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Children behave if they are not given candy.

Angela is a child who is given candy.

Therefore, Angela will misbehave.

I don't think it's true nor false because there's not enough information given.

I tried using modus tollens and got,

If children misbehave, they are given candy.

She was probably given candy, but not necessarily. Could it be there are other reasons why Angela is misbehaving?

Community
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Jossie Calderon
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4 Answers4

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Your job is not to decide whether the conclusion is true; it might not be, as some premise could be false. Your job is to tell whether the argument is valid. As you suspected, it isn't. This is more obvious if reword the first premise as, "If children aren't given candy, they behave". (This isn't even using a contrapositive, or anything like that; we're just using the usual if-before-then structure logicians like.) In other words, we're told nothing about what happens if Angela is given candy.

J.G.
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From a strictly logical viewpoint we only make a statement about children who are not given candy. We don't know what children who are given candy will or will not do. Therefore the deduction that Angela will misbehave is wrong.

That doesn't mean that Angela will behave though. As you said, there is not enough information.

T. Som
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Let $C = \{\text{The set of children that are not given candy.}\}$

We know a property of set $C$: $$\forall c \in C: \quad c \text{ will behave.}$$

We also know that $\text{Angela} \notin C$.

Therefore, we do not have information about Angela behaving or misbehaving.

We can easily find another mathematical example where such a thing holds:

Let $C := \{\text{The set of primes that are } \ge 3\}$

We know a property of set $C$:

$$\forall c \in C: \quad c \text{ is odd.}$$

Take a number (Angela) or $a$, that is not part of $C$. $(a \notin C)$

We have no information about whether or not $a$ is even or odd, similarly to having no information about Angela behaving or misbehaving.

Daniel P
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In a (hypothetical) universe where all children behave, the first premise is true, nothing logically prevents the second premise from being true as well, but the conclusion is false.

Hagen von Eitzen
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