The conjunction operator

Now that we have seen what a proposition is, we can connect propositions together to make more complicated logic expressions.

The conjunction operator

The statement “I like coffee and I like cake” is known as a compound proposition as it is made up of two simpler statements, namely the statement “I like coffee”, and the statement “I like cake”. The word “and” in the compound proposition has a special purpose. It conveys the concept of conjunction. The word “and” is known as a conjunction operator.

Consider the compound proposition again, this time more carefully. Consider how you would interpret the compound proposition if a friend said it to you. Firstly, unless your friend was not telling the truth, you would infer that your friend likes coffee, and you would also infer that your friend likes cake. In other words, the compound proposition would be true.

So when is the compound proposition “I like coffee and I like cake” false? The compound statement is false if your friend is not telling the truth, that is, when your friend is lying to you. If you knew your friend was lying to you, and your friend said “I like coffee and I like cake” then you would be able to infer one of three possibilities:

1. that your friend does not like coffee, or
2. that your friend does not like cake, or
3. that your friend neither likes coffee or cake.

To summarise, a compound proposition (that is the conjunction of two propositions) is true when both statements are true, and is false otherwise.

The conjunction operator symbol

In logic, we have a symbol that represents the conjunction operator.

• Let (p) represent the proposition “I like coffee”.
• Let (q) represent the proposition “I like cake”.
• The compound proposition “I like coffee and I like cake” can be represented as (p land q), which is read as “p and q”.

Now let’s do a quick quiz in the next step to check your understanding so far.