Formal Semantics: Conditionals
We have come to recognize that conditional sentences of English provide another example of logical form. And since we believe that logical form is what makes an argument valid or invalid, we also recognized that the presence of conditional phrases in an argument will affect the validity of that argument. We’ve adapted our logical grammar to include a symbol for conditional phrases, so these phrases have a way of showing up in our symbolic translations of English sentences. And now, once we have the hang of getting the form of arguments with conditionals, we want to go ahead and test that form. We now want to test English arguments using conditional phrases, to see if such arguments are valid.
In order to use a semantic test on an argument form – for example, the truth table or truth tree test – we need semantic rules for each sort of sentence in our language of form. So now that we’ve added a new sort of sentence to our logical language – conditional sentences – we’ll need a semantic rule for that type of sentence. Having expanded our logical syntax (grammar) to include conditionals, and recalling that every syntactic rule has been matched by a semantic rule, we realize that we need to add a semantic rule for conditional sentences.
As we’ve seen in the previous chapter, each semantic rule will perfectly match the corresponding syntactic rule. (That’s why a grammatical tree is such a good guide when we’re building the truth table for a sentence: the truth table will follow the grammatical structure of that sentence, move for move.) And conditionals will be no exception: our semantic rule for conditionals will simple mirror the syntactic (grammatical) rule.
The new grammatical rule for conditionals was nothing special.
This rule
is basically like the grammatical rules for conjunctions or disjunctions:
two smaller sentences are the inputs; and the output is these smaller sentences
glued together by a connective, and that whole thing wrapped in parentheses.
So our semantic rule for conditionals should exhibit the same steps: take in two smaller sentences as inputs (the antecedent and the consequent), and give, as output, the whole conditional sentence.
And then we need to run through all the possibilities for the two little sentences, and decide what value the whole conditional will have in each possible situation.
Since we have two different little sentences here, and (according to Bivalence) each one can be either true or false, we know we’ll need 4 valuations to go through all the possibilities.
Now, how will we decide what truth value the whole conditional sentence has, in each of these situations? Well, remember that the arrow is intended as our formal counterpart to English language conditional phrases like “if… then,” “if,” and their relatives. So we expect the arrow to faithfully mirror the behavior of these English language conditional phrases. And since, as native speakers of English, we’ve been using these conditional phrases most of our lives, we’re the world's foremost authority on how English language conditional phrases behave. We just need to think of concrete examples of English conditionals, and when they’d be true, when false.
So consider a concrete example of an English conditional.
That’s the kind of conditional we might really utter in certain situations. For example, if our friends were visiting from out of town, and were leaving to drive back home today, we might wonder if they got stuck in rush-hour traffic as they drove by some big city. And I might say: “If they left before noon, then they avoided the rush hour traffic.” Maybe you’d disagree, and say that’s not the case at all. We might even bet some money on who’s right about this. So this evening, after Logic class, we’ll settle the bet: we’ll call our friends, and see if my conditional was true or not.
Remember that this conditional (like every conditional) has two parts: the antecedent and the consequent. In this example, “They left before noon” is the antecedent, and “They avoided the rush-hour traffic” is the consequent.
To translate this conditional into formal language, we assign a sentence letter to each of these subject matter sentences.
So the English language form of this sentence is:
And of course the whole conditional will be translated as
The first valuation is simple enough: this is a situation where it’s true that they left before noon, and it’s true that they avoided the rush-hour traffic.
And I said: “If they left before noon, then they avoided the rush-hour traffic.” Clearly, in this situation, what I said was true.
Now, for convenience, I want to jump down to the third valuation, and get that one out of the way as well. In the third valuation, the antecedent is true, but the consequent is false. In this situation, it’s true that they left before noon, but it’s false that they avoided the rush-hour traffic.
The conditional claim I made was: “If they left before noon, then they avoided the rush-hour traffic.” Now, if it turns (like in 3) that they did leave before noon, but still didn’t avoid the rush-hour traffic, then clearly my conditional claim was false.
Those are the two easy valuations. But now consider valuation 2: here, it’s false that they left before noon (but still, luckily, true that they avoided the rush-hour traffic).
What do we say about the whole conditional in this case? If we bet money on whether the conditional was true or false, who wins the bet? The bet was about what would happen if they left before noon. But valuation 2 is a situation where they didn’t leave before noon. It looks like that’s just not the sort of situation we were betting on. If we ended up in a situation like valuation 2 – if in fact they didn’t leave before noon – then we’d probably say that the bet was off; nobody won the bet and nobody lost it.
And the same is true for valuation 4: here again, it’s false that they left before noon. So here again, we’d probably say that the bet is off over whether the conditional is true or false.
So shall we just leave a blank for the whole conditional in these cases, stating that the conditional is neither true nor false?
No, we can’t do that – because that would be a violation of Bivalence. The Principle of Bivalence says that in any possible situation, this sentence is either true in that situation, or false; the sentence can’t be both true and false, and it can’t be neither true nor false.
Bivalence demands that the conditional either be true in 2, or false; and likewise with 4. So which will it be?
Think about it like this: if we did bet on my conditional, and it turned out they hadn’t left before noon at all, we wouldn’t say that such a situation made my conditional false. So if we have to choose – and Bivalence says we do – it’s at least a little less offensive to say that the whole conditional is true when the antecedent is false. So let’s say that in valuations 2 and 4, the whole conditional is false (in a kind of don’t-really-care way).
And with this example finished, we end up with a quite simple semantic rule for conditionals: a conditional will only be false when the antecedent is true, and the consequent is false.
Now, before we go on to start applying this semantic rule to conditionals, I want to answer some lingering doubts you may have about this rule. Maybe you aren’t convinced that the conditional really should be true when the antecedent is false (in valuations 2 and 4). Maybe you think that my ‘argument’ for that was pretty weak, and that I’m wrong about it being true in those cases.
But, I answer, if the conditional isn’t true in valuations 2 and 4, then what will it be? The Principle of Bivalence only leaves one option: it will have to be false in 2 and 4.
OK, you say, bravely: let it be so. Let’s have the conditional be false in valuations 2 and 4. I’ll bet that’s still better than your lousy true-in-2-and-4 rule, Beakley!
But now I’ll convince you that this alternative rule for the conditional will be a disaster: it will make claims and predictions about truth and meaning that, you’ll have to admit, are obviously false. (So really, I’m going to make an indirect argument that this alternate rule is wrong: I’ll say, “Suppose, for the sake of argument, that we do use this rule”; and then I’ll show how that will blow up in your face.)
Here’s my argument: compare this alternative rule for the conditional with our semantic rule for conjunctions.
Notice that, if we accept this alternative rule for conditionals, we’ll end up saying that conditionals and conjunctions are true in all the same situations, and false in all the same situations. (In technical terms: we’ll end up saying that the two sentences are truth functionally equivalent.)
But since you’re a speaker of English, and an expert on when simple “if” and “and” sentences are true, ask yourself this: do you really believe that conditionals and conjunctions are true in all the same situations? For example, consider these two sentences.
The two sentences both use the exact same subject matter sentences as parts (“I won the lottery” and “I’m a millionaire”). The only difference is that the first sentence is a conditional, while the second is a conjunction.
But we don’t have to stretch our imaginations very much to see that these two sentences are not true in all the same situations. The actual situation we’re sitting in right now is a situation where, yes, if I did win the lottery, then I would indeed be a millionaire. In the actual world, that conditional is true. But in the actual world, it’s definitely false to say that I won the lottery and I’m a millionaire. (Proof: if I were a millionaire, I would have quit my teaching job, and I wouldn’t be spending my time writing these notes on Logic.)
So in the actual world, the conditional is true, but the conjunction is false. And that’s solid evidence that conditionals and conjunctions are not true and false in all the same situations. So the radical alternative rule for conditional is not making correct predictions about when conditionals are true, and when they’re false. So the radical alternative rule for conditionals is clearly a failure.
So we stick with our original semantic rule for conditionals: a conditional is only false when the antecedent is true, but the consequent is false.
So our semantic rule for conditionals should exhibit the same steps: take in two smaller sentences as inputs (the antecedent and the consequent), and give, as output, the whole conditional sentence.
And then we need to run through all the possibilities for the two little sentences, and decide what value the whole conditional will have in each possible situation.
Since we have two different little sentences here, and (according to Bivalence) each one can be either true or false, we know we’ll need 4 valuations to go through all the possibilities.
Now, how will we decide what truth value the whole conditional sentence has, in each of these situations? Well, remember that the arrow is intended as our formal counterpart to English language conditional phrases like “if… then,” “if,” and their relatives. So we expect the arrow to faithfully mirror the behavior of these English language conditional phrases. And since, as native speakers of English, we’ve been using these conditional phrases most of our lives, we’re the world's foremost authority on how English language conditional phrases behave. We just need to think of concrete examples of English conditionals, and when they’d be true, when false.
So consider a concrete example of an English conditional.
If they left before noon, then they avoided the rush-hour traffic.
That’s the kind of conditional we might really utter in certain situations. For example, if our friends were visiting from out of town, and were leaving to drive back home today, we might wonder if they got stuck in rush-hour traffic as they drove by some big city. And I might say: “If they left before noon, then they avoided the rush hour traffic.” Maybe you’d disagree, and say that’s not the case at all. We might even bet some money on who’s right about this. So this evening, after Logic class, we’ll settle the bet: we’ll call our friends, and see if my conditional was true or not.
Remember that this conditional (like every conditional) has two parts: the antecedent and the consequent. In this example, “They left before noon” is the antecedent, and “They avoided the rush-hour traffic” is the consequent.
“If they left before noon, then they avoided the rush-hour traffic.”
Antecedent: They left before noon
Consequent: They avoided the rush-hour traffic
To translate this conditional into formal language, we assign a sentence letter to each of these subject matter sentences.
P: They left before noon
Q: They avoided the rush-hour traffic
So the English language form of this sentence is:
If P, then Q.
And of course the whole conditional will be translated as
Now let’s go through each of the possibilities for the antecedent and consequent, and decide what the whole conditional will be (true or false) in each situation.
The first valuation is simple enough: this is a situation where it’s true that they left before noon, and it’s true that they avoided the rush-hour traffic.
And I said: “If they left before noon, then they avoided the rush-hour traffic.” Clearly, in this situation, what I said was true.
Now, for convenience, I want to jump down to the third valuation, and get that one out of the way as well. In the third valuation, the antecedent is true, but the consequent is false. In this situation, it’s true that they left before noon, but it’s false that they avoided the rush-hour traffic.
The conditional claim I made was: “If they left before noon, then they avoided the rush-hour traffic.” Now, if it turns (like in 3) that they did leave before noon, but still didn’t avoid the rush-hour traffic, then clearly my conditional claim was false.
Those are the two easy valuations. But now consider valuation 2: here, it’s false that they left before noon (but still, luckily, true that they avoided the rush-hour traffic).
What do we say about the whole conditional in this case? If we bet money on whether the conditional was true or false, who wins the bet? The bet was about what would happen if they left before noon. But valuation 2 is a situation where they didn’t leave before noon. It looks like that’s just not the sort of situation we were betting on. If we ended up in a situation like valuation 2 – if in fact they didn’t leave before noon – then we’d probably say that the bet was off; nobody won the bet and nobody lost it.
And the same is true for valuation 4: here again, it’s false that they left before noon. So here again, we’d probably say that the bet is off over whether the conditional is true or false.
So shall we just leave a blank for the whole conditional in these cases, stating that the conditional is neither true nor false?
No, we can’t do that – because that would be a violation of Bivalence. The Principle of Bivalence says that in any possible situation, this sentence is either true in that situation, or false; the sentence can’t be both true and false, and it can’t be neither true nor false.
Bivalence demands that the conditional either be true in 2, or false; and likewise with 4. So which will it be?
Think about it like this: if we did bet on my conditional, and it turned out they hadn’t left before noon at all, we wouldn’t say that such a situation made my conditional false. So if we have to choose – and Bivalence says we do – it’s at least a little less offensive to say that the whole conditional is true when the antecedent is false. So let’s say that in valuations 2 and 4, the whole conditional is false (in a kind of don’t-really-care way).
And with this example finished, we end up with a quite simple semantic rule for conditionals: a conditional will only be false when the antecedent is true, and the consequent is false.
Now, before we go on to start applying this semantic rule to conditionals, I want to answer some lingering doubts you may have about this rule. Maybe you aren’t convinced that the conditional really should be true when the antecedent is false (in valuations 2 and 4). Maybe you think that my ‘argument’ for that was pretty weak, and that I’m wrong about it being true in those cases.
But, I answer, if the conditional isn’t true in valuations 2 and 4, then what will it be? The Principle of Bivalence only leaves one option: it will have to be false in 2 and 4.
OK, you say, bravely: let it be so. Let’s have the conditional be false in valuations 2 and 4. I’ll bet that’s still better than your lousy true-in-2-and-4 rule, Beakley!
But now I’ll convince you that this alternative rule for the conditional will be a disaster: it will make claims and predictions about truth and meaning that, you’ll have to admit, are obviously false. (So really, I’m going to make an indirect argument that this alternate rule is wrong: I’ll say, “Suppose, for the sake of argument, that we do use this rule”; and then I’ll show how that will blow up in your face.)
Here’s my argument: compare this alternative rule for the conditional with our semantic rule for conjunctions.
Notice that, if we accept this alternative rule for conditionals, we’ll end up saying that conditionals and conjunctions are true in all the same situations, and false in all the same situations. (In technical terms: we’ll end up saying that the two sentences are truth functionally equivalent.)
But since you’re a speaker of English, and an expert on when simple “if” and “and” sentences are true, ask yourself this: do you really believe that conditionals and conjunctions are true in all the same situations? For example, consider these two sentences.
(i) If I won the lottery, then I’m a millionaire.
(ii) I won the lottery, and I’m a millionaire.
The two sentences both use the exact same subject matter sentences as parts (“I won the lottery” and “I’m a millionaire”). The only difference is that the first sentence is a conditional, while the second is a conjunction.
But we don’t have to stretch our imaginations very much to see that these two sentences are not true in all the same situations. The actual situation we’re sitting in right now is a situation where, yes, if I did win the lottery, then I would indeed be a millionaire. In the actual world, that conditional is true. But in the actual world, it’s definitely false to say that I won the lottery and I’m a millionaire. (Proof: if I were a millionaire, I would have quit my teaching job, and I wouldn’t be spending my time writing these notes on Logic.)
So in the actual world, the conditional is true, but the conjunction is false. And that’s solid evidence that conditionals and conjunctions are not true and false in all the same situations. So the radical alternative rule for conditional is not making correct predictions about when conditionals are true, and when they’re false. So the radical alternative rule for conditionals is clearly a failure.
So we stick with our original semantic rule for conditionals: a conditional is only false when the antecedent is true, but the consequent is false.