Article: Material equivalence (Dr. Van Cleave)

By Matthew Van Cleave

"Unless"

The English term “unless” can be tricky to translate.  For example,

If we use the constant “R” to stand for the atomic proposition, “the Reds will win” and “S” to stand for the atomic proposition, “the Reds’ starting pitcher is injured,” how would we translate this sentence using truth functional connectives?  Think about what the sentence is saying (think carefully).  Is the sentence asserting that the Reds will win?  No; it is only saying that

“As long as” denotes a conditional statement.  In particular, what follows the “as long as” phrase is a sufficient condition, and as we have seen, a sufficient condition is always the antecedent of a conditional.  But notice that the sufficient condition also contains a negation.  Thus, the correct translation of this sentence is:

One simple trick you can use to translate sentences which use the term “unless” is just substitute the phrase “if it’s not the case that” for the “unless.”  But another trick is just to substitute an “or” for the “unless.”  Although it may sound strange in English, a disjunction will always capture the truth functional meaning of “unless.”  Thus, we could also correctly translate the sentence like this:

We will show how we can prove that these two sentences are equivalent using a truth table.

Material equivalence

As we saw in the last section, two different symbolic sentences can translate the same English sentence.  In the last section I claimed that “~S ⊃ R” and “S v R” are equivalent.  More precisely, they are equivalent ways of capturing the truth-functional relationship between propositions.  Two propositions are materially equivalent if and only if they have the same truth value for every assignment of truth values to the atomic propositions.  That is, they have the same truth values on every row of a truth table.  The truth table below demonstrates that “~S ⊃ R” and “S v R” are materially equivalent.

R	S	~S ⊃ R	S v R
T	T	F   T	T
T	F	T   T	T
F	T	F   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, as far as propositional logic goes. 

Let’s demonstrate material equivalence with another example.  We have seen that we can translate “neither nor” statements as a conjunction of two negations.  So, a statement of the form, “neither p nor q” can be translated:

But another way of translating statements of this form is as a negation of a disjunction, like this:

We can prove these two statements are materially equivalent with a truth table (below).

p	q	~p ⋅ ~q	~(p v q)
T	T	F   F   F	F    T
T	F	F   F   T	F    T
F	T	T   F   F	F    T
F	F	T   T   T	T    F

 

Again, as you can see from the truth table, the truth values under the main operators of each sentence are identical on every row (i.e., for every assignment of truth values to the atomic propositions).

In fact, there is a fifth truth functional connective called “material equivalence” or the “biconditional” that is defined as true when the atomic propositions share the same truth value, and false when the truth values different.  Although we will not be relying on the biconditional, I provide the truth table for it below.  The biconditional is represented using the symbol “≡” which is called a “tribar.”

p	q	p ≡ q
T	T	T
T	F	F
F	T	F
F	F	T

Some common ways of expressing the biconditional in English are with the phrases “if and only if” and “just in case.”  If you have been paying close attention (or do from now on out) you will see me use the phrase “if and only if” often.  It is most commonly used when one is giving a definition, such as the definition of validity and also in defining the “material equivalence” in this very section.  It makes sense that the biconditional would be used in this way since when we define something we are laying down an equivalent way of saying it.

Do Quiz 9b before moving further in the course.