How can we prove this argument is invalid?


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?

Sazzad Hissain Khan

Posted 2020-10-15T07:49:48.540

Reputation: 236

Change the first premise to "All men are mortal" and you have the counterexample: Socrates is not Swedish. – Mauro ALLEGRANZA – 2020-10-15T07:51:46.237

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 – 2020-10-15T15:10:51.030



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.


Posted 2020-10-15T07:49:48.540

Reputation: 33 575

But is it allowed to make different interpretation of Z for making a counterexample other than what I meant in the original argument? – Sazzad Hissain Khan – 2020-10-15T08:26:56.167

@SazzadHissainKhan - what is your def of valid argument ? – Mauro ALLEGRANZA – 2020-10-15T08:30:53.793

1What I know, an argument is valid when for all cases if the premises are true then the conclusion must be true. – Sazzad Hissain Khan – 2020-10-15T08:33:25.850

@SazzadHissainKhan - perfect. You have found a case when the conclusion is false. – Mauro ALLEGRANZA – 2020-10-15T08:34:47.113

Didn't get you. For that specific argument, I cannot show any single counterexample from observations where THAT conclusion is false. :( – Sazzad Hissain Khan – 2020-10-15T08:37:50.243

"Socrates is Swedish" is a different conclusion. Which is not related to my argument. – Sazzad Hissain Khan – 2020-10-15T08:39:52.657

@SazzadHissainKhan - Last try: if two examples of an argument (I mean having the same logical form) one has a false conclusion and one has a true conclusion means that we cannot satisfy the definition of valid argument: when for all cases if the premises are true then the conclusion must be true. – Mauro ALLEGRANZA – 2020-10-15T08:43:36.363

OK. I understand. Your point is as the Z is none of P or Q, we can put any class to it and thus sometimes it produces false conclusion. Hence the argument is invalid. Is that? – Sazzad Hissain Khan – 2020-10-15T08:51:06.147

@SazzadHissainKhan Here your def for valid argument is 'for all cases if the premises are true then the conclusion must be true'. So according to your premises, Socrates cannot be logically able to think. What if this is the new-born Socrates who does not have the ability to think? And what if this is the dying Socrates who has just had the poison? Now you see there are two counterexamples which do not break the premises you give but also fail to satisfy the conclusion. – Ether Lin – 2020-10-15T08:55:47.333

@EtherLin I can show you other conclusion where you can't find it false. How about a conclusion, Therefore, the Sun is hot? – Sazzad Hissain Khan – 2020-10-15T09:05:34.010

@SazzadHissainKhan To prove an argument is invalid, we only need to find a case where the premises are all true but the conclusion is false. Here your argument has no premise and one conclusion 'Therefore, the Sun is hot'. We should note that all of the conclusion must be derived from your logic rules and premises. However, the Sun is not inherently hot. To be more general, the sun will be cold when all the gas is depleted. – Ether Lin – 2020-10-15T09:17:55.930

@SazzadHissainKhan Our intuition tells us that this argument is valid because we tend to subliminally add the premise 'The Sun is hot'. Thus, the argument is valid in this way. However, as I stated, 'all of the conclusion must be derived from your logic rules and premises'. Since this premise is not indicated explicitly in your argument, we should not take it into account automatically. – Ether Lin – 2020-10-15T09:26:23.353

@EtherLin I know by intuition we can understand that the argument is invalid and a non sequitur, but I was looking for mathematical proof by contradictions. "Sun is hot" was the conclusion of the previous premises. And if you try to bring conditions like inherently hot or not, that can also be applied in all valid arguments to prove they are invalid so, it's not relevant here. Anyway, from Mauro's answer I have got the main point. Z can be interpreted to anything as there is no information provided. Thanks – Sazzad Hissain Khan – 2020-10-15T10:34:36.720

1@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? – Sazzad Hissain Khan – 2020-10-15T11:51:15.263