
Introduction to Logic

Tools for Thought

Intuitively, we think of two sentences as being equivalent if they say the same thing, i.e. they are true in exactly the same worlds. More formally, we say that a sentence φ is logically equivalent to a sentence ψ if and only if every truth assignment that satisfies φ satisfies ψ and every truth assignment that satisfies ψ satisfies φ.
The sentence ¬(p ∨ q) is logically equivalent to the sentence (¬p ∧ ¬q). If p and q are both true, then both sentences are false. If either p is true or q is true, then the disjunction in the first sentence is true and the sentence as a whole false. Similarly, either p is true or q is true, then one of the conjuncts in the second sentence is false and so the sentence as a whole is false. Since both sentences are satisfied by the same truth assignments, they are logically equivalent.
By contrast, the sentences (p ∧ q) and (p ∨ q) are not logically equivalent. The first is false when p is true and q is false, while in this situation the disjunction is true. Hence, they are not logically equivalent.

One way of determining whether or not two sentences are logically equivalent is to check the truth table for the proposition constants in the language. This is called the truth table method. (1) First, we form a truth table for the proposition constants and add a column for each of the sentences. (2) We then evaluate the two expressions. (3) Finally, we compare the results. If the values for the two sentences are the same in every case, then the two sentences are logically equivalent; otherwise, they are not.

As an example, let's use this method to show that ¬(p ∨ q) is logically equivalent to (¬p ∧ ¬q). We set up our truth table, add a column for each of our two sentences, and evaluate them for each truth assignment. Having done so, we notice that every row that satisfies the first sentence also satisfies the second. Hence, the sentences are logically equivalent.
p 
q 
¬(p ∨ q) 
¬p ∧ ¬q 
1 
1 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
1 

Now, let's do the same for (p ∧ q) and (p ∨ q). We set up our table as before and evaluate our sentences. In this case, there is only one row that satisfies first sentence while three rows satisfy the second. Consequently, they are not logically equivalent.
p 
q 
p ∧ q 
p ∨ q 
1 
1 
1 
1 
1 
0 
0 
1 
0 
1 
0 
1 
0 
0 
0 
0 

One of the interesting properties of logical equivalence is substitutability. If a sentence φ is logically equivalent to a sentence ψ, then we can substitute φ for ψ in any Propositional Logic sentence and the result will be logically equivalent to the original sentence. (Note that this is not quite true in Relational Logic, as we shall see when we cover that logic.)

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