Implication operator

So far we have looked at the following operators:

• Conjunction operator
• Disjunction operator
• Exclusive disjunction operator
• Negation operator

In this step, we will introduce the implication operator.

The implication operator

The statement “If I win the lottery then I will retire” is an example of an implication. Let us consider this statement carefully, what is this statement saying? The statement says that on the condition that I win the lottery then I definitely will retire. The statement says nothing about what happens on the condition that I don’t win the lottery. Also, it does not say that the only condition for me to retire is when I win the lottery.

In this example, it is natural to identify that there is a causal relationship between the condition (antecedent) (“I win the lottery”) and the consequence (“I will retire”). Essentially, the reason why I might retire is because I have won the lottery and now have sufficient financial security to live comfortably without having to work. In logic, this idea that there is a causal relationship between the condition (antecedent) and the consequence does not exist.

For example, the proposition “If I eat cake, then I like cats” is a valid position but it is very unlikely that the reason I like cats is because I eat cake.

Logical implication is a concept in logic that describes the relationship between two propositions, where one statement (the antecedent) implies the truth of the other statement (the consequent). In other words, if the antecedent is true, then the consequent must also be true.

The logical implication is usually denoted by the symbol “(rightarrow)“. For example, let (p) represent the proposition “it is raining” and let (q) 
represent the proposition “the street is wet” can be written as (p rightarrow q), which is read as “if it is raining, then the street is wet”, or alternatively “it is raining implies the street is wet”.

The statement that comes after the arrow is the consequent, and it is said to follow logically from the antecedent. If the antecedent is false, the implication is considered to be true (since a false antecedent cannot prove or disprove the truth of the consequent). However, if the consequent is false, the implication is considered to be false.

Consider another example, “If it is sunny outside, then the temperature is warm.” This statement implies that if it is sunny outside (the antecedent), then the temperature is warm (the consequent). If it is indeed sunny outside, then we would expect the temperature to be warm. If it is not sunny, however, we cannot infer anything about the temperature.

The tabular description of the implication operator is provided below.

More ways of expressing the implication

The implication is very important in mathematics, computer science and in spoken English when you are putting forward an argument. As a result there are a variety of ways that implication might be expressed in written or spoken language. All of the following are equivalent to (p rightarrow q). Have this in mind when you attempt using logic on your own in future steps.

• “if (p), then (q)“
• “(p) implies (q)“
• “if (p), (q)“
• “(p) only if (q)“
• “(p) is sufficient for (q)“
• “a sufficient condition for (q) is (p)“
• “(q) if (p)“
• “(q) whenever (p)“
• “(q) when (p)“
• “(q) is necessary for (p)“
• “a necessary condition for (p) is (q)“
• “(q) follows from (p)“
• “(q) unless (neg p)“

Work carefully through each of these equivalent ways of communicating the implication and ensure that you understand that they really do convey the implication operator as we have described it here. It may be easier if you try to construct a sentence using each of the variations.

For example, consider one of the more challenging variations to reason about. Consider “(p) only if (q)“, it says that (p) can be true only if (q) is true, which is to say that when (q) is false, (p) must also be false. This does not say anything about (p) if (q) is true. Consider the following example, “The shower is on only if the tap is open”. If the shower is on, then the tap must be open, since if the tap is not open then no water will flow and the shower is not on. However, If the tap is open it does mean the shower is on as there could be some other plumbing issue.

What have we learned so far?

So far in this activity, we have introduced the conjunction, disjunction, exclusive disjunction, negation and implication operators. These are all the essential building blocks of propositional logic.

Later on this week, we will look at how we can use these operators to make complicated expressions that model real world problems and how we can then use logic to solve them.