Patrick C. answered • 10/26/21
Recent McGill University grad looking to help students succeed!
In logic we can write if-then statements in the form 'P → Q'. We can symbolize any English-language statement that can have a truth-value (T or F) with a single letter.
Let's re-write "If you finish your homework by 5pm, then you go out to dinner" as H → D, where H = finishing your homework by 5pm and D = going out to dinner, each of which can have the value of T or F depending on the information given. To make sense of these symbolizations in plain English we might read 'H → D' as 'it is the case that if H then D'.
We don't know whether it is or is not the case that if H then D though, we're asked to figure that out given our conditions.*
- You finish your homework by 5pm (this makes H true) AND you go out to dinner (this makes D true). In the case where both P and Q are true, a sentence of the form 'P → Q' will also be true. Thus our sentence H → D is true.
- You finish your homework by 4pm AND you go out to dinner. Let's take a second with this one. Does finishing our homework by 4pm make 'finishing our homework by 5pm' true? We should probably say yes if this is on an assignment, because finishing our homework any time before 5 could, in a sense, count as finishing it by 5pm. But according to logic we don't necessarily have to interpret it this way, we could say 'finishing your homework by 5pm' is only true when you finish your homework sometime between the hour of 4pm and 5pm. Anyway, the second part of our statement (the part following " 'and' ") makes D true. Again, in the case where both P and Q are true, a sentence of the form 'P → Q' will also be true. Thus our sentence H → D is true.
- You finish your homework by 5pm (this makes H true) AND you do not go out to dinner (this makes D false). In the case where P is true and Q is false any statement of the form 'P → Q' will be false. Thus, under these conditions, our sentence H → D is false.
- If 89 is divisible by two then 89 is an even number
- Converse (flip the order of what's on either side of the conditional around, e.g. 'P → Q' becomes 'Q → P'): If 89 is an even number then 89 is divisible by 2
- Inverse (negate both P and Q for statements of the form 'P → Q' e.g. 'P → Q' becomes 'not-Q → not-P'): If 89 is not divisible by two then 89 is not an even number.
- Contrapositive (flip the order of what's on either side of the conditional around and negate both P and Q for statements of the form 'P → Q' e.g. 'P → Q' becomes 'not-Q → not-P').
Bonus info (IGNORE IF THIS BEGINS TO CONFUSE YOU): *I've noticed something a bit unusual about this question: instead of taking the whole 'H → D' as true, we're asked to determine when it is true given the set of conditions. It's unusual because technically, any time D is is true, the whole statement 'H → D' is true. Also, when we have a statement with the conjunction 'and' and we know it is true, each individual statement separated by an 'and' must be true on its own. This is to say that for conditions 1. and 2. D is true 'separately' from H being true. This is counter to our common understanding of conditional statements but demonstrates the meaning they take on in logic.
