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N01SLSyntax - Vocabulary and Syntax of SL.pdf

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James Hildebrand

THE VOCABULARY AND SYNTAX OF SL Introduction Vocabulary Syntactic Rules Variables Demonstrations Conventions Main Connectives, Immediate Components, and Forms Formal Properties INTRODUCTION The study of modern formal deductive logic begins with whole sentences. We do not attempt to break down sentences into parts such as subjects and predicates, or identify the forms made up of parts of sentences. Our inquiry is into what forms or arrangements of whole sentences make more compound sentences logically true, false, or indeterminate; pairs of sentences logically equivalent or non-equivalent; sets of sentences logically consistent or inconsistent; and arguments deductively valid or invalid. We take a simple or, as we will call it, atomic sentence to be a sentence that cannot be divided into parts that are themselves sentences. In contrast, a compound sentence is a sentence that has at least one part that is already a sentence (and that is not identical to that sentence itself). “All whales are mammals,” is a simple sentence. It cannot be divided into parts that are themselves sentences. “Either the roof of University College is brown or the roof of University College is green” is a compound sentence. It can be divided into two parts that are themselves sentences. In addition to two simple sentences, the compound sentence, “Either the roof of University College is brown or the roof of University College is green” contains a “conne ctive” phrase that puts those two sentences together into one, the phrase, “Either … or …”. “It is not the case that the roof of University College is green,” is also a compound sentence. Though it cannot be divided into two parts that are themselves sentences, it does have a smaller part that is already a sentence. The remaining part, “It is not the case that,” is not a sentence, but it can be considered to a limiting case of a connective phrase. Whereas “Either … or …” can be considered a “binary connective” that connects two sentences together, “It is not the case that …” can be considered a “unary connective” that “connects” to one sentence. You may consider it to be an abuse of the term “connect” to apply it to something that is attached to just one thing, but it is common practice in logic and mathematics to extend the application of concepts to the “1-place” or even the “0-place” case, as you earlier saw us do with the notion of an “empty” set — which considered from one point of view is an absurd notion (how can you have a set if you don’t have anything to make it up?). In addition to unary and binary connectives, English uses trinary connectives, such as “Either … or … or …,” “All of … and … and …,” or “It is that case that … unless … in which case …” and that English has the facility to use even higher place connectives, e.g., “At least two but no more than seven of … [imagine any arbitrarily larger number than seven different things being listed here]” We represent the forms of English sentences by using capital letters (A, B, C, …Z), with or without numerical subscripts (A B 1 C27 1,032 …) to stand for whole sentences, special symbols to stand for connectives, and parentheses to group sentences according to the level of connective. No parentheses are used with unary connectives; parentheses around two sentences are used with binary connectives, and as it turns out we will not have to worry about higher level connectives. We have the option of using numerical subscripts with capital letters because there are infinitely many different sentences of English, and we want to cover the case where we would need an arbitrarily large number of symbols for different sentences. However, to keep things simple, we will only use symbols for one unary connective and four binary connectives, even though the English language contains many more than just five connectives, most of them of higher levels than just two. As it turns out, we don’t need any more than five connectives to represent all the English connectives, of however many places. The logical meaning of all the remaining connectives can be captured using combinations of just the five connectives that we will bother to symbolize. In fact, we can get by with fewer than five. Our symbolic language for representing the forms of simple sentences is therefore provably “complete.” It does not leave any sentential forms out. What I have just outlined is the vocabulary and the grammar or syntax for a “formal language” that we will use to represent the forms of sentences, sets of sentences, and arguments of English. Here is a slightly more rigorous restatement of the specifics of what we will call the formal language for sentential logic or SL. The Formal Language SL (A formal language for sentential logic) The Vocabulary of SL consists of: Sentence letters of SL A, B, C, ..., A , A , A , ... 1 2 3 Connectives of SL ~ , &, v, ⊃ , ≡ Punctuation Marks: ( , ) When the sentence letters of SL are taken to stand for simple sentences of English, and the connectives are taken to stand for connectives of English, the sentences of SL become the forms of groups of sentences of English. This is what makes SL a formal language for English (and other natural languages). However, we will not worry just yet about how to interpret the connectives of SL in English or how to use SL to represent the forms of English sentences or, as it is said, “translate” English sentences. That is the job of semantics. We will take some time working just with the grammar or syntax of the language SL before turning to its semantics. The syntax of SL specifies the rules for how vocabulary elements can be strung together to create well-formed strings that we will call “sentences of SL.” We consider any string of vocabulary elements to be an expression of SL. We interpret this very strictly. Symbols as “{,” “∪,” and “Γ,” that you will often see mixed in with vocabulary elements of SL in expressions used in the text or the notes are not themselves in the vocabulary of SL, so strings of symbols including those symbols are not expressions of SL. Even letters of the English alphabet that are in lower case, bold face, or italics such as “a,” “A,” and “A ” are not in the vocabulary, and therefore any string containing them is not an expression of SL. Even with this restriction, not every expression of SL is a sentence of SL. An expression of SL is a sentence of SL if and only if the symbols have been strung together in the right way. The permissible ways of stringing symbols together are provided by the syntactic rules of SL, which constitute what is in effect the grammar of the language and the rules in accord with which sentences are built up from vocabulary elements. SYNTACTIC FORMATION RULES 1. Every sentence letter of SL is a sentence of SL. 2. If P is a sentence of SL, then ~P is a sentence letter of SL. 3. If P and Q are two (not necessarily distinct) sentences of SL, then (P & Q), (P v Q), (P ⊃ Q), and (P ≡ Q) are sentences of SL. 4. Nothing is a sentence of SL unless it can be formed by repeated applications of rules 1- 3. METAVARIABLES In stating these rules I have used bold italicized letters as “variables” to stand for spots that could be filled in with sentences of SL. I could just as well have used blanks, e.g., “If ___ and _ _ _ are sentences of SL then (___ & _ _ _) is a sentence of SL,” but that procedure has drawbacks. It costs time and keystrokes and takes up space. It is also much harder on the eyes and thereby on the mind. Because the blanks are of different sorts and take up so much space, they rapidly fill the line and it becomes harder to see what is going on at a glance. The use of bold italicized letters to stand for blanks rectifies this defect, but you need to keep in mind that the letters do not necessarily stand for atomic sentences of SL. They could stand for arbitrarily compound sentences. Using the same letter indicates that the same sentence of SL should be understood to go wherever that letter occurs. Note that in rule 3 it is specified that P and Q need not refer to different sentences of SL. So here, even though different bold italicized letters are used, we are told that we can consider them to refer to either different sentences of SL or the same sentence of SL. This is always the case, even when not explicitly stated. P and Q are “metavariables” that can stand for any expression of SL. Insofar as P can stand for any expression of SL, it can stand for the same expression that Q stands for. So, while different occurrences of the same metavariable must be understood to refer to the same expression of SL, different occurrences of different metavariables need not be assumed to refer to different expressions of SL. As already mentioned, the bold italicized letters are not part of the vocabulary of SL. They are special symbols that we use when we want to refer to expressions of SL without stating precisely what expressions we have in mind, the way we use “John Doe” to refer just to someone, without further specifying whom. APPLICATIONS According to the rules, sentences of SL are built up in stages from parts as follows: You must start with a sentence letter, whether plain or sub-scripted. Each sentence letter is considered to already be a sentence of SL by itself. You may put a “~” immediately in front of a sentence letter to build a compound sentence. More generally, you can always put a “~” (tilde) in front of any sentence to generate a longer sentence. So, starting with “A”, you can proceed to build “~A” , “~~A” , “~~~A” , and so on. Note that you may never put a “~” after a sentence, only in front of one. Note also t hat you never use parentheses when you add tildes, no matter how many tildes you add. If you ever see a parenthesis before or after a tilde, it did not get to be there by being added along with the tilde. It can only have come to be there by being added along with some binary connective. Parentheses are only ever used along with binary connectives. If you have already built two sentences, you may put any one of the binary connectives (&, v, ⊃, ≡) between those two sentences and enclose the result in parentheses. Note that now, you must use parentheses. A binary connective may only be put between two sentences and when it is put between two sentences, parentheses must be put on either side of those sentences. Conversely, the only place parentheses are to be used is on either side of the two sentences connected by a binary connective. So, for example, given two sentences, “A” and “B” , you may proceed to build such sentences as “(A & B)” , “(B & A)” , and “(A ⊃ B).” The two sentences you put together need not be distinct. (They may be two occurrences of the same sentence.) So, can also build sentences like “(A & A)” . As with the “~” , the sentences you put together can be longer sentences that you have built previously. So, given the sentences, “A” and “~A” , you can proceed to build sentences like “(~A ⊃ A)” and “(A & ~A)” . And given the sentences, “(~A ⊃ A)” and “(A & ~A)” , you can proceed to build sentences like “~(A & ~A)” , “(~A ⊃ A) ≡ ~(A & ~A)” , and “(~((~A ⊃ A) ≡ ~(A & ~A)) v (~A ⊃ A))” . (These examples illustra
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