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Here are two inferences A and B, one of which is valid, but the other not valid.

Inference A

  1. Every human is mortal.
  2. Socrates is human.
  3. Therefore Socrates is mortal.

Inference B

  1. Every hero likes women.
  2. Taro likes women.
  3. Therefore Taro is a hero.

Which one is correct, and in what sense it is correct ?

Can somebody give the hints to solve it ?

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    $\begingroup$ Which one seems to make sense? $\endgroup$ – MJD May 21 '13 at 16:07
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    $\begingroup$ I watched Taro flee in fright from a horde of Mongolians. That's no hero! $\endgroup$ – Asaf Karagila May 21 '13 at 16:21
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    $\begingroup$ He sure does love the ladies though! $\endgroup$ – MJD May 21 '13 at 16:27
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    $\begingroup$ A general hint for this kind of problem is that if the inference is valid, then it must hold even if you replace all the names with other names (e.g. with "user77788" or with "Richard Nixon") $\endgroup$ – Trevor Wilson May 21 '13 at 17:09
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    $\begingroup$ @Trevor: But what if someone thinks of Nixon as their hero? $\endgroup$ – Asaf Karagila May 21 '13 at 17:45
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Draw a circle. Draw another circle, inside the first. Draw a point inside one of the circles and see if being inside one the circles implies being inside the other.

The difference is like between necessary and sufficient conditions. Being a human is sufficient to be mortal. Being mortal is necessary but not sufficient to be human. Similarly, being a lover is necessary for being a hero but not sufficient.

In probability, this fallacy is know as Confusion of inverse

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  • $\begingroup$ ohh!!! i see ! Thank you very much $\endgroup$ – user77788 May 21 '13 at 16:13
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    $\begingroup$ Yes, A -> B means that A is inside B. The set of humans is a subset of mortals. Socrates is a point inside humans. Being Socrates -> being in the ring of humen -> being inside mortals. Heroes are inside women lovers. But, being inside a big ring of human lovers does not imply that you are inside a smaller ring of heroes. $\endgroup$ – Val May 21 '13 at 16:22
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Hint: Is everyone who likes women necessarily a hero?

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Hint

Inference B

$\varphi(x)$ mean $x$ likes women

$Hero(p)$ means $p$ is a hero

1) $\forall p:Hero(p)\rightarrow \varphi(p)$

2) $\varphi(taro)$

3) $\varphi(taro)\rightarrow Hero(taro)$

So the second inference is

$(P\rightarrow Q)\rightarrow (Q\rightarrow P) $

Is this always true?

Inference A

But when we talk about socrate we have a different thing:

$\varphi(x)$ mean $x$ is mortal

$Human(p)$ means $p$ is a human

1) $\forall p:Human(p)\rightarrow \varphi(x)$

2) $Human(socrate)$

3) $Human(socrate)\rightarrow \varphi(socrate)$

that is very different.

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