Consider the statement "If you work hard then you will succeed". What is the right way of writing the converse and contrapositive of this statement? Should they be "If you will not succeed then you do not work hard." and "If you will succeed then you work hard." or "If you will not succeed then you did not work hard." and "If you will succeed then you worked hard."?
Verb tenses when writing the converse and contrapositive of an implication
logiclogic-translationpredicate-logic
Related Solutions
Recall,
Statement: if $p$ then $q.$
Converse: if $q$ then $p.$
Inverse: if not $p$ then not $q.$
Contrapositive: if not $q$ then not $p.$
So, yes, your answers are indeed correct.
Classical logic (Logic from more than a thousand years ago) does not include notions of time (past, present, future, etc...).
The following are examples of a few rules of inference from classical logic:
Rule of Inference: Implication Operator From Parent to Left Child
For all $P$ and $Q$ in the set $\{true, false\}$, the following is true:
$\mathtt{NOT}\,(\mathtt{if}\, P \,\, \mathtt{then} \,\, Q) \implies P $
Rule of Inference: Implication Operator From Parent to Right Child
For any $P$ and $Q$ in the set $\{true, false\}$, the following is true:
$\mathtt{NOT}\,(\mathtt{if}\, P \,\, \mathtt{then} \,\, Q) \implies Q $
Rule of Inference: DeMorgan's Law
For any $P$ and $Q$ in the set $\{true, false\}$, the following is true:
$[\mathtt{NOT}\,(P \,\, \mathtt{OR} \,\, Q)] \longleftrightarrow [\text{ } (\mathtt{NOT}\, P) \text{ } \mathtt{AND} \text{ } (\mathtt{NOT}\, Q) \text{ } ]$
Note that none of the rules of inference in classical logic talk about time.
It is not easy to express any of the following English sentences in logic from hundreds of years ago:
- "I will water my plants in the future"
- "I watered my plants in the past"
- "I currently water the plants inside my house."
I would stay that it is incorrect to use classical logic to express the English sentence "If I water my plants then they will grow" as "$\mathtt{if}\, P \,\, \mathtt{then} \,\, Q$" where $P$ is "I water my plants"
Even an operation as simple as negation is not the same in English as it is in classical logic.
Perhaps, you would say that the following are equivalent:
- $\mathtt{NOT}$
I do water my plants I do$\mathtt{NOT}$water my plants
The statements about plants might be okay. However, consider the two following two statements:
- $\mathtt{NOT}$
My wife's cousin Hector's pet monkey does like to eat strawberries My wife's cousin Hector's pet monkey does$\mathtt{NOT}$like to eat strawberries
There are some issues, not the least of which are:
- I do not have a wife
- If I had a wife, I am not sure if she would have a cousin Hector or not
- Even if I was married, and my wife had a cousin named Hector it is doubtful that Hector would own a pet monkey.
In English, you are not allowed to move the word "$\mathtt{NOT}$" further and further to the left, or further to the right.
It is helpful to learn simple logic from ye olden days before you learn more advanced contemporary logic. However, I really wish that your teacher would not give you examples in English. English is really too complicated to force into the classical logic model.
In mathematics, time is sometimes modeled as follows:
If there exists at least one time in the future at which statement $P$ is true, then we write:
◇$P$
For example:
- ◇
I am watering my plants- There exists a time in the future such that the following will be true at that time:
I am watering my plants
I still think that this is an abuse of the English language. However, using the symbols ◇ and □ to say something about time makes for a better model of English, relatively speaking, than using classical logic alone.
If something is true for all times in the future then we might write:
□$P$
For example:
- For all times in the future
I am watering my plants- □
I am watering my plants
I am not sure that modal logic using the symbols ◇ and □ is great for English, but it does help in math sometimes.
For example, the following are equivalent:
- Eventually, $F(x)$ is always zero.
- ◇□ $F(x) = 0$
- There exists a number $x$ such that for all $y \geq x, F(y) = 0$
Suppose that:
- $P$ is the string
I water my plants- $Q$ is the string
they will grow.
Well then, we have:
| Thing 1 | Thing 2 |
|---|---|
| $\mathtt{IF} \text{ } P \,\, \mathtt{THEN} \text{ } Q$ | $\mathtt{IF}$ I water my plants $\mathtt{THEN}$ they will grow |
| $P \,\, \mathtt{AND} \,\, (\mathtt{NOT} \,\, Q)$ | (I water my plants) $\mathtt{AND}$ ($\mathtt{NOT} \,$ they will grow) |
| $\mathtt{IF} \text{ } (\mathtt{NOT} Q) \mathtt{THEN} \text{ } (\mathtt{NOT} \text{ } P)$ | $\mathtt{IF}$ ($\mathtt{NOT} \,\,$ they will grow) $\mathtt{THEN} \text{ }$ ($\mathtt{NOT} \,\, \text{ }$ I water my plants) |
If some of the things written in the table above do not sound like English it is because, the things written in the table are not English.
You cannot model English sentences so easily using classical logic.
In English, the following two sentences have the same meaning:
- If I water my plants then they will grow.
- If I water my plants then my plants will grow.
With classical logic alone, you are not allowed to do something as simple as understand that the word "they" can be replaced with the phrase "my plants" while retaining the meaning.
Best Answer
For the sake of math I don't think it matters.
Technically it is "If you work hard (in the present) then you will succeed (in the future)" and the contrapositive would be "If you will not succeed (in the future) then you do not work hard (in the present)".
But that makes for fairly unnatural and awkward english. We frequently talk of current causes having potential effect in the future, or present situations having being caused by conditions in the past, but we seldom talk about future known situations being caused by present (but unknown) causes. The rare exception might be a precautionary "If you are to fail to succeed, it is only because you do not work hard now".
I think it is understood that this statement applies for any time frame. If your grandfather worked hard in 1960 than he succeeded in 1970. If you grand daughter works hard in 2045, then she will succeed in 2055.
I think "If you do not succeed, then you didn't work hard" is the more natural statement. Although "If you will not succeed, then you do not work hard" is more technically accurate.