By Mims III F.M.

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**Example text**

Firstly, we need to explain the notation used. In a fragment of reasoning presented in this way, the (implicit) claim is that the statement below the line follows logically from the statements above the line. To ask whether the reasoning is valid is not to ask whether or not the individual statements are true. Rather, it is asking whether the conclusion, which is the statement below the line following ‘Therefore . . ’, necessarily follows from the previous statements. In this case, each of the three individual statements is true, but the reasoning itself is not valid.

This observation allows us to complete the ﬁnal column of the truth table. P Q R ¬R A P ∧ ¬R P ∨R B Q ∧ (P ∨ R) A⇔B T T T T F F F F T T F F T T F F T F T F T F T F F T F T F T F T F T F T F F F F T T T T T F T F T T F F T F F F F T T F F T T T 27 Logic and natural language As we have seen, natural language is more expressive than propositional logic as it is able to express nuances that we cannot capture using logical expressions. For example, each of the two English sentences Max played for the ﬁrst team and he scored the wining try and Max played for the ﬁrst team although he was nursing an injury is logically a conjunction.

It is not unusual to break a proof down into separate cases and consider each separately. The only prerequisite knowledge for the proof is the evenness and oddness properties and some basic algebra. Compare this with the following, much shorter proof. Proof 2. Let n be an integer. Then n2 + n = n(n + 1). Now n and n + 1 are consecutive integers, so one of them is even and the other is odd (but we don’t know which is which). Hence their product is the product of an even and an odd integer, which is even.