‘All’, ‘none’, ‘only’ and ‘some’ arguments.

Have a look at these arguments:

Every triangle has exactly three sides. This is a triangle, so it must have exactly three sides.

All bachelors are unmarried men. This is a Bachelor, so he must be an unmarried man.

These arguments are universal arguments, because they depend on universal premises concerning properties something possesses. For instance, being a bachelor universally has the property of being an unmarried man, and a triangle universally has the property of having exactly three sides.

Consider the next two arguments:

No bachelors are married. This man is married. Thus, he is not a bachelor.

No triangle has four sides. This shape has four sides. Hence, this is not a triangle.

These arguments are called negations. In this case, what is being asserted is that that certain things contain no particular properties. For instance, being married or having four sides.

Now on to the next arguments:

Only dogs eat dog food. My cat is not a dog. So, my cat does not eat dog food.

Only Astronauts can go into outer space. I am not an Astronaut. Hence, I can’t go into outer space.

These arguments which contain the word ‘only’, are saying that if only one kind of thing possesses a particular property, such as dogs having the property of eating dog food, then anything that is not that kind of thing does not possess that property. I like to call these ‘constriants’

And finally, consider these arguments:

Some meat is poisonous. Meat is food. Therefore, some food is poisonous.

This dog is vicious, so some dogs are vicious

These arguments which include the word ‘some’ are called existentials. This is where there is at least one kind of thing that possesses a particular property. In the above case, the property of being posionous or vicious. Please note, in philosophy, the word ‘some’ does not mean ‘some but not all’ or ‘more than one but not all’. It means ‘at least one’.

These are styles of a kind of reasoning called ‘cateogorical syllogisms’, because they all involve contrasting of different categories with the properties they possess.

Valid and invalid universals

Our first argument in our first tutorial involving humans eventually dying is an example of a valid universal, which was:

• 1. All humans will eventually die
• 2. I am human
• Therefore,
• 3. I’ll eventually die

We have established in earlier tutorials that this is valid, because if all/every kind of thing possesses some kind of certain property, then if something is that kind of thing then it would have to possess that property. In this case, the property of eventually dying. However, we also established that just possessing the property of eventually dying does not allow us to conclude that we are that kind of thing, which is being human. Hence, other valid universals can be:

• 1. All humans will eventually die
• 2. I will not eventually die
• Therefore,
• 3. I am not human

This is valid because if it is true that all humans will eventually die, then there are no humans that will not eventually die. So, if it is true that I do not share the property of eventually dying, then it has to be the case that I am not human.

Valid and invalid negations

Consider the negation in the argument raised earlier:

• 1. No bachelors are married
• 2. This man is married
• Therefore,
• 3. This man is not a bachelor

This argument is valid, it is valid because the negation is saying that there is not a single bachelor that is married. So, if there is a man who is married, it must be the case that he is not a bachelor, since no bachelors are married. Logical form of such arguments would be:

• 1. No a are b.
• 2. B
• Therefore,
• 3. Not a

Now consider the next argument:

• No bachelors are married
• I am not married
• Therefore,
• I am a bachelor

This is invalid, because from the argument alone, just knowing that no bachelors are married and I am not married, does not mean I must be a bachelor. I could be an unmarried woman, and unmarried women are not bachelors. Unmarried men are bachelors. So, this invalid logical form looks like this:

• 1. No a are b
• 2. Not b
• Therefore,
• 3. A

Valid and invalid constraints

Let’s look at this argument:

• Only dogs eat dog food.
• My cat is not a dog
• Therefore,
• My cat does not eat dog food

This argument is valid, because if it is true that only dogs eat dog food, then anything that is not a dog does not eat dog food. And since my cat is not a dog, it would have to be the case that my cat does not eat dog food. So, this valid logical form of constraints would be:

• 1. Only a are b
• 2. Not a
• Therefore,
• 3. Not b

Now let’s look at the next argument:

• Only dogs eat dog food
• This is a dog
• Therefore,
• This dog eats dog food

This is invalid, because even if it is true that only dogs eat dog food, it does not mean that all dogs eat dog food. This dog could be a wild dog that hunts for its food, or has pet owners that feeds it scraps rather than dog food. All we can say about constraints is that only one kind of thing possesses a certain property, not that all of those kind of thing possess that property. The logical form of this invalid argument is:

• 1. Only a are b
• 2. A
• Therefore,
• 3. B

Valid and invalid existentials

And finally, this argument:

• 1. Some meat is poisonous
• 2. Meat is food
• Therefore,
• 3. Some food is poisonous

As mentioned earlier, in philosophy, ‘some’ means at least one. So, to say that some meat is poisonous is to say that there is at least one kind of meat that is poisonous. So, if it is the case that there is at least one kind of meat that is poisonous, and meat is food, then it would have to be the case that there is at least one kind of food is poisonous. Hence, this is a valid argument. In logical form, this can be expressed as:

• 1. Some a is b
• 2. A is c
• Therefore
• 3. Some c is b

Now, this argument:

• 1. Some meat is poisonous
• Therefore,
• 2. Some meat is healthy

This argument is invalid, when we say that some meat is poisonous, which is to say that at least one kind of meat is poisonous, it could be the case that all meat is poisonous, because if it is true that all meat is poisonous, it would be also true that at least one kind of meat is poisonous. Also, it could be true that all other kinds of meat are not poisionous, but unhealthy. This invalid form of existentials is:

• 1. Some a is b
• Therefore,
• 2. Some a is c

Soundness and counterexamples

Universals

For universals, to show that the universal premise is false, is to provide a counterexample. A counterexample can be either an existing or possible case where it is not true that all of one kind of thing possess that property. For example, a counterexample to ‘all humans will eventually die’ could be that in the future we may invent technology that will make us immortal, or there may be a human hiding somewhere who has lived forever and we have never discovered them. This counterexample does reveal another dispute in philosophy called ‘the problem of induction’, but this will be discussed in later tutorials. There is some dispute on whether appealing to possible circumstances count as a counterexample. I hold the view that it does regarding being skeptical about a universal, but does not regarding rejecting the universal, because if we cannot appeal to an existing example, it is still possible that the universal is true, but if we can appeal to a possible example, we cannot know that the universal is true.

Negations

Counterexamples to negations are when we can show a single existing or possible circumstance where that kind of thing does possess that property. For instance, consider the negation ‘no dogs are vicious’. We can reject the negation by appealing to a situtation where a dog was vicious. Regarding possible examples and real examples, I hold the same view as universals.

Constraints

Counterexample to constraints are when we can appeal to a single existing or possible example where there are other things that also possess a particular property that the original thing also possesses. Take for example ‘only heroes wear masks’. We can show this if false by demonstrating that villians also wear masks.

Existentials

To reject an existential is to say that there are absolutely none of those kinds of things that possess that property. So, to say that ‘some meat is poisonous’ is false, is to say that there are no meats that are poisionous.

Test questions

Try out these test questions and try to see if they are sound. Furthermore, to really test yourself try to see which ones use premises that are universals, negations, constraints, or existentials.

1: All cars produce carbon emissions. This produces carbon emissions. Therefore, this is a car.

2: Some Christians eat won’t eat red meat on Fridays. Christians are religious people. Therefore, some religious people won’t eat red meat on Fridays.

3. No pets are harmed by their owners. Hence, this pet has not been harmed by its owner.

4: Only police can arrest people on suspicion. This is man is not a police officer, so he cannot arrest me on suspicion.

If you wish to ask any questions, seek clarification, raise some objections, or check how you went on the test questions, please write them in the comments section and I will try respond as soon as I can.

I highly recommend purchasing the book ‘Understanding arguments’ by Walter Sinnott-Armstrong and Robert Fogelin, which is available for purchase in the link below. If you do purchase the book via this link, you are helping support this webpage. Thank you.