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All teachers are intelligent.

All teachers are well-paid.

From the Star Test, we can deduce that the argument must be invalid with whatever conclusion (according to the classical syllogism figures), because

All T* is I.

All T* is W.

Here T is double-starred, so no conclusion follows.

But I find the conclusion

∴Some I is W

is also reasonable. So there are some limitations to the Star Test. How can we specify them?

Conifold
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    "we can deduce that this argument must be invalid" *What* argument? All you've done so far is list two hypotheses ("All teachers are intelligent" and "All teachers are well-paid"); to be an argument you *also* need a claimed conclusion (e.g. "All intelligent people are well-paid" - which would *not* be a valid argument, of course). – Noah Schweber Jul 13 '20 at 05:58
  • That is to say, no conclusion can follow. Whatever the conclusion is, the argument is invalid. –  Jul 13 '20 at 06:10
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    That's not the case. We can always draw *some* conclusion - e.g. the conjunction of the hypotheses themselves. There is no set of hypotheses which does not let us draw any conclusion at all. If nothing else, we could simply look at an argument which ignores the hypotheses entirely and just derives a tautology. The star test is about certain types of argument in particular, not all arguments in general. – Noah Schweber Jul 13 '20 at 06:13
  • However, the star test is a certain technique used in syllogisms, and every sentence is wff. No rule is break. –  Jul 13 '20 at 06:30
  • And it is possible that in the syllogisms no conclusion follows. –  Jul 13 '20 at 06:31
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    It is not possible that literally "no conclusion follows", one can simply repeat one of the premises, and that will be a valid conclusion. What is meant is that no "meaningful" conclusion follows by applying the valid figures of syllogism. *Your* conclusion does not follow unless we ascribe [existential import](https://philosophy.stackexchange.com/a/55972/9148) to "All", as it requires that teachers exist. If we do ascribe existential import then yours *is* a valid figure, but the star test has to be adjusted. – Conifold Jul 13 '20 at 07:13
  • So is the statement 'some I is W' true without applying the star test, suppose the teachers do exist? –  Jul 13 '20 at 07:18
  • With existential import, yes, the figure is called [*Darapti*](https://philosophy.stackexchange.com/a/21895/9148). – Conifold Jul 13 '20 at 07:21
  • In modern logic, we always suppose there is the existential import, so this conclusion must be true in the modern view. What confuses me is that the result of the star test says that it cannot follow a meaningful conclusion. And is the existential import a necessary condition for the star test? –  Jul 13 '20 at 07:35
  • In modern logic we *never* ascribe existential import to "All", that is how translation into quantifiers works, see the *Darapti* link. A necessary condition for the star test to work is the *lack* of existential import. – Conifold Jul 13 '20 at 07:41
  • So since we don't need to have existential import in the star test, why does the reasoning fail in this circumstance? I think those who are true in modern logic must be true in the star test. –  Jul 13 '20 at 07:53
  • But it does not fail. *Darapti* is invalid just as the star test predicted. If there are no teachers then "some intelligent people are well paid" does not follow. – Conifold Jul 13 '20 at 08:01
  • You can see Gensler's [original article](https://projecteuclid.org/euclid.ndjfl/1093891100). You have to consider the well-known issue of [Existential import](https://en.wikipedia.org/wiki/Syllogism#Existential_import). – Mauro ALLEGRANZA Jul 13 '20 at 08:06
  • According to Aristotle's original theory, [Darapti](https://en.wikipedia.org/wiki/Syllogism#Darapti_(AAI-3)) is a valid figure, But Gensler's method is designed to test a categorical syllogism **on the modern interpretation**. – Mauro ALLEGRANZA Jul 13 '20 at 08:08

1 Answers1

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Gensler's star test is a simplified method for determining the validity of a syllogism proposed in 1973. According to the test, one stars (asterisks) the first (capital) letter after "All", and all letters after "not" or "No". The syllogism is valid if and only if every capital letter is starred exactly once and there is exactly one star on the right-hand side.

The test has limitations, however. It only applies to categorical syllogisms, and even to those only if we make a modern assumption that "All" does not carry existential import. In other words, "All A are B" does not imply that there are any A-s. But the OP example uses the Darapti syllogism form: All A are B, all A are C, therefore, some B are C, which is only valid with existential import, see Why is the darapti syllogism invalid? Indeed, if there are no teachers we can not conclude that some intelligent people are well paid based on the OP premises. So the Darapti form is invalid, just as the star test predicts.

In modern logic we translate "All A are B" into ∀x(A(x) → B(x)). If there are no A-s then A(x) is always false, hence the implication is always true, and so is the quantified statement. In other words, modern logic does not give existential import to "All", and Gensler's test adopts this modern interpretation. It was different in classical times, Aristotle himself considered Darapti a valid form.

If we go with Aristotle, Darapti is not the only example where the star test fails. The same applies to some other forms of the third and fourth figure. Yildirim gives a detailed analysis of the star test and limits of its validity in Gensler's Star Test and Some Examples of its Application:

"Gensler’s star test is a useful and functional method for checking the validity of syllogisms. But we should note that this method is useless in the conditional or the hypothetical syllogism, compound syllogisms such as the compound conditional or the compound syllogism «involving a contradiction» (qıyas al-khalf), and Darapti or Felapton types of syllogisms which are third figure (Middle terms of premises are subjects of the premises). And also we can add Fesapo or Bramantip types of syllogisms of fourth figure, enthymeme, etc."

An alternative to Gensler's test is the use of Venn diagrams. Grennan describes how to modify their use when the existential import for "All" is assumed in Informal Logic, p.119.

Conifold
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  • Although some concepts do confuse me, I can now understand this difference. Many thanks! –  Jul 13 '20 at 08:28