7.2 Equivalence (DeM and DN)
Tautology, tautological falsity, and contingency are concepts that apply to individual sentences.
Equivalence is a property that applies to a pair of sentences. Two sentences are equivalent if they always have the same truth value. In other words, they co-vary in truth value: if one is true, then the other is true; if one is false, then the other is false.
Since the connectives in BOOL are truth functional, we are focused here on one type of equivalence: tautological equivalence, that is, equivalence on the basis of truth functionality.
We figure out if sentences are tautologically equivalent by computing their truth functions. If the truth functions are the same, they are tautologically equivalent. Otherwise, they're not tautologically equivalent.
Since "tautologically equivalent" is a mouthful, we'll just say "equivalent" for short, until we learn other logical systems.
Let's try it out.
Remember both of those sentences are good translations of neither P nor Q. Now we can see why that is: they are equivalent, because their truth tables are identical.
~(PvQ) ⇔ ~P&~Q
~(P&Q) ⇔ ~Pv~Q
This equivalence is so important it has a name: DeMorgan's Law.
The same principle holds if we swap the conjunction and disjunction, which is the other version of the law: ~(P&Q) ⇔ ~Pv~Q.
The symbol "... ⇔ ..." deserves explanation. It means "... is logically equivalent to ..."
Though it may look like a formal symbol, don't be fooled: ⇔ is not in the language of BOOL.
Instead, it is just shorthand for the English words "logically equivalent".
Here's a bit of terminology logicians use to make this point. When we define a formal language as part of a logical system, that language is called the object language, since it is the object we are talking about. When we talk about that language, though, we must use another language, called the metalanguage ("meta" means "about").
Metalanguage: the natural language we use to create the formal language.
In this case, we are using English as our metalanguage. What we can say, then, is that "⇔" is not part of the object language. Instead, it is metalinguistic shorthand for "is logically equivalent".
Let's try another case. Assess whether these two sentences are equivalent:
If you are wondering how to do that, just follow this plan. First, construct a joint truth table for those sentences. A joint truth table is a truth table for several sentences together.
When making a joint truth table, be sure to give each sentence its own column on the right, like this:
After you compute the truth function for each sentence, you can see if they have the same function in order to figure out if the sentences are equivalent.
Now apply the notion of equivalence to the concepts you learned in the last chapter.
Remember, contingent just means a sentence has at least one T and one F.
Both P&Q and PvQ are contingent. But they aren't equivalent, since they are not the same truth function.
Here's a slightly harder question.
This question is tricky because we said every atomic sentence has a truth function like this:
That makes it sound like they are all equivalent. But that's not so.
Each atomic sentence on its own has a truth function like that, but in a joint truth table they become reference columns with all the different combinations represented.
That is what we want: atomic sentences represent independent truths, like "Pia is guilty", "Quinn is guilty", etc.
Those different sentences aren't equivalent. Logic doesn't guarantee that they are both guilty or not guilty.
Since atomic sentences always make a grid of values in a joint truth table, it follows that no atomic sentence is ever equivalent to any other atomic sentence. When we learn other systems later in this book, we'll return to this issue to deepen our understanding of it.
For now, we want you to figure this out.
P is not equivalent to any other atomic sentence, but it is equivalent to many complex sentences.
Since negation just inverts the truth value of the input sentence, the negation of a negation gets you back to where you started.
This principle is called the Law of Double Negation: P ⇔ ~~P.
Once you think about it, what holds for two negations also holds for any even number of negations.
In this chapter you've learned about the concept of equivalence; you've learned how to assess sentences for equivalence; and you've learned two key principles of equivalence, DeMorgan's and Double Negation.