Conditionals Part 2

1. “Unless” Statements

Let’s look at this argument, and translate it into logical terms.

The Reds will win unless their starting pitcher is injured.

How would you write this in logical notation? Recalling the translation steps, we first identify atomic sentences and assign them constants.

R = The Reds will win.

S = Starting pitcher is injured.

Next, we look for truth-functional connectives. This is where things get tricky. There are no conjunctions or disjunctions, but we do have the word “unless.” “Unless” indicates what might or might not happen, which is similar to other truth-functional connectives we know already.

In fact, there are two hidden in the word “unless,” which is why this translation might be perplexing. First, it contains a conditional, so it can be paraphrased as an “if…then” statement:

If the starting pitcher for the Reds isn’t injured, then the Reds will win.

But note that this paraphrase also contains a negation, “isn’t.” The Reds will only win if the pitcher isn’t injured. Having identified these two logical connectives, we can now translate our original sentence as such:

¬S → R

Thus, we can understand sentences with “unless” as containing both a conditional and a negation.

Is the condition on the right-hand side of “unless” in the initial example a necessary or sufficient condition? This is a bit harder to determine without having more context or knowing the speaker’s intent. The speaker could mean a healthy pitcher is sufficient to get a win:

The Reds will surely win unless their starting pitcher is injured, but even then they could pull it off.

The speaker could also mean that a healthy pitcher is necessary to get a win:

The Reds will win unless their starting pitcher is injured, in which case they’re doomed.

Given the principle of charity, we might side with the more plausible argument, where a healthy pitcher is sufficient to get a win. In other words, the Reds can’t lose if their pitcher is healthy, and it’s anybody’s game if he isn’t. The necessity condition interpretation makes a stronger claim; the Reds are doomed if their starting pitcher is injured, and undefeatable if he isn’t. Remember, when we’re uncertain, the principle of charity guides us in choosing the most plausible (i.e., weaker) interpretation.

try it

What is the best way to interpret the following statement? What additional information might you need to interpret the statement most effectively?

Eleanor says she won’t go to the picnic unless she can take the dog.

Check Your Answer

We could translate the statement like this:

E = Eleanor will go to the picnic.

D = Eleanor can bring her dog.

¬D → ¬E

This is the most charitable way to translate it. However, this suggests that even if she can bring the dog, Eleanor might be a no-show. The additional information you might need to ask is, is Eleanor guaranteeing that she will come if she can bring the dog? If so, you might translate it like this:

D → E

However, this implies that Eleanor might come even if the dog can’t, which is precisely what we were told wouldn’t happen! This is where we might use the “iff” operator, or biconditional. Eleanor will come to the picnic iff (if and only if) she can bring her dog.

D ↔ E

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2. Material Equivalence

Recall this argument from the last section:

The Reds will win unless their starting pitcher is injured.

We explained that “unless” had two truth-functional meanings: a conditional and a negation. You can see this by rephrasing the statement like this:

If the starting pitcher for the Reds isn’t injured, then the Reds will win.

Which could then be put in symbolic notation like this:

S = The starting pitcher for the Reds is injured.

R = The Reds will win.

¬S → R

But here’s another way to rephrase the statement, and then translate it into formal logic.

Either the Reds will win or their starting pitcher is injured.

Using the same constants for our atomic sentences, we could then translate it into logical form like this:

R ∨ S

Although it may seem surprising, a disjunction will also capture the truth-functional meaning of “unless,” at least if we interpret the right-hand side of “unless” as a sufficient condition. Remember that disjunction is only false if both disjuncts are false, so the only way R ∨ S is false is if both R and S are false. Also remember that a sentence with → is only false if the antecedent is true and the consequent is false. Since the antecedent (¬S) is negated, we can say that ¬S → R is only false if S is false (¬S is true) and R is false.

Let’s demonstrate this with a truth table.

R	S	¬S	¬S → R	R ∨ S
T	T	F	T	T
F	T	F	T	T
T	F	T	T	T
F	F	T	F	F

If you look at the truth values under the main operators of each sentence, you can see that their truth values are identical on every row. That means the two statements are materially equivalent and can be used interchangeably in logical form because they always have the same truth value. Material equivalence is indicated using the biconditional, ↔. Sentences that are materially equivalent have the same logical meaning.

Let’s look at another kind of material equivalence. We have seen that we can translate “neither nor” statements as the conjunction of two negations. Remember that we “push the negation” (inside or outside of the parentheses) and “flip the sign” (turning a conjunct into a disjunct or vice versa). So, a statement of the form “neither p nor q” can be translated in either of these ways:

¬p ∧ ¬q

¬(p ∨ q)

Let’s look at an example with a truth table.

Cherish can neither drive to work nor work from home.

D = Cherish can drive to work.

W = Cherish can work from home.

¬D ∧ ¬W

¬(D ∨ W)

D	W	¬D	¬W	¬D ∧ ¬W	D ∨ W	¬(D ∨ W)
T	T	F	F	F	T	F
F	T	T	F	F	T	F
T	F	F	T	F	T	F
F	F	T	T	T	F	T

As you can see from the truth table, the truth values under the columns for ¬D ∧ ¬W and ¬(D ∨ W) are the same in every row.

We use the biconditional to show equivalence between complex sentences:

(¬S → R) ↔ (S ∨ R)

(¬D ∧ ¬W) ↔ ¬(D ∨ W)

try it

Construct a truth table to determine whether the following sentences are materially equivalent.
A → B and ¬B → ¬A

Check Your Work

The two sentences are materially equivalent, as we can see from this truth table. Remembering the rule of the conditional, A → B is only false if A is true and B is false. Similarly, ¬B → ¬A is only true if ¬B is true (B is false) and ¬A is false (A is true).

A	B	A → B	¬B → ¬A
T	T	T	T
F	T	T	T
T	F	F	F
F	F	T	T

term to know

Material Equivalence

When two sentences have the same truth value in all circumstances.