both the grammar of sentential logic and how to recognise and express the logical form of English sentences, which in turn will allow us to ''translate'' sentences of English into formulae of sentential logic and vice versa. Goals for this Chapter: Understand the logical structure of English sentences. Learn how to symbolise, i.e., to translate English sentence into formulae of sentential logic. Learn the formal grammar of sentential logic. Atomic Formulae and Logical Connectives One of the first things you probably learned about English grammar, way back in grade school, had to do with the basic parts of speech, that is, the different categories of words, such as nouns, verbs, adjectives, and so on. We're going to start in exactly the same fashion with the syntax of sentential logic. In English there are many different categories of words to worry about, and you probably still don't even know what all of them are, or what exactly words in each category are supposed to do. What exactly is a gerund, anyway? The good news is, in sentential logic there are only two different categories of words or basic expressions: ATOMIC FORMULAE and LOGICAL CONNECTIVES. Okay, so what are atomic formulae and logical connectives? Atomic formulae are, well, formulae that happen to have no interesting parts, at least from a particular point of view. Logical connectives, on the other hand, are things that serve to connect formulae together in order to create new and more complex formulae. The atomic formulae of sentential logic correspond to certain kinds of sentences of English (or any other language, for that matter), namely, sentences that express statements. The logical connectives correspond to certain words and phrases of English that we call LOGICAL OPERATORS., for example, "and", "or", "if then." The big question about atomic formulae as we saw in chapter 1, has to do with what sorts of English sentences we want to consider as atomic formulae of sentential logic, rather than how we deal with atomic formulae within sentential logic itself—that's the easy part. The answer here should be reasonably obvious if you think about the fact that the only basic expressions we have to deal with in sentential logic are just atomic formulae and logical connectives—atomic formulae are going to turn out to be just those formulae that don't involve any logical connectives, so the English sentences we consider to correspond to atomic formulae will be those that do not involve any logical operators. With this in mind, we should probably say a little bit more about the logical connectives and their corresponding operators in English. In our system of sentential logic, we have precisely four logical connectives, called CONJUNCTION, DISJUNCTION, THE CONDITIONAL, and NEGATION. These connectives correspond to logical operators in English as follows: Conjunction corresponds to the word 'and'; disjunction corresponds to 'or' (or the phrase 'either...or...'); the conditional corresponds to (the phrase) 'if...then....'; and negation corresponds to 'not.' Of course, when we say that a connective corresponds to a particular word or phrase in English, we aren't trying to claim that the connective should be considered to have exactly the same meaning as the English word. All we are trying to claim is that the connective captures an important part of the meaning —the TRUTH-FUNCTIONAL part. The logical connectives thus represent logical IDEALISATIONS of the corresponding words and phrases in English—idealisations that allow us to focus on the logical structure of English sentences.