1. Logical equivalence

2. Logical equivalence
At the end of our last session we briefly touched on the idea of logical equivalence. As we said then, the basic idea is that two sentences are logically equivalent if they are true and false in exactly the same cases.
Or, for any assignment of truth values, if the one sentence comes out with the value t, the other will as well, and if it comes out as f, the other will as well too.

3. Logical equivalence and truth tables
We can think of this in terms of truth tables. Imagine we have two sentences of sentence logic (lets call them X and Y). We write out a truth table for them both.
Remember that a truth table tells us what truth value a sentence will have for every possible combination of truth values of the atomic sentence letters that comprise it.

4. Logical equivalence and truth tables
So all we need to do to check whether a particular pair of sentences are logically equivalent is to write out a truth table, and check the columns for both. If they always take the same value for every line of the truth table, they are logically equivalent.
Because a line of a truth table is just a way of graphically representing an assignment of truth values. So if they take the same truth value for every line of the table, they will have the same truth value for every different assignment of truth values.

5. Logical equivalence and truth tables
Let’s take an example. ~(A ∧ B) and ~A ∨ ~B.
A B ~(A ∧ B) ~A ∨ ~B t t f f t f t t f t t t f f t t

6. Logical equivalence and truth tables
The two sentences are true and false in exactly the same cases. So they’re logically equivalent.
In fact, what we’ve just done there is prove one of De Morgan’s Laws: for any two sentences X and Y, ~(X ∧ Y) is logically equivalent to ~X ∨ ~Y.

7. Logical equivalence and truth tables
In the textbook Teller uses the same method to prove the Law of Double Negation.
The Law of Double Negation says that for any sentence X, X is logically equivalent to ~~X.

8. Logical equivalence
With this in mind, let’s now propose a more specific definition of logical equivalence.
Two sentences of sentence logic are logically equivalent if and only if in each possible case (for each assignment of truth values to sentence letters) the two sentences have the same truth value.

9. Logical equivalence and Venn diagrams
So we just looked at how to check for logical equivalence using truth tables.
We can also check for it using Venn diagrams, as Teller demonstrates in the textbook.

10. Logical equivalence and Venn diagrams
The basic idea is simple. You draw a box, and then a circle within that box. The circle represent some sentence of sentence logic, call it X. Now, every point inside X represents cases in which X is true. And every point outside represents a case in which X is false. X

11. Logical equivalence and Venn diagrams
We can add in another circle to represent a different sentence. Call it Y.
Points at the intersection represent cases in which both X and Y are true, points outside both circles represent cases in which neither is true, and so on. X Y

12. Logical equivalence and Venn diagrams
So let’s check our De Morgan’s Law. First ~(X ∧ Y):
This is going to be true whenever it is not the case that both X and Y are true – i.e. everywhere within the shaded area. X Y

13. Logical equivalence and Venn diagrams
What about ~X ∨ ~Y? Well that’s going to be true whenever ~X is true or whenever ~Y is true. So, let’s shade the entire area outside the X circle (which represents ~X remember), and then everywhere outside the Y circle. X Y X Y

14. Logical equivalence and Venn diagrams
We end up with the same diagram! And that’s because the two sentences are logically equivalent. They are true and false in exactly the same cases, so if the shading represents the cases for which they are true and false, their shaded Venn diagrams will be identical.
So Venn diagrams can be handy visual ways of depicting and proving logical equivalence claims.

15. Some laws
So we’ve seen the law of double negation (X is logically equivalent to ~~X) and one of De Morgan’s laws (~(X ∧ Y) is logically equivalent to ~X ∨ ~Y).
There’s another De Morgan law, which says that ~(X ∨ Y) is logically equivalent to ~X ∧ ~Y.

16. Some laws There are also the two laws of distribution.
For any three sentences, X, Y, and Z, X ∧ (Y ∨ Z) is logically equivalent to (X ∧ Y) ∨ (X ∧ Z).
For any three sentences, X, Y, and Z, X ∨ (Y ∧ Z) is logically equivalent to (X ∨ Y) ∧ (X ∨ Z).

17. Some laws
These laws can all be proved, both by the truth table method and the Venn diagram method.
They will come in handy a lot later down the line – so remember them!