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I'm new to logic and I'm trying to figure out the correct way to make a counterexample to an argument.

Let's say that I'm given an argument

(A)

no cats are dogs

no dogs are cows


/ no cats are cows

(the "/" symbol means "therefore" and designates the conclusion)

My understanding is I would first convert it to its argument form. The rules for this are that identical descriptive terms must be assigned the same variable and no distinct descriptive terms can share the same variable. This means I would get the argument form:

(F)

no X are Y

no Y are Z


/ no X are Z

Because

X = cats, Y = dogs, Z = cows

From here I would then make a substitution instance to produce a counterexample. This next one is with none of the variables sharing the same descriptive terms:

X = cats, Y = dogs, Z = felines,

(1)

no cats are dogs

no dogs are felines


/ no cats are felines

And this one is one sharing descriptive terms:

(2)

X = cats, Y = dogs, Z = cats,

no cats are dogs

no dogs are cats


/ no cats are cats

Assuming my methods are correct (if not, please let me know) my questions are:

When creating a counterexample for an argument, does the counterexample have to have its descriptive terms be distinct for each variable as in (1), or is (2) ok? What about if I am producing a counterexample for an argument form that was not directly derived from an argument? Let's pretend I created (F) arbitrarily and it didn't come from (A). Does it matter if the form is derived from an actual argument when determining if the variables have to have distinct terms assigned to them? And what's the reasoning behind your answer? Thank you.

ryang
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Justin
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 20 '22 at 18:01

1 Answers1

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Your intuitions are right. (A) and (F) are essentially the same thing, since for the purpose of evaluating the logical validity of argument (A), "cats", "dogs", and "cows" are just arbitrary symbols. For the same reason, we have no reason to expect that the symbols have distinct interpretations (i.e. they could be multiple words for the same thing, like "cats" and "felines", in which case writing "cats" in both places would be equivalent). If the given argument were valid, both of your substitution instances would be too.

Karl
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