Our task is to give precise meanings to all of the well-formed formulas of SL. We will refer to these, quite sensibly, as “sentences of SL”. Some of this task is already complete. We know something about the meanings to the 26 capital letters: they stand for simple English sentences of our choosing. While the semantics for a natural language like English is complicated (What is the meaning of a sentence? Its truth-conditions? The proposition expressed? Are those two things the same? Is it something else entirely? Ugh.), the semantics for SL sentences is simple: all we care about is truth-value. A sentence in SL can have one of two semantic values: true or false. That’s it.
This is one of the ways in which the move to SL is a taming of natural language. In SL, every sentence has a determinate truth-value; and there are only two choices: true or false. English and other natural languages are more complicated than this. Of course, there’s the issue of non- declarative sentences, which don’t express propositions and don’t have truth-values at all. (Pausing briefly to note, once again, that this talk of sentences, rather than the propositions that they express, having truth-values is a bit fast and loose. Reaffirming our earlier stance on this: not a big deal.) But even if we restrict ourselves to declarative English sentences, things don’t look quite as simple as they are in SL. Consider the sentence ‘Napoleon was short’. You may not be aware that the popular conception of the French Emperor as diminutive in stature has its roots in British propaganda at the time. As a matter of fact, he was about 5’ 7”. Is that short? Well, not at the time (late 18th, early 19th centuries); people were shorter back then (nutrition wasn’t what it is these days, e.g.), and so Napoleon was about average or slightly above. People are taller now, though, so 5’ 7” might be considered short. At least, short for a man. A grown man, that is. I mean, a grown man who’s not a dwarf. Er, also, a grown non-dwarf man of French extraction (he’d be a tall man in Cambodia, for example, where the average height is only 5’ 4”). The average height for a modern Frenchman is 5’ 9.25”. Napoleon is 2.25 inches shorter than average. Is that short? Heck, I don’t know!
The problem here is that relative terms like ‘short’ have borderline cases; they’re vague. It’s not clear how to assign a truth-value to sentences like ‘Napoleon is short’. So, in English, we might say that they lack a truth-value (absent some explicit specification of the relevant standards). Logics that are more sophisticated than our SL have developed ways to deal with these sorts of cases. Instead of just two truth-values, some logics add more. There are three-values logics, where you have true, false, and neither. So we could say ‘Napoleon is short’ is neither. There are logics with infinitely many truth-values between true and false (where false is zero and true is 1, and every real number in between is a degree of truth); in such a system, we could assign, I don’t know, .62 to the proposition that Napoleon is short. The point is, English and other natural languages are messy when it comes to truth-value. We’re taming them in SL by assuming that every SL sentence has a determinate truth-value, and that there are only two truth-values: true and false—which we will indicate, by the way, with the letters ‘T’ and ‘F’.
Our task from here is to provide semantics for the five operators: tilde, dot, wedge, triple-bar, and horseshoe (we start with tilde because it’s the simplest, and we save horseshoe for last because it’s quite a bit more involved). We will specify the meanings of these symbols in terms of their effects on truth-value: what is the truth-value of a compound sentence featuring them as the main operator, given the truth-values of the components? The semantic values of the operators will be truth- functions: systematic accounts of the truth-value outputs (of the compound sentence) resulting from the possible truth-value inputs (of the simpler components).
Because tilde is a one-place operator, this is the simplest operator to deal with. The general form of a negation is ~ p, where ‘p’ is a variable standing for any generic SL sentence, simple or compound. As a lower-case letter, ‘p’ is not part of our language (SL); rather, it’s a tool we use to talk about our language—to refer to generic well-formed constructions within it.
We need to give an account of the meaning of the tilde in terms of its effect on truth-value. Tilde, as we said, is the SL equivalent of ‘not’ or ‘it is not the case that’. Let’s think about what happens in English when we use those terms. If we take a true sentence, say ‘Edison invented the light bulb’, and form a compound with it and ‘not’, we get ‘Edison did not invent the light bulb’—a falsehood. If we take a false sentence, like ‘James Brown is alive’, and negate it, we get ‘James Brown is not alive’—a truth.
Evidently, the effect of negation on truth-value is to turn a truth into a falsehood, and a falsehood into a truth. We can represent this graphically, using what we’ll call a “truth-table.” The following table gives a complete specification of the semantics of tilde:
In the left-hand column, we have ‘p’, which, as a variable, stands for a generic, unspecified SL sentence. Since it’s unspecified, we don’t know its truth-value; but since it’s a sentence in SL, we do know that there are only two possibilities for its truth-value: true or false (T or F). So in the first column, we list those two possibilities. In the second column, we have ‘~ p’, the negation of whatever ‘p’ is. We can compute the truth-value of the negation based on the truth-value of the sentence being negated: if the original sentence is true, then its negation is false; if the original sentence is false, then the negation is true. This is what we represent when we write ‘F’ and ‘T’ underneath the tilde (the operator that effects the change in truth-value) in the second column, in the same rows as their opposites.
Tilde is a truth-functional operator. Its meaning is specified by a function: if you input a T, the output is an F; if you input an F, the output is a T. The other four operators will also be defined in terms of the truth-function they represent. This is exactly analogous, again, to arithmetic. Addition, with its operator ‘+’, is a function on numbers. Input 1 and 3, and the output is 4. In SL, we only have two values—T and F—but it’s the same kind of thing. We could just as well use numbers to represent the truth-values: 0 for false and 1 for true, for example. In that case, tilde would be a function that outputs 0 when 1 is the input, and outputs 1 when 0 is the input.
Our rough-and-ready characterization of conjunctions was that they are ‘and’-sentences— sentences like ‘Beyoncé is logical and James Brown is alive’. Since these sorts of compound sentences involve two simpler components, we say that dot is a two-place operator. So when we specify the general form of a conjunction using generic variables, we need two of them. The general form of a conjunction in SL is p • q. The questions we need to answer are these: Under what circumstances is the entire conjunction true, and under what circumstances false? And how does this depend on the truth-values of the component parts?
We remarked earlier that when someone utters a conjunction, they’re committing themselves to both of the conjuncts. If I tell you that Beyoncé is wise and James Brown is alive, I’m committing myself to the truth of both of those alleged facts; I am, as it were, promising you that both of those things are true. So, if even one of them turns out false, I’ve broken my promise; the only way the promise is kept is if both of them turn out to be true.
This is how conjunctions work, then: they’re true just in case both conjuncts are true; false otherwise. We can represent this graphically, with a truth-table defining the dot:
Since the dot is a two-place operator, we need columns for each of the two variables in its general form—p and q. Each of these is a generic SL sentence that can be either true or false. That gives us four possibilities for their truth-values as a pair: both true, p true and q false, p false and q true, both false. These four possibilities give us the four rows of the table. For each of these possible inputs to the truth-function, we get an output, listed under the dot. T is the output when both inputs are Ts; F is the output in every other circumstance.
Our rough characterization of disjunctions was that they are ‘or’-sentences—sentences like ‘Beyoncé is logical or James Brown is alive’. In SL, the general form of a disjunction is p ∨ q. We need to figure out the circumstances in which such a compound is true; we need the truth-function represented by the wedge.
At this point we face a complication. Wedge is supposed to capture the essence of ‘or’ in English, but the word ‘or’ has two distinct senses. This is one of those cases where natural language needs to be tamed: our wedge can only have one meaning, so we need to choose between the two alternative senses of the English word ‘or’.
‘Or’ can be used exclusively or inclusively. The exclusive sense of ‘or’ is expressed in a sentence like this: ‘Clinton will win the election or Trump will win the election’. The two disjuncts present exclusive possibilities: one or the other will happen, but not both. The inclusive sense of ‘or’, however, allows the possibility of both. If I told you I was having trouble deciding what to order at a restaurant, and said, “I’ll order lobster or steak,” and then I ended up deciding to get the surf ‘n’ turf (lobster and steak combined in the same entrée), you wouldn’t say I had lied to you when I said I’d order lobster or steak. The inclusive sense of ‘or’ allows for one or the other—or both.
We will use the inclusive sense of ‘or’ for our wedge. There are arguments for choosing the inclusive sense over the exclusive one, but we will not dwell on those here. (As was the case when we had to make a choice about the word ‘some’ in Aristotelian logic, the argument makes the case that the inclusive sense is the core meaning of ‘or’, and the exclusive sense is a meaning that’s often, but not always, conveyed when we use ‘or’ in particular circumstances—an implicature. This line of reasoning has both adherents and detractors.) We need to choose a meaning for wedge, and we’re choosing the inclusive sense of ‘or’. As we will see later, the exclusive sense will not be lost to us because of this choice: we will be able to symbolize exclusive ‘or’ within SL, using a combination of operators.
So, wedge is inclusive ‘or’. It’s true whenever one or the other—or both—conjuncts is true; false otherwise. This is its truth-table definition:
As we said, biconditionals are, roughly, ‘if and only if’-sentences—sentences like ‘Beyoncé is logical if and only if James Brown is alive’. ‘If and only if’ is not a phrase most people use in everyday life, but the meaning is straightforward: it’s used to claim that both components have the same truth-value, that one entails the other and vice versa, that they can’t have different truth- values. In SL, the general form of a biconditional is p ≡ q. This is the truth-function:
The triple-bar is kind of like a logical equals-sign (it even resembles ‘=’): the function delivers an output of T when both components are the same, F when they’re not.
While the truth-functional meaning of triple-bar is now clear, it still may be the case that the intuitive meaning of the English phrase ‘if and only if’ remains elusive. This is natural. Fear not: we will have much more to say about that locution when we discuss translating between English and SL; a full understanding of biconditionals can only be achieved based on a full understanding of conditionals, to which, as the names suggest, they are closely related. We now turn to a specification of the truth-functional meaning of the latter.
Our rough characterization of conditionals was that they are ‘if/then’ sentences—sentences like ‘If Beyoncé is logical, then James Brown is alive’. We use such sentences all the time in everyday speech, but is surprisingly difficult to pin down the precise meaning of the conditional, especially within the constraints imposed by SL. There are in fact many competing accounts of the conditional—many different conditionals to choose from—in a literature dating back all the way to the Stoics of ancient Greece. Whole books can be—and have been—written on the topic of conditionals. In the course of our discussion of the semantics for horseshoe, we will get a sense of why this is such a vexed topic; it’s complicated.
The general form of a conditional in SL is p ⊃ q. We need to decide for which values of p and q the conditional turns out true and false. To help us along (by making things more vivid), we’ll consider an actual conditional claim, with a little story to go along with it. Suppose Barb is suffering from joint pain; maybe it’s gout, maybe it’s arthritis—she doesn’t know and hasn’t been to the doctor to find out. She’s complaining about her pain to her neighbor, Sally. Sally is a big believer in “alternative medicine” and “holistic healing”. After hearing a brief description of the symptoms, Sally is ready with a prescription, which she delivers to Barb in the form of a conditional claim: “If you drink this herbal tea every day for a week, then your pain will go away.” She hands over a packet of tea leaves and instructs Barb in their proper preparation.
We want to evaluate Sally’s conditional claim—that if Barb drinks the herbal tea daily for a week, then her pain will go away—for truth/falsity. To do so, we will consider various scenarios, the details of which will bear on that evaluation.
Scenario #1: Barb does in fact drink the tea every day for a week as prescribed, and, after doing so, lo and behold, her pain is gone. Sally was right! In this scenario, we would say that the conditional we’re evaluating is true.
Scenario #2: Barb does as Sally said and drinks the tea every day for a week, but, after the week is finished, the pain remains, the same as ever. In this scenario, we would say that Sally was wrong: her conditional advice was false.
Perhaps you can see what I’m doing here. Each of the scenarios represents one of the rows in the truth-table definition for the horseshoe. Sally’s conditional claim has an antecedent—Barb drinks the tea every day for a week—and a consequent—Barb’s pain goes away. These are p and q, respectively, in the conditional p ⊃ q. In scenario #1, both p and q were true: Barb did drink the tea, and the pain did go away; in scenario #2, p was true (Barb drank the tea) but q was false (the pain didn’t go away). These two scenarios are the first two rows of the four-row truth tables we’ve already seen for dot, wedge, and triple-bar. For horseshoe, the truth-function gives us T in the first row and F in the second:
All that’s left is to figure out what happens in the third and fourth rows of the table, where the antecedent (p, Barb drinks the tea) is false both times and the consequent is first true (in row 3) and then false (in row 4). There are two more scenarios to consider.
In scenario #3, Barb decides Sally is a bit of a nut, or she drinks the tea once and it tastes awful so she decides to stop drinking it—whatever the circumstances, Barb doesn’t drink the tea for a week; the antecedent is false. But in this scenario, it turns out that after the week is up, Barb’s pain has gone away; the consequent is true. What do we say about Sally’s advice—if you drink the tea, the pain will go away—in this set of circumstances?
In scenario #4, again Barb does not drink the tea (false antecedent), and after the week is up, the pain remains (false consequent). What do we say about the Sally’s conditional advice in this scenario?
It’s tempting to say that in the 3rd and 4th scenarios, since Barb didn’t even try Sally’s remedy, we’re not in a position to evaluate Sally’s advice for truth or falsity. The hypothesis wasn’t even tested. So, we’re inclined to say ‘If you drink the tea, then the pain will go away’ is neither true nor false. But while this might be a viable option in English, it won’t work in SL. We’ve made the simplifying assumptions that every SL sentence must have a truth-value, and that that the only two possibilities are true and false. We can’t say it has no truth-value; we can’t add a third value and call it “neither”. We have to put a T or an F under the horseshoe in the third and fourth rows of the truth table for that operator. Given this restriction, and given that we’ve already decided how the first two rows should work out, there are four possible ways of specifying the truth-function for horseshoe:
These are our only options (remember, the top two rows are settled; scenarios 1 and 2 above had clear results). Which one captures the meaning of the conditional best?
Option 1 is tempting: as we noted, in rows 3 and 4, Sally’s hypothesis isn’t even tested. If we’re forced to choose between true and false, we might as well go with false. The problem with this option is that this truth-function—true when both components are true; false otherwise—is already taken. That’s the meaning of dot. If we choose option 1, we make horseshoe and dot mean the same thing. That won’t do: they’re different operators; they should have different meanings. ‘And’ and ‘if/then’ don’t mean the same thing in English, clearly.
Option 2 also has its charms. OK, we might say, in neither situation is Sally’s hypothesis tested, but at least row 3 has something going for it, Sally-wise: the pain does go away. So let’s say her conditional is true in that case, but false in row 4 when there still is pain. Again, this won’t do. Compare the column under option 2 to the column under q. They’re the same: T, F, T, F. That means the entire conditional, p ⊃ q, has the same meaning as its consequent, plain old q. Not good. The antecedent, p, makes no difference to the truth-value of the conditional in this case. But it should; we shouldn’t be able to compute the truth-value of a two-place function without even looking at one of the inputs.
Option 3 is next. Some people find it reasonable to say that the conditional is false in row 3: there’s something about the disappearance of the pain, despite not drinking the tea, that’s incompatible with Sally’s prediction. And if we can’t put an F in the last row too (this is just option 1 again), then make it a T. But this fails for the same reason option 1 did: the truth-function is already taken, this time by the triple-bar. ‘If and only if’ is a much stronger claim than the mere ‘if/then’;biconditionals must have a different meaning from mere conditionals.
That leaves option 4. This is the one we’ll adopt, not least because it’s the only possibility left. The conditional is true when both antecedent and consequent are true—scenario 1; it’s false when the antecedent is true but the consequent false—scenario 2; and it’s true whenever the antecedent is false—scenarios 3 and 4. This is the definition of horseshoe:
It’s not ideal. The first two rows are quite plausible, but there’s something profoundly weird about saying that the sentence ‘If you drink the tea, then the pain will go away’ is true whenever the tea is not drunk. Yet that is our only option. We can perhaps make it a bit more palatable by saying— as we did about universal categorical propositions with empty subject classes—that while it’s true in such cases, it’s only true vacuously or trivially—true in a way that doesn’t tell you about how things are in the world.
What can also help a little is to point out that while rows 3 and 4 don’t make much sense for the Barb/Sally case, they do work for other conditionals. The horror author Stephen King lives in Maine (half his books are set there, it seems). Consider this conditional: ‘If Stephen King is the Governor of Maine, then he lives in Maine’. While a prominent citizen, King is not Maine’s governor, so the antecedent is false. He is, though, as we’ve noted, a resident of Maine, so the consequent is true. We’re in row 3 of the truth-table for conditionals here. And intuitively, the conditional is true: he’s not the governor, but if he were, he would live in Maine (governors reside in their states’ capitals). And consider this conditional: ‘If Stephen King is president of the United States, then he lives in Washington, DC’. Now both the antecedent (King is president) and the consequent (he lives in DC) are false: we’re in row 4 of the table. But yet again, the conditional claim is intuitively true: if he were president, he would live in DC.
Notice the trick I pulled there: I switched from the so-called indicative mood (if he is) to the subjunctive (if he were). The truth of the conditional is clearer in the latter mood than the former. But this trick won’t always work to make the conditional come out true in the third and fourth rows. Consider: ‘If Stephen King were president of the United States, then he would live in Maine’ and ‘If Stephen King were Governor of Maine, then he would live in Washington, DC’. These are third and fourth row examples, respectively, but neither is intuitively true.
By now perhaps you are getting a sense of why conditionals are such a vexed topic in the history of logic. A variety of approaches, with attendant alternative logical formalisms, have been developed over the centuries (and especially in the last century) to deal with the various problems that arise in connection with conditional claims. Ours is the very simplest approach, the one with which to begin. As this is an introductory text, this is appropriate. You can investigate alternative accounts of the conditional if you extend your study of logic further.
Computing Truth-Values of Compound SL Sentences
With the truth-functional definitions of the five SL operators in hand, we can develop a preliminary skill that will be necessary to deploy when the time comes to test SL arguments for validity. We need to be able to compute the truth-values of compound SL sentences, given the truth-values of their simplest parts (the simple sentences—capital letters). To do so, we must first determine what type of compound sentence we’re dealing with—negation, conjunction, disjunction, conditional, or biconditional. This involves deciding which of the operators in the SL sentence is the main operator. We then compute the truth-value of the compound according to the definition for the appropriate operator, using the truth-values of the simpler components. If these components are themselves compound, we determine their main operators and compute accordingly, in terms of their simpler components—repeating as necessary until we get down to the simplest components of all, the capital letters. A few examples will make the process clear.
Let’s suppose that A and B are true SL sentences. Consider this compound:
~ A ∨ B
What is its truth-value? To answer that question, we first have to figure out what kind of compound sentence we’re dealing with. It has two operators—the tilde and the wedge. Which of these is the main operator; that is, do we have a negation or a disjunction? We answered this question earlier, when we were discussing the syntax of SL. Our convention with tildes is that they negate the first well-formed construction immediately to their right. In this case, ‘A’ is the first well-formed construction immediately to the right of the tilde, so the tilde negates it. That means wedge is the main operator; this is a disjunction, where the left-hand disjunct is ~ A and the right-hand disjunct is B. To compute the truth-value of the disjunction, we need to know the truth-values of its disjuncts. We know that B is true; we need to know the truth-value of ~ A. That’s easy, since A is true, ~ A must be false. It’s helpful to keep track of one’s step-by-step computations like so:
I’ve marked the truth-values of the simplest components, A and B, on top of those letters. Then, under the tilde, the operator that makes it happen, I write ‘F’ to indicate that the left-hand disjunct, ~ A, is false. Now I can compute the truth-value of the disjunction: the left-hand disjunct is false, but the right hand disjunct is true; this is row 3 of the wedge truth-table, and the disjunction turns out true in that case. I indicate this with a ‘T’ under the wedge, which I highlight (with boldface and underlining) to emphasize the fact that this is the truth-value of the whole compound sentence:
When we were discussing syntax, we claimed that adding parentheses to a compound like the last one would alter its meaning. We’re now in a position to prove that claim. Consider this SL sentence (where A and B are again assumed to be true):
Now the main operator is the tilde: it negates the entire disjunction inside the parentheses. To discover the effect of that negation on truth-value, we need to compute the truth-value of the disjunction that it negates. Both A and B are true; this is the top row of the wedge truth-table— disjunctions turn out true in such cases:
So the tilde is negating a truth, giving us a falsehood:
The truth-value of the whole is false; the similar-looking disjunction without the parentheses was true. These two SL sentences must have different meanings; they have different truth-values.
It will perhaps be useful to look at one more example, this time of a more complex SL sentence. Suppose again that A and B are true SL simple sentences, and that X and Y are false SL simple sentences. Let’s compute the truth-value of the following compound sentence:
~ (A • X) ⊃ (B ∨ ~ Y)
As a first step, it’s useful to mark the truth-values of the simple sentences:
Now, we need to figure out what kind of compound sentence this is; what is the main operator? This sentence is a conditional; the main operator is the horseshoe. The tilde at the far left negates the first well-formed construction immediately to its right. In this case, that is (A • X). ~ (A • X) is the antecedent of this conditional; (B ∨ ~ Y) is the consequent. We need to compute the truth- values of each of these before we can compute the truth-value of the whole compound.
Let’s take the antecedent, ~ (A • X) first. The tilde negates the conjunction, so before we can know what the tilde does, we need to know the truth-value of the conjunction inside the parentheses. Conjunctions are true just in case both conjuncts are true; in this case, A is true but X is false, so the conjunction is false, and its negation must be true:
So the antecedent of our conditional is true. Let’s look at the consequent, (B ∨ ~ Y). Y is false, so ~ Y must be true. That means both disjuncts, B and ~ Y are true, making our disjunction true:
Both the antecedent and consequent of the conditional are true, so the whole conditional is true:
One final note: sometimes you only need partial information to make a judgment about the truth- value of a compound sentence. Look again at the truth table definitions of the two-place operators:
For three of these operators—the dot, wedge, and horseshoe—one of the rows is not like the others. For the dot: it only comes out true when both p and q are true, in the top row. For the wedge: it only comes out false when both p and q are false, in the bottom row. For the horseshoe: it only comes out false when p is true and q is false, in the second row.
Noticing this allows us, in some cases, to compute truth-values of compounds without knowing the truth-values of both components. Suppose again that A is true and X is false; and let Q be a simple SL sentence the truth-value of which is a mystery to you (it has one, like all of them must; I’m just not telling you what it is). Consider this compound:
A ∨ Q
We know one of the disjuncts is true; we don’t know the truth-value of the other one. But we don’t need to! A disjunction is only false when both of its disjuncts are false; it’s true when even one of its disjuncts is true. A being true is enough to tell us the disjunction is true; Q doesn’t matter.
Consider the conjunction:
X • Q
We only know the truth-value of one of the conjuncts: X is false. That’s all we need to know to compute the truth-value of the conjunction. Conjunctions are only true when both of their conjuncts are true; they’re false when even one of them is false. X being false is enough to tell us that this conjunction is false.
Finally, consider these conditionals:
Q ⊃ A and X ⊃ Q
They are both true. Conditionals are only false when the antecedent is true and the consequent is false; so they’re true whenever the consequent is true (as is the case in Q ⊃ A) and whenever the antecedent is false (as is the case in X ⊃ Q).
Compute the truth-values of the following compound sentences, where A, B, and C are true; X, Y, and Z are false; and P and Q are of unknown truth-value.