Logical reasoning exercise on conditional statements and their forms.

Let's analyze the logic of the conditional statement and its related forms: inverse, converse, and contrapositive.

We start with the original conditional statement:

> If the temperature is warm, then you can play outside.

In logic, this is written as:
If P, then Q, where
- P = "The temperature is warm"
- Q = "You can play outside"

Now let’s examine each of the related statements and determine whether they are logically equivalent or necessarily true.

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1. Original Conditional Statement:

If the temperature is warm, then you can play outside.
✔ This is assumed to be true in the context (we're testing what follows from it).

But note: This doesn't say anything about what happens when the temperature is *not* warm — so we must be careful.

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2. Inverse:

If the temperature is not warm, then you cannot play outside.

This is: If not P, then not Q

⚠️ The inverse is not logically equivalent to the original statement.
Just because being warm allows playing outside doesn’t mean that only warm weather allows it.
For example, maybe you can play outside on a cool day too (e.g., wearing a jacket). So the inverse might be false even if the original is true.

👉 Is this true? ✘ Not necessarily.
It could be false. So no, the inverse is not guaranteed to be true.

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3. Converse:

If you can play outside, then the temperature is warm.

This is: If Q, then P

Again, this is not logically equivalent to the original.
The original says warm → can play, but the converse says can play → warm.
But you might be able to play outside for other reasons (e.g., cold but sunny, or no rain). So this could be false.

👉 Is this true? ✘ Not necessarily.
So the converse is not guaranteed to be true.

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4. Contrapositive:

If you cannot play outside, then the temperature is not warm.

This is: If not Q, then not P

✔ This is logically equivalent to the original conditional statement.
In logic:
> If P → Q is true, then ¬Q → ¬P is also true (and vice versa).

So if you cannot play outside, then the temperature must not be warm (based on the original rule).
This is the only one that is always true if the original is true.

👉 Is this true? ✔ Yes, it is logically true if the original is true.

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Summary:

| Statement | Logical Form | Equivalent to Original? | Always True? |
|------------------|--------------------|--------------------------|--------------|
| Original | P → Q | Yes | Assumed true |
| Inverse | ¬P → ¬Q | No | ✘ Not necessarily |
| Converse | Q → P | No | ✘ Not necessarily |
| Contrapositive | ¬Q → ¬P | Yes | ✔ Yes |

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Final Answer:

- Original: True (assumed)
- Inverse: ✘ Not necessarily true
- Converse: ✘ Not necessarily true
- Contrapositive: ✔ True (logically equivalent)

✔ Only the contrapositive must be true if the original statement is true.

So, to answer the question "Can it be true?":

- The contrapositive must be true.
- The inverse and converse may not be true — they can be false even if the original is true.

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Conclusion:

> Yes, the contrapositive can be true — in fact, it must be true if the original is true.
> But the inverse and converse do not have to be true.