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If the argument

(1) Some dogs are carnivores.
(2) Some dogs are mortal.
(c) Some dogs that are carnivores are mortal.

is given, do those 'some dogs' refer to two different groups (dogs might or might not be carnivores and mortal at the same time) or the same (all of the 'some dogs' are automatically both carnivores and mortal). Or more simply, is the statement valid?

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    Compare with : (1) Some numbers are odd. (2) Some numbers are even. (therefore ???) Some numbers that are odd are even. Oct 26, 2016 at 18:22
  • Search the term plural quantification perhaps helps to clarify... Jun 11, 2022 at 3:38

5 Answers 5

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The deduction is not valid.

The modern translation of 'Some X are...' is 'There exists an (meaning at least one) X such that...'

So, in modern parlance, you are saying: There exists a vegetarian dog. There exists a carnivorous dog. Therefore, there exists a vegetarian, carnivorous dog.

The latter statement is false, since a being cannot be both vegetarian and carnivorous. But the premises are both true. (Sadly so, for the poor vegetarian dog.) Any syllogistic form with a real counterexample can't be valid. So this one is not a reliable form.

And, yes, those three phrases all refer to different groups, represented by different choices that satisfy the 'exists' statement. You know the first two don't coincide because the dog that represents one group cannot be in the other, and therefore the third group is empty.

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    sorry, i think this is wrong. the reason the inference fails is because of the form of the syllogism, not because of the meanings of the terms. "Some dogs are that are carnivores are vegetarians" is an invalid conclusion. it does not follow that no dogs are carnivorous vegetarians. just replace "carnivores" and "vegetarians" with X and Y and it becomes obvious.
    – user20153
    Oct 26, 2016 at 19:43
  • Paragraph 4, I read it as stating "the conclusion is false due to the fact that the words carnivorous and vegetarian refer to disjoint sets" presumably by virtue of their meaning; whether or not this is true is irrelevant for the logical structure of the OP's question.
    – Dave
    Oct 27, 2016 at 19:20
  • Ok, then since several readers miss that point, it might be better if you made it explicit in the main body.
    – Dave
    Oct 27, 2016 at 19:46
  • @Dave I have now gone out of my way to state the obvious, for those who want to read more into the statement than it can possibly actually be taken to mean by any careful reader.
    – user9166
    Oct 27, 2016 at 20:20
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The argument is:

(1) Some dogs are carnivores.

(2) Some dogs are mortal.

(therefore) Some dogs that are carnivores are mortal.

No valid conclusion follows; neither premise distributes the middle term (dogs). In a valid syllogism, the middle term must be distributed in at least one premise.

Note also that because "dog" is the middle term, it is going to disappear in the conclusion. Because the predicate of the conclusion is in the major premise, and the subject is in the minor premise, the conclusion is, “Some mortals are carnivores.”

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"Some" just means a sub-set with at least one member.

If "all dogs" are thought of visually (like a Venn diagram) as a circle, then "some dogs that are carnivorous" would be a smaller circle inside the "all dogs" circle. Likewise, "some dogs that are vegetarian" would also be a smaller circle inside the "all dogs" circle.

Do the two smaller circles inside the "all dogs" circle overlap?

Well, if we take carnivorous to mean "eats meat" and vegetarian to mean "eats vegetables" in an exclusive sense of "eats only meat" and "eats only vegetables" then they do not overlap. If we take carnivorous and vegetarian to describe the predominant diet, then they could overlap at different times of at least one dogs life, but this is stretching things a bit far to make a contradiction plausible. Also, where they might overlap, would we not say the dog is omnivorous?

As for whether or not each premise is true, there is significant evidence supporting the claim that some dogs are indeed carnivorous. While dogs are generally considered to be omnivorous, I don't think there are many considered vegetarian, tho it is certainly arguable that there is at least one instance of a vegetarian dog somewhere.

So, if we take premise one to be true and premise two to be true, are we correct in deducing that there is a dog that is both vegetarian and carnivorous? In the exclusive sense of "only eats meat" or "only eats vegetables" this is obviously a contradiction, so no, the two smaller circles do not overlap. In the loose sense of "predominant diet" then the truth value is still relative to the time of evaluation and predominance implies that one diet or the other is the main source of nutrition, so, again, no: there are no dogs that are both carnivorous and vegetarian.

Of note, exclusively vegetarian diets are not at all a good idea for your doggy. Please talk to a licensed veterinarian for nutritional guidelines and breed risk assessment if you are feeding your dog like so.

As for logic, if you dig this kind of stuff, check out Logic-Cola by Harry Gensler - it is a fun companion to his book, Introduction to Logic.

Let's look at your initial premises:

Premise 1) Some X are A (i.e. at least one X is A)

Premise 2) Some X are B (i.e. at least one X is B)

Here I think you might want an additional step, stating the relationship of A and B.

Premise 3) All A are not B (i.e. No A is B)

Therefore your

Conclusion) Some X are A and B (i.e. at least one X is A and B)

is false because it contradicts the third premise.

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the reasoning is fallacious. if you want to know why, google "term logic", and pay attention to the "square of opposition" and the 4 types of proposition (A, I, E O). For a contemporary treatment search for Fred Sommers.

the point being that "some" has no place in modern logic. it comes from traditional Aristotlean (i.e. term) logic.

see section 4.3 of http://plato.stanford.edu/entries/aristotle-logic/

square of opposition: http://plato.stanford.edu/entries/square/

note that Aristotlean "some" is not equivalent to modern existential quantification. there is no quantification in Aristotlean logic. of course we can come up with quantificational representations of Aristotlean reasoning, but it does not follow that his logic was quantificational.

Clarification

It's important to be clear about the differences between logical validity, truth, and the meanings of terms. Syllogisms (inferences, arguments) may be valid, but true and false to not apply to them. Similarly statements may be true or false, but neither valid nor invalid. In determining the validity of an argument, only its form is taken into consideration, not the meanings of its extra-logical terms like "dog" and "carnivorous". If a syllogism is invalid, its conclusion may yet be either true or false. Similarly, a syllogism could be valid, yet its conclusion still either true or false, as in the case where a premise is false but the reasoning is valid. If the premises are true and the reasoning is valid then the conclusion must be true (this is often called a "sound" argument to distinguish it from a merely valid one.)

So your premises are probably true (that's for empirical science to determine), but the syllogism is invalid, so the conclusion - "Some dogs that are carnivores are vegetarians" - may be either true or false as a matter of logic, where the meanings of "dogs", "carnivores", and "vegetarians" are irrelevant. It may be false as a matter of science, based on the meanings of those terms, but that is a separate question that has nothing to do with logic.

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  • There is no reference to Aristotle in the OP, and 'Some' is a word in the English language that is reasonably represented via quantification. So this technical rant is totally irrelevant to the question.
    – user9166
    Oct 27, 2016 at 20:56
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What exactly does 'Some' mean in Logic?

The expression "some dogs" refers to at least one dog or possibly to all dogs.

It does so without specifying which particular dogs.

The dogs in the second premise are possibly not the same as the dogs in the first premise.

Some dogs are carnivores;

Some dogs are mortal;

So, some dogs that are carnivores are mortal.

This obviously is not valid. The conclusion is possibly false. Whether it is or not, This falsifies the argument.

Interestingly, we can make a similar argument with Augustus De Morgan's syllogistic form:

Most Greeks like feta;

Most Greeks like ouzo;

So, some Greek likes feta and ouzo.

"Most Greeks" does not specify which Greeks, so it is similar in this respect to Like "some Greeks".

And the Greeks in the second premise are not necessarily the same as the Greeks in the first premise.

Yet, unlike your dog argument, this one is valid.

Obviously valid.

Congratulation to Augustus.

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