Let's say below is the argument,

Premise 1: All men are mortal
Premise 2: Socrates is a man
Conclusion: Therefore, Socrates can think

Now, to prove an argument is invalid we usually need counterexample. What counterexample we can use in this case to prove that the argument is invalid? Any other way to prove an argument invalid?

  • Change the first premise to "All men are mortal" and you have the counterexample: Socrates is not Swedish. Oct 15 '20 at 7:51
  • Who told you that to prove an argument invalid you MUST or NEED a counter example? A counter example is SUFFICIENT but NOT NECESSARY. There are rules of categorical syllogisms that you seem to ignore or not aware of. The conclusion should NOT HAVE terms outside the premises. The conclusion MUST use terms that are not the middle terms of the premises. Your example, has language out of the blue aka random. I could just conclude anything if this were allowed. Thus there is not validity. I would be able to use true premises while the conclusion would be false.This is the definition of invalidity.
    – Logikal
    Oct 15 '20 at 15:10

In logic an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

Valid arguments must be clearly expressed by means of symbolic sentences called formulas.

The validity of an argument can be tested using the corresponding formulas: if some "interpretation" of the formal version of the argument produces true premises and false conclusion, then the argument is invalid and the interpretation provides a counter-example.

Thus, consider your argument: it has the following form:

All P are Q

s is P

Therefore s is Z.

You have provided an interpretation of the argument that replace the symbolic formulas with statements:

All men (P) are mortal (Q)

Socrates (s) is a man (P)

therefore, Socrates (s) can think (Z).

With this interpretation, the two premises are True and also the conclusion is.

But you have provided also another interpretation, where the conclusion is:

therefore, Socrates (s) is Swedish (Z).

In this case the conclusion is a False.

This is a counterexample showing that the argument form above is invalid.

If we consider the syllogistic structure of the argument, it violates the definition of syllogism:

an inferences with two premises, each of which is a categorical sentence, having exactly one term in common, and having as conclusion a categorical sentence the terms of which are just those two terms not shared by the premises. Aristotle calls the term shared by the premises the middle term and each of the other two terms in the premises an extreme.

In your example, the term Z ("think" and "Swedish") is not shared with any premise.

For more examples, you can see some introductory textbook, like e.g.:

Petr Smith, An Introduction to Formal Logic (Cambridge UP, 2020), Ch.2 Validity and soundness.

  • But is it allowed to make different interpretation of Z for making a counterexample other than what I meant in the original argument? Oct 15 '20 at 8:26
  • @SazzadHissainKhan - what is your def of valid argument ? Oct 15 '20 at 8:30
  • 1
    What I know, an argument is valid when for all cases if the premises are true then the conclusion must be true. Oct 15 '20 at 8:33
  • @SazzadHissainKhan - perfect. You have found a case when the conclusion is false. Oct 15 '20 at 8:34
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    @MauroALLEGRANZA Each of the premises has one term in common with the conclusion.. Lest say another argument, P1- all P are Q, P2- all Q are R, P3- all R are S, therefore, C- all S are P. This is still a valid argument but we see the P2- all Q are R has no common term with the conclusion C- all S are P. Is this an invalid argument then? Oct 15 '20 at 11:51

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