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According to wikipedia faulty generalization belongs to the class of informal fallacies. Also, a fallacy is called informal if it

originates in a reasoning error other than a flaw in the logical form of the argument.

But I'm wondering why is it assumed it cannot be formalized. Isn't the following proposition in FOL a formalization of faulty generalization?

∃x: F(x) ∴ ∀x: F(x)

Here F(x) is an arbitrary function where a variable x is involved, but other variables can be used as well. An informal example would be: "There are sighted people. Therefore, all people are sighted." In formal way this is ∃x: Px → Sx ∴ ∀x: Px → Sx. Px means "x belongs to people" and Sx means "x is sighted".

The replacement of a quantifier is a formal fallacy, isn't it? So, why is faulty generalization assumed to be an informal fallacy?

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    The FOL sequent is not well-formed. Did you maybe mean: ∃x(Px→Qx) .: ∀x(Px→Qx)? Or even more simply ∃xPx .: ∀xPx? Both are invalid, of course, but I’m not sure either is a good formalisation of Faulty Generalisation. Usually, when such a generalisation is made, it starts from specific instances and not just an existential quantification, no? So, perhaps we could try: Pa, Pb, Pc .: ∀xPx - which is still invalid. – MarkOxford Jul 28 '18 at 9:07
  • The formalisation of ‘There are sighted people’ is ∃x(Px ∧ Sx), with ∧ instead of →. ∃x(Px → Sx) says that there is something that is S if it is P. This sentence is true in all structures where nothing is P. – MarkOxford Jul 28 '18 at 12:27
  • @MarxOxford, at first I thought so, but then it would mean "Any x is a sighted human", not the one I want to have. I want to have "Every x who is human is sighted". The truth table of the implication is appropriate one. – rus9384 Jul 28 '18 at 12:51
  • I was talking about the premise, “There are sighted people”. The conclusion, “All people are sighted”, must be formalised with → (and ∀), as you did. – MarkOxford Jul 28 '18 at 12:56
  • @MarkOxford, I'm not sure why we can't use implication. "Some x who are humans are sighted" which cannot be reversed. If we put ∧ we get "Some x who are humans are sighted and some x who are sighted are humans", which is irrelevant. – rus9384 Jul 28 '18 at 13:05
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The reason faulty generalization is an informal fallacy is contained in the very link you provide. The fallacy lies in insufficient empirical evidence, but the same form of argument could become convincing and cogent with enough empirical evidence. For example, suppose someone argues, "I've taken two cab rides and both of the times the driver was rude. Therefore, most cab drivers are rude." The reasoner has gone wrong because two instances is not enough to draw a general conclusion about most cab drivers. But if you gathered enough empirical evidence you could make the argument convincing. By contrast, formal fallacies are invalid and could never become valid no matter what empirical evidence exists.

The formalization you have provided is in fact deductively invalid. But that is not the form of faulty generalization. Faulty generalization has the form of an inductive argument which gathers a number of instances and draws a general conclusion. Consider again the example, "I've taken two cab rides and on both the driver was rude, therefore most cab drivers are rude." This is a faulty generalization, but it doesn't have the form you have given, since it is a conclusion about "most" cab drivers and not "all." Where the reasoning has gone wrong has to do with the fact that the reasoner does not have enough empirical instances to draw her conclusion, but if she had enough instances she could be warranted in drawing the conclusion.

By contrast, a formal fallacy consists in a form of argument that could never be sound or cogent no matter the empirical evidence. For example, if someone says, "All birds are animals. This is an animal. Therefore, it is a bird." No abundance of empirical evidence could make this a sound argument because it is formally invalid.

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