Confusion between implication with its converse.

Question

I have given two propositions

p: You drive over 65 mph
q: You get a speeding ticket

Here are two natural language statements

1. Driving over 65 mph is sufficient for getting a speeding ticket (answer key: $p \to q$)
2. Whenever you get a speeding ticket, you are driving over 65 mph (answer key: $q \to p$)

Based on my understanding, anything that is sufficient, is consequent therefore in the 1^st one, it should be $q \to p$. And, anything that comes after when/whenever is the antecedent, it should be $p \to q$. Where I am wrong here?

 

 

Answer 1

Dont go by what comes after or before certain keywords, think it logically
Think of P $\rightarrow$ Q as IF P(is true) THEN Q(must be too)

Taking P : “Driving over 65” and Q : “Getting a speeding ticket”

Statement 1 can be interpreted as IF you drive over 65 THEN you get a speeding ticket. That’s why it is P $\rightarrow$ Q

The preposition before the arrow gives you What has happened, the one AFTER will tell you the action happening as its consequence. 
Similarly, The second statement translates to IF you get a speeding ticket THEN you must’ve been driving above 65.

So that is Q $\rightarrow$ P

Comments

As a follow up question, would you help me with understanding necessary and sufficient conditions? Its’ getting convoluted for me.

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Also I may me overfitting it, but I think it is ok to link antecedent with sufficieny and conclusion with necessity.

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Yeah sure tell me

Answer 2

Your understanding is correct, and you've correctly identified the implications of the given natural language statements.

1. "Driving over 65 mph is sufficient for getting a speeding ticket" translates to p→q, where p is "You drive over 65 mph," and q is "You get a speeding ticket." This is because driving over 65 mph is enough to guarantee that you will get a speeding ticket.

2. "Whenever you get a speeding ticket, you are driving over 65 mph" translates to q→p, where q is "You get a speeding ticket," and p is "You drive over 65 mph." This is because getting a speeding ticket implies that you were driving over 65 mph.

Your understanding aligns with the correct interpretations of these statements. It's a good practice to carefully consider the logical structure of the statements to determine the correct relationships between the propositions.