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All goats have a beard. Karl Marx had a beard. So, Karl Marx is a goat.

Here,

  • first premise = "All goats have a beard"
  • second premise = "Karl Marx had a beard"
  • conclusion = "Karl Marx was a goat"

Why is this deductive reasoning incorrect?

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  • 7
    Because the premises are True and the concusion in False. Commented Nov 3, 2019 at 16:49
  • 7
    To get this conclusion the first premise should be the converse:"All bearded creatures are goats". Confusing a premise with its converse is called the fallacy of affirming the consequent.
    – Conifold
    Commented Nov 3, 2019 at 17:37
  • The characterization of the reasoning as "incorrect" suggests that there was a goal to achieve validity and that a particular effort failed to achieve the goal. However, the reasoning could be designed to achieve a goal that is neither of the following: 1. To reach a conclusion that is a logical consequence of the premises; 2. To provide an example of valid, deductive reasoning Although a student may not initially be inclined to accept the reasoning, if it seems that teachers, professors, textbook writers, and other educational authorities accept the reasoning, then students may interpret beli Commented Nov 3, 2019 at 19:28
  • 2
    The middle term (beard) is undistributed. Without a distributed middle term, there is nothing that links the two premises. Check the rules of the syllogism. Commented Nov 3, 2019 at 22:08
  • This is Barbara's evil twin ... Commented Nov 3, 2019 at 22:25

5 Answers 5

5

It's incorrect because its logical form is incorrect. If we use notation similar to what's used in Tarski's world (a good program to learn the basics of first-order logic, see the lecture notes on it here, especially the first one on atomic sentences), the general form of the first two premises would be something like this:

  1. For all x, Property1(x) -> Property2(x)
  2. Property2(a)

Here, Property1 stands for "is a goat", Property2 stands for "has a beard", and "a" represents Karl Marx. The key is that these two premises do not logically imply the conclusion Property1(a), i.e. the statement that Karl Marx is a goat--the rules of inference of first-order logic (see here and here) don't give you any way to deduce the conclusion Property1(a) from the two premises. This can be seen by looking at the truth table for the material condition, where the statement "Property1(a) -> Property2(a)" will be true if the atomic sentence Property1(a) is false and the atomic sentence Property2(a) is true.

Another good way to analyze logical statements in a more abstract form is by use of Venn diagrams, where the statement "all goats have a beard" would be represented by a smaller circle representing the class of "things that are goats" lying within a larger circle representing the class of "things having beards". This makes it visually clear that there can be points that are within the circle "things having beards" which do not lie within the circle "things that are goats".

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This is, I believe, the Fallacy of the Undistributed Middle, one of the classical syllogistic fallacies.

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∀x(x∈G->x∈B); k∈b, therefore k∈G

I hope you see this is basically HOL version of affirming the consequent:

P->Q; Q, therefore P.

Every car has 4 wheel, a cart has 4 wheel; therefore, is a cart a car?

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Just because all goats have beards doesn't mean that if you are not a goat you cannot have a beard: it could be that lots of different things have beards: goats, but also pigs and salamanders. Therefore, if I tell you that Karl Marx has a beard, Karl Marx need not be a goat. Maybe Karl Marx is a salamander!

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Let's imagine the set A defined by the predicate "having a beard" and the subset B defined by "being a goat". From 1. you can infer that B is included into A. From 2. you can infer that Karl Marx is an element of A.

And that's it. With the visualization below you can clearly see that the exact position of Karl Marx into A (i.e. whether if he is an element of B) cannot be determined. enter image description here

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