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4.2: Syntax of Sentential Logic

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    24335
  • 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.

    We can distinguish, in English, between two types of (declarative) sentences: simple and compound. A simple sentence is one that does not contain any other sentence as a component part. A compound sentence is one that contains at least one other sentence as a component part. (We will not give a rigorous definition of what it is for one sentence to be a component part of another sentence. Rather, we will try to establish an intuitive grasp of the relation by giving examples, and stipulate that a rigorous definition could be provided, but is too much trouble to bother with.) ‘Beyoncé is logical’ is a simple sentence; none of its parts is itself a sentence. (You might think ‘Beyoncé is’ is a part of the sentence that qualifies as a sentence itself—a sentence claiming that she exists, maybe. But that won’t do. The word ‘is’ in the original sentence is the “‘is’ of predication”—a mere linking verb; ‘Beyoncé is’ only counts as a sentence if you change the meaning of ‘is’ to the “‘is’ of existence”. Anyway, stop causing trouble. This is why we didn’t give a rigorous definition of ‘component part’; we’d get bogged down in these sorts of arcane distinctions.) ‘Beyoncé is logical and James Brown is alive’ is a compound sentence: it contains two simple sentences as component parts—namely, ‘Beyoncé is logical’ and ‘James Brown is alive’.

    In SL, we will use capital letters—‘A’, ‘B’, ‘C’, ..., ‘Z’—to stand for simple sentences. Our practice will be simply to choose capital letters for simple sentences that are easy to remember. For example, we can choose ‘B’ to stand for ‘Beyoncé is logical’ and ‘J’ to stand for ‘James Brown is alive’. Easy enough. The hard part is symbolizing compound sentences in SL. How would we handle ‘Beyoncé is logical and James Brown is alive’, for example? Well, we’ve got capital letters to stand for the simple parts of the sentence, but that leaves out the word ‘and’. We need more symbols.

    We will distinguish five different kinds of compound sentence, and introduce a special SL symbol for each. Again, at this stage we are only discussing the syntax of SL—the rules for combining its symbols into well-formed constructions. We will have some hints about the semantics of these new symbols—hints about their meanings—but a full treatment of that topic will not come until the next section.

    Conjuctions

    The first type of compound sentence is one that we’ve already seen. Conjunctions are, roughly, ‘and’-sentences—sentences like ‘Beyoncé is logical and James Brown is alive’. We’ve already decided to let ‘B’ stand for ‘Beyoncé is logical’ and to let ‘J’ stand for ‘James Brown is alive’. What we need is a symbol that stands for ‘and’. In SL, that symbol is a “dot”. It looks like this: •.

    To form a conjunction in SL, we simply stick the dot between the two component letters, thus:

    B • J

    That is the SL version of ‘Beyoncé is logical and James Brown is alive’.

    A note on terminology. A conjunction has two components, one on either side of the dot. We will refer to these as the “conjuncts” of the conjunction. If we need to be specific, we might refer to the “left-hand conjunct” (‘B’ in this case) or the “right-hand conjunct” (‘J’ in this case).

    Disjunction

    Disjunctions are, roughly, ‘or’-sentences—sentences like ‘Beyoncé is logical or James Brown is alive’. Sometimes, the ‘or’ is accompanied by the word ‘either’, as in ‘Either Beyoncé is logical or James Brown is alive’. Again, we let ‘B’ stand for ‘Beyoncé is logical’ and let ‘J’ stand for ‘James Brown is alive’. What we need is a symbol that stands for ‘or’ (or ‘either/or’). In SL, that symbol is a “wedge”. It looks like this: ∨.

    To form a conjunction in SL, we simply stick the wedge between the two component letters, thus:

    B ∨ J

    That is the SL version of ‘Beyoncé is logical or James Brown is alive’.

    A note on terminology. A disjunction has two components, one on either side of the wedge. We will refer to these as the “disjuncts” of the disjunction. If we need to be specific, we might refer to the “left-hand disjunct” (‘B’ in this case) or the “right-hand disjunct” (‘J’ in this case).

    Negations

    Negations are, roughly, ‘not’-sentences—sentences like ‘James Brown is not alive’. You may find it surprising that this would be considered a compound sentence. It is not immediately clear how any component part of this sentence is itself a sentence. Indeed, if the definition of ‘component part’ (which we intentionally have not provided) demanded that parts of sentences contain only contiguous words (words next to each other), you couldn’t come up with a part of ‘James Brown is not alive’ that is itself a sentence. But that is not a condition on ‘component part’. In fact, this sentence does contain another sentence as a component part—namely, ‘James Brown is alive’. This can be made more clear if we paraphrase the original sentence. ‘James Brown is not alive’ means the same thing as ‘It is not the case that James Brown is alive’. Now we have all the words in ‘James Brown is alive’ next to each other; it is clearly a component part of the larger, compound sentence. We have ‘J’ to stand for the simple component; we need a symbol for ‘it is not the case that’. In SL, that symbol is a “tilde”. It looks like this: ~.

    To form a negation in SL, we simply prefix a tilde to the simpler component being negated:

    ~J

    This is the SL version of ‘James Brown is not alive’.

    Conditionals

    Conditionals are, roughly, ‘if/then’ sentences—sentences like ‘If Beyoncé is logical, then James Brown is alive’. (James Brown is actually dead. But suppose Beyoncé is a “James Brown-truther”, a thing that I just made up. She claims that James Brown faked his death, that the Godfather of Soul is still alive, getting funky in some secret location. (Play along.) In that case, the conditional sentence might make sense.) Again, we let ‘B’ stand for ‘Beyoncé is logical’ and let ‘J’ stand for ‘James Brown is alive’. What we need is a symbol that stands for the ‘if/then’ part. In SL, that symbol is a “horseshoe”. It looks like this: ⊃ .

    To form a conditional in SL, we simply stick the horseshoe between the two component letters (where the word ‘then’ occurs), thus:

    B ⊃ J

    That is the SL version of ‘If Beyoncé is logical, then James Brown is alive’.

    A note on terminology. Unlike our treatment of conjunctions and disjunctions, we will distinguish between the two components of the conditional. The component to the left of the horseshoe will be called the “antecedent” of the conditional; the component after the horseshoe is its “consequent”. As we will see when we get to the semantics for SL, there is a good reason for distinguishing the two components.

    Biconditional

    Biconditionals are, roughly, ‘if and only if’-sentences—sentences like ‘Beyoncé is logical if and only if James Brown is alive’. (This is perhaps not a familiar locution. We will talk more about what it means when we discuss semantics.) Again, we let ‘B’ stand for ‘Beyoncé is logical’ and let ‘J’ stand for ‘James Brown is alive’. What we need is a symbol that stands for the ‘if and only if’ part. In SL, that symbol is a “triple-bar”. It looks like this: ≡ .

    To form a biconditional in SL, we simply stick the triple-bar between the two component letters, thus:

    B ≡ J

    That is the SL version of ‘Beyoncé is logical if and only if James Brown is alive’.

    There are no special names for the components of the biconditional.

    Punctuation - Parentheses

    Our language, SL, is quite austere: so far, we have only 31 different symbols—the 26 capital letters, and the five symbols for the five different types of compound sentence. We will now add two more: the left- and right-hand parentheses. And that’ll be it.

    We use parentheses in SL for one reason (and one reason only): to remove ambiguity. To see how this works, it will be helpful to draw an analogy between SL and the language of simple arithmetic. The latter has a limited number of symbols as well: numbers, signs for the arithmetical operations (addition, subtraction, multiplication, division), and parentheses. The parentheses are used in arithmetic for disambiguation. Consider this combination of symbols:

    2 +3 x 5

    As it stands, this formula is ambiguous. I don’t know whether this is a sum or a product; that is, I don’t know which operator—the addition sign or the multiplication sign—is the main operator. (You may have learned an “order of operations” in grade school, according to which multiplication takes precedence over addition, so that there would be no ambiguity in this expression. But the order of operations is just a (mostly arbitrary) way of removing ambiguity that would be there without it. The point is, absent some sort of disambiguating convention—whether it’s parentheses or an order of operations—the meanings of expressions like this are indeterminate.) We can use parentheses to disambiguate, and we can do so in two different ways:

    (2 + 3) x 5
    or
    2 + (3 x 5)

    And of course, where we put the parentheses makes a big difference. The first formula is a product; the multiplication sign is the main operator. It comes out to 25. The second formula is a sum; the addition sign is the main operator. And it comes out to 17. Different placement of parentheses, different results.

    This same sort of thing is going to arise in SL. We use the same term we use to refer to the addition and multiplication signs—‘operator’—to refer to dot, wedge, tilde, horseshoe, and triple-bar. (As we will see when we look at the semantics for SL, this is entirely proper, since the SL operators stand for mathematical functions on truth-values.) There are ways of combining SL symbols into compound formulas with more than one operator; and just as is the case in arithmetic, without parentheses, these formulas would be ambiguous. Let’s look at an example.

    Consider this sentence: ‘If Beyoncé is logical and James Brown is alive, then I’m the Queen of England’. This is a compound sentence, but it contains both the word ‘and’ and the ‘if/then’ construction. And it has three simple components: the two that we’re used to by now about Beyoncé and James Brown, which we’ve been symbolizing with ‘B’ and ‘J’, respectively, and a new one—‘I’m the Queen of England’—which we may as well symbolize with a ‘Q’. Based on what we already know about how SL symbols work, we would render the sentence like this:

    B • J ⊃ Q

    But just as was the case with the arithmetical example above, this formula is ambiguous. I don’t know what kind of compound sentence this is—a conjunction or a conditional. That is, I don’t know which of the two operators—the dot or the horseshoe—is the main operator. In order to disambiguate, we need to add some parentheses. There are two ways this can go, and we need to decide which of the two options correctly captures the meaning of the original sentence:

    (B • J) ⊃ Q
    or
    B • (J ⊃ Q)

    The first formula is a conditional; horseshoe is its main operator, and its antecedent is a compound sentence (the conjunction ‘B • J’). The second formula is a conjunction; dot is its main operator, and its right-hand conjunct is a compound sentence (the conditional ‘J  Q’). We need to decide which of these two formulations correctly captures the meaning of the English sentence ‘If Beyoncé is logical and James Brown is alive, then I’m the Queen of England’.

    The question is, what kind of compound sentence is the original? Is it a conditional or a conjunction? It is not a conjunction. Conjunctions are, roughly (again, we’re not really doing semantics yet), ‘and’-sentences. When you utter a conjunction, you’re committing yourself to both of the conjuncts. If I say, “Beyoncé is logical and James Brown is alive,” I’m telling you that both of those things are true. If we construe the present sentence as a conjunction, properly symbolized as ‘B • (J ⊃ Q)’, then we take it that the person uttering the sentence is committed to both conjuncts; she’s telling us that two things are true: (1) Beyoncé is logical and (2) if James Brown is alive then she’s the Queen of England. So, if we take this to be a conjunction, we’re interpreting the speaker as committed to the proposition that Beyoncé is logical. But clearly she’s not. She uttered ‘If Beyoncé is logical and James Brown is alive, then I’m the Queen of England’ to express dubiousness about Beyoncé’s logicality (and James Brown’s status among the living). This sentence is not a conjunction; it is a conditional. It’s saying that if those two things are true (about Beyoncé and James Brown), then I’m the Queen of England. The utterer doubts both conjuncts in the antecedent. The proper symbolization of this sentence is the first one above: ‘(B • J) ⊃ Q’.

    Again, in SL, parentheses have one purpose: to remove ambiguity. We only use them for that. This kind of ambiguity arises in formulas, like the one just discussed, involving multiple instances of the operators dot, wedge, horseshoe, and triple-bar.

    Notice that I didn’t mention the tilde there. Tilde is different from the other four. Dot, wedge, horseshoe, and triple-bar are what we might call “two-place operators”. There are two simpler components in conjunctions, disjunctions, conditionals, and biconditionals. Negations, on the other hand, have only one simpler component; hence, we might call tilde a “one-place operator”. It only operates on one thing: the sentence it negates.

    This distinction is relevant to our discussion of parentheses and ambiguity. We will adopt a convention according to which the tilde negates the first well-formed SL construction immediately to its right. This convention will have the effect of removing potential ambiguity without the need for parentheses. Consider the following combination of SL symbols:

    ~ A ∨ B

    It may appear that this formula is ambiguous, with the following two possible ways of disambiguating:

    ~ (A ∨ B)
    or
    (~ A) ∨ B

    But this is not the case. Given our convention—tilde negates the first well-formed SL construction immediately to its right—the original formula—‘~ A ∨ B’—is not ambiguous; it is well-formed. Since ‘A’ is itself a well-formed SL construction (of the simplest kind), the tilde in ‘~ A ∨ B’ negates the ‘A’ only. This means that we don’t have to indicate this fact with parentheses, as in the second of the two potential disambiguations above. That kind of formula, with parentheses around a tilde and the item it negates, is not a well-formed construction in SL. Given our convention about tildes, the parentheses around ‘~ A’ are redundant.

    The first potential disambiguation—‘~ (A ∨ B)’—is well-formed, and it means something different from ‘~ A ∨ B’. In the former, the tilde negates the entire disjunction, ‘A ∨ B’; in the latter, it only negates ‘A’. That makes a difference. Again, an analogy to arithmetic is helpful here. Compare the following two formulas:

    - (2 + 5)
    vs.
    -2 + 5

    In the first, the minus-sign covers the entire sum, and so the result is -7; in the second, it only covers the 2, so the result is 3. This is exactly analogous to the difference between ‘~ (A ∨ B)’ and ‘~ A ∨ B’. The tilde has wider scope in the first formula, and that makes a difference. The difference can only be explained in terms of meaning—which means it is time to turn our attention to the semantics of SL.

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