- [[biconditional implication]], [[logical implication]] # Idea For any [[logical implication]] $p \rightarrow q$, we can translate by saying the following. $p$ is a *sufficient* condition for $q$. - For $p$, it is *sufficient* that $q$ - $p$ being true guarantees that $q$ is true $q$ is a *necessary* condition for $p$. - For $p$, it is *necessary* that $q$, which can be translated to $q \rightarrow p$. - For $q$ to be true, $p$ must also/always be true. $q$ cannot be true unless $p$ is true. *Necessity* and *sufficiency* describe a conditional or implicational relationship between two statements. *Sufficiency* is represented by the *direct implication*, $p \rightarrow q$. *Necessity* is represented by the *converse implication*, $q \rightarrow p$. ## Why is $q$ necessary for $p$ in the statement $p \rightarrow q$? ![[20231230110837.png]] Interpreting the [[truth table]]: - When $p$ is true and $q$ is true, $p \rightarrow q$ is true. - When $p$ is true and $q$ is false, $p \rightarrow q$ is false. - When $p$ is false (regardless of $q$), $p \rightarrow q$ is true. From the table, for $p$ to be true (column 1, rows 1 and 2), it is *necessary* for $q$ to be true (row 1) if $p \rightarrow q$ is also true (row 1). In other words, for $p \rightarrow q$ to hold true, whenever $p$ is true, $q$ must necessarily be true. Thus, when $p \rightarrow q$, for $q$ must (necessarily) be true if $p$ is true. **Important point/distinction**: It also doesn't assert that $q$ alone makes $p$ true, *but rather that $qs falsity makes $p$ false* (see row 4). This point is critical! The necessity of $q$ for $p$ means $p$ is reliant or *conditionally dependent* on $q$ being true to be true itself. Without $q$, $p$ lacks its *essential* support. ### Concrete examples $p \rightarrow q$: If a structure is standing $p$, the foundation $q$ must be present. If the foundation $q$ is absent, the structure $p$ cannot exist. The necessity of $q$ for $p$ means that $q$ is an *essential requirement* for the truth or $p$. If we want $p$ to be true when $p \rightarrow q$, $q$ has to be true (see rows 1 and 2). In $p \rightarrow q$, $ps truth is *conditionally dependent* on $q$. If $q$ is not true, then the condition for $p$ being true cannot be met. It doesn't assert that $q$ alone makes $p$ true, *but rather that $qs falsity makes $p$ false* (see rows 2 and 4). This point is critical! ## Necessary but not sufficient For $p$, it is *necessary* but *not sufficient* that $q$, can be written as the following [[propositional logic systems|propositional logic]]: $ \underbrace{q \rightarrow p }_{\text {it is neccessary }} \underbrace{\land}_{\text {and/but }}\underbrace{\neg(p \rightarrow q)}_{\text {it is not sufficient }} $ It captures the idea that if $p$ is true, then $q$ must also be true, but the truth of $q$ does not guarantee the truth of $p$. ![[20231229160436.png]] ## Example Write using logical connectives (from [[propositional logic systems]]): - $p$: Grizzly bears have been seen in the area. - $q$: Hiking is safe on the trail. - $r$: Berries are ripe along the trail. For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area. $ \underbrace{q \rightarrow (\neg r \land \neg p) }_{\text {it is neccessary }} \underbrace{\land}_{\text {and/but }}\underbrace{\neg(\neg r \wedge \neg p) \rightarrow q}_{\text {it is not sufficient }} $ # References - [logic - Writing Propositions With Propositional Variables - Mathematics Stack Exchange](https://math.stackexchange.com/questions/257283/writing-propositions-with-propositional-variables) - [Mathematical Logic: Mathematical logic](https://gateoverflow.in/60011/mathematical-logic) - [Necessity and sufficiency - Wikipedia](https://en.wikipedia.org/wiki/Necessity_and_sufficiency)