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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?

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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 at 5:58
  • That is to say, no conclusion can follow. Whatever the conclusion is, the argument is invalid. – Ether Lin Jul 13 at 6:10
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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 at 6:13
  • However, the star test is a certain technique used in syllogisms, and every sentence is wff. No rule is break. – Ether Lin Jul 13 at 6:30
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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 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 at 7:13
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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.

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  • Although some concepts do confuse me, I can now understand this difference. Many thanks! – Ether Lin Jul 13 at 8:28

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