Implication to Disjunction and Negation of Implication

[jpanknin]

Can anyone explain the "Implication to Disjunction" concept intuitively (perhaps even non-mathematically initially) or provide some resources? I've seen that the truth tables for [imath]p \implies q[/imath] and [imath]\neg p \lor q[/imath] are equivalent, but I don't see how the statements themselves are related. Or are they related?

Implication to disjunction

These statements rephrased in English don't make sense:

If we say p is "it rains" and q is "the grass is wet"

• For [imath]p \implies q[/imath] we have "If it rains, then the grass is wet" - totally makes sense
• For [imath]\neg p \lor q[/imath] we have "It did not rain or the grass is wet" - doesn't make sense to me

Is there a way the sentences are supposed to follow from one another or is it just that their truth tables are equivalent? In our exercises, these two are used interchangeably to prove or simplify statements. The other propositional logic laws make sense, but this just doesn't to me.

Same goes for the negation law below.

Negation of the implication


Any help would be appreciated.

 

[MaxWong]

When I first learnt it, I tried to make sense of [imath]\neg (p \rightarrow q) \equiv p \wedge \neg q[/imath] first, and then recover the identity [imath]p \rightarrow q \equiv \neg p \vee q[/imath] using De Morgan's Law.

Using your example, if p means "it rains" and q means "the grass is wet", [imath]p \wedge \neg q[/imath] is saying that "It rains, but the grass is not wet". This means raining does not cause the grass to be wet, i.e., p has no implication on q. Therefore [imath]p \wedge \neg q[/imath] is logically the same as [imath]\neg (p \to q)[/imath].

To recover the identity [imath]p \rightarrow q \equiv \neg p \vee q[/imath], take [imath]\neg[/imath] on both sides. Then we have

[math] \begin{array}{rcl} p \to q &\equiv& \neg(p \wedge \neg q)\\ &\equiv& \neg p \vee \neg(\neg q)\\ &\equiv& \neg p \vee q \end{array} [/math]

 

[blamocur]

There is already a thread with this discussion. You might find some of the posts there informative.