Week 1

Implications

There is a multitude of meanings when it comes to the use of the word "implication". There is a good detailed explanation in the script. To provide an overview consider the following:

Implications between Mathematical statements

The arrow "$\implies$" can be used between mathematical statements to form a new mathematical statement. For example, let $S$ and $T$ be mathematical statements. Then we can form the mathematical statement $S \implies T$. The new statement can be translated into natural language by saying that "that if $S$ is true, then $T$ is also true" (which is a statement that can be false). It does not mean that because $S$ is true, $T$ is also true.

For example, consider the following statements:

• $S$: "$2$ is an even number"
• $T$: "$5 + 10 = 15$"

In this case, $S \implies T$ is true. This may seem counterintuitive, but it is a true mathematical statement. The statement $S \implies T$ does not mean that there is a derivation from $S$ to $T$.

Additionally, it can be quite counterintuitive to consider the statement $S \implies T$ when you know that the statement $S$ is actually false. This automatically makes $S \implies T$ true because we essentially don't care about the truth value of the statement $T$. Since $S$ is false, the implication "if $S$ is true, then $T$ is also true" is vacuously true because $S$ is not true — we don't need to check whether $T$ is true.

For example, consider the following statements:

• $S$: "$2$ is an odd number"
• $T$: "$3$ is an even number"

If we now consider $S \implies T$, we would find that $S \implies T$ is indeed a true statement because we know $S$ is false.

Note that the only case where the statement $S \implies T$ is false is when $S$ is a true statement and $T$ is a false statement.

Derivation step arrow

$\overset{\cdot}{\implies}$

Implication as formula operator

$\to$

Logical consequence

$\vDash$

EBNF: Statements vs Formulas

Exercises