Associativity of Connectives:

In order to tell us brackets are useful when to perform calculations in arithmetic and when to evaluate the truth of sentences in logic. Imagine we want to add 10, 7 and 5. We could do this: (10 + 7) + 5 = 22. Alternatively, we could do this: 10 + (7 + 5) = 22. We, in this case, can alter the bracketing and the answer still comes out the same. We say that addition is associative because it fulfills this property with respect to bracketing.

The ∧ and ∨ connectives are associative. This can be true, because the order in which we examine truth values does not matter when we are working with sentences only involving or only involving ∨. For example, assume we wanted to know the truth of P ∧ (Q ∧ R). To do this, we just have to examine that every proposition is true, in which case the complete sentence will be true, or else the entire sentence will be false. Due to this reason, it does not matter how the brackets are arranged, and hence the ∧ is associative.

Similarly, assume we wanted to work out the truth of:

(P ∨ Q) ∨ (R ∨(X ∨Z))

Then all we have to do is examine whether 1 of these propositions is true, and the bracketing is beside the point. As equivalences, the 2 associativity results are then:

(P ∧ Q) ∧ R = P ∧ (Q ∧ R)

(P ∨ Q) ∨ R = P ∨ (Q ∨ R)