Module 4: Sentential Logic
4.2 Syntax of SL
The syntax of a language comprises the rules governing what counts as a well-formed construction within that language; that is, syntax is the language’s grammar. Syntax is what tells me that ‘What a handsome poodle you have there.’ is a well-formed English construction, while ‘Poodle a handsome there you what have.’ is not. This treatment of syntax will give us some clues about the relationship between SL and English, but a full accounting of that relationship awaits us in the next section on semantics.
4.2.1 An Overview of “Sentences”
We can distinguish, in English, between these 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.
Note that 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. ‘Beyoncé is logical’ is a simple sentence; none of its parts is itself a sentence. ’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 to choose a capital letters for any simple sentence that is 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 more intricate 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 pieces of the compound sentence, but that does not address the word ‘and’. We need more symbols!
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4.2.2 Compound Sentence Types and Their 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 occur in the next section (4.3).
Conjunctions
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 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).
Disjunctions
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 the “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 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 the “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’ (or ‘It is not the case that James Brown is 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”, an imagined thing — play along. She claims that James Brown faked his death, that the Godfather of Soul is still alive, getting funky in some secret location. 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’.
Unlike our treatment of conjunctions and disjunctions where a distinction is possible but not necessary, we must 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.
Biconditionals
Biconditionals are, roughly, ‘if and only if’ sentences—sentences like ‘Beyoncé is logical if and only if James Brown is alive’. We will talk more about what ‘if and only if” 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.
4.2.3 Parentheses as Punctuation
Our language, SL, is quite basic: so far, we have only 31 different symbols—the 26 capital letters, and the five symbols for the five different types of compound sentences. We will now add two more: the left-hand and right-hand parentheses. And that will 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 to remove ambiguity. Consider this combination of symbols:
2 + 3 x 5
As it stands, this formula is ambiguous. I don’t know whether the main operation 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.
Note: 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 a mostly arbitrary way of removing the 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 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 produces different results.
This same sort of thing is going to arise in SL. We will use the same term—‘operator’—we use for arithmetic’s addition and multiplication signs to refer to our five SL symbols: dot, wedge, tilde, horseshoe, and triple-bar. 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 will 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 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 they are 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, they are not. They 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 use them only for that. This kind of ambiguity arises in expressions, like the one just discussed, involving multiple instances of these four operators: dot, wedge, horseshoe, and triple-bar.
Notice that the tilde is not included in that list of operations. 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 can 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 to disambiguate this:
~ (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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