Logic - 2
This lecture is very similar to COMP111’s truth values lecture. View that lecture for all truth tables.
Truth Values
Interpretations are a way of assigning values to propositions which may vary depending on the situation or person who answers them.
An interpretation $I$ is a function which assigns to any atomic proposition $p_i$ a truth value:
\[I(p_i)\in \{0,1\}\]- If $I(p_i)=1$, then $p_i$ is called true under the interpretation $I$.
- If $I(p_i)=0$, then $p_i$ is called false under the interpretation $I$.
Given an assignment $I$ we can compute the truth value of compute formulas step by step using so-called truth tables.
Implication
The implication $(P\Rightarrow Q)$ of $P$ and $Q$: \(\text{If } P \text{ then } Q\)
Truth Table
| $P$ | $Q$ | $(P\Rightarrow Q)$ |
|---|---|---|
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 1 |
Consider this table with the proposition $P$ being a promise. If you don’t make a promise but you fulfil it anyway then you aren’t breaking that promise.
Another example would be the following statement:
- If a number is divisible by 6 then it is divisible by 3.
- If it is not divisible by 6 but still divides by 3, this does not invalidate the proposition.