And/or: Conjunctions, disjunctions and contradictions

So for we have been dealing with arguments that have singular claims in the premise, such as all men will eventually die and some dogs are vicious. But how do we deal with premises that have more than one claim? Such claims are conjunctions and disjunctions.

Conjunctions

Conjunctions are claims which use the word ‘and’ in between two (or sometimes more) claims. Here is an example:

I was born in Australia and I was raised in Australia.

This is a conjunction, because the word ‘and’ separates the two claims that are:

  1. I was born in Australia
  2. I was raised in Australia

When a conjunction is used, it is saying that both claims are true. So, for a conjunction to be true, both claims in the conjunction to be true. If it is not true that I was born in Australia, the conjunction is false. If it is not true that I was raised in Australia, the conjunction is false. In standard form, the official way to express this is:

  • 1. I was born in Australia and I was raised in Australia
  • 2. Anyone who was born and raised in Australia can call themselves Australian
  • Therefore,
  • 3. I can call myself Australian

Law of non contradiction

In philosophy, we can know that something is false if we can prove that it depends upon something that is contradictory, which is called ‘the law of non contradiction. Consider this conjunction:

I was born in Australia and I was not born in Australia.

This is a contradiction, and it can prove that the conjunction has to be false. This is because if it is true that I was born in Australia, then it is false that I was not born in Australia, and if it was true that I was not born in Australia, then it would not be true that I was born in Australia. So, one of the claims in the conjunction must be false, and since one must be false, the conjunction must be false, because that conjunction needs both claims to be true. The contradiction is the expression of a conjuction where one of the claims the the negation of the other claim, which in logical form would be:

  1. a and not a.

These kind of contradictions we would very rarely come across. However, there are other ways something can be contradictory but needs more work to make it obvious. Consider the next conjunction:

I was born in Australia and I was born in England.

We know this is contradictory, but it does not (on its own) hold the logical form of ‘a and not a’, it is actually the form ‘a and b’. But some extra work, using conditionals, can demonstrate that this does involve the contradiction of ‘a and not a’, which is:

  1. If I was born in Australia then I was not born in England
  2. I was born in Australia
  3. Therefore,
  4. I was not born in England

And also,

  1. If I was born in England then I was not born in Australia
  2. I was born in England
  3. Therefore,
  4. I was not born in Australia

So now it can be shown that if it is true that I was born in Australia then I was not born in England, and vice versa, then it can be shown that acceptance of one of the claims necessarily requires the negation of that claim, so it would be the original expression of ‘a and not a’.

There is another way contradictions can come about, consider this conjuction:

Sam is taller than Mary and Mary is taller than Josh and Josh is taller than Sam

So, lets look at each claim separately:

  1. Sam is taller than Mary
  2. Mary is taller than Josh
  3. Josh is taller than Sam

If it is the case that Sam is taller than Mary, then anyone Mary is taller than would have to be shorter than Sam, since Sam is taller than Mary. But since the last claim in the conjunction says that Josh is taller than Sam, this is the contradiction because if Sam is taller than Mary, and Mary is taller than Josh, then it would have to be the case that Josh is not taller than Sam. Once again, the conditionals bear this out when we consider what is implied by being ‘taller’.

Disjunctions

Disjunctions are when we use the word ‘or’ to separate two claims, but in this case it does not necessarily need to have both claims to be true. Take this example:

I can go to the movies this weekend or I can go the Aquarium this weekend.

For this to be true, there are three possibilities: First, if I can go to the movies but can’t go to the Aquarium, it is true, because it is not saying that I can do both, just that I can do either. Second, if I can go to the Aquarium but can’t go to the movies, it is true for the same reason. Obviously if I can’t do either then the disjunction is false because I had no option available. Thirdly, if I can go to the movies and go to the Aquarium on the weekend, the disjunction is true because both options are available.

The third way just mentioned has some controversy. This is called the ‘inclusive’ treatment of the word ‘or’, where both can be true. Another is the ‘exclusive’ treatment, which treats ‘or’ as ‘one or the other, but not both’. Most philosophers use it in the inclusive sense. I personally think that there are legitimate uses of it in an exclusive sense. If I went on a plane, and the flight attendent asked if I wanted ‘chicken or fish’ for dinner, and I said ‘both please’, the attendent’s reaction would be of confusion. However, if I were to order desert at a resturant, and my choices were ‘cake or ice cream’, and I was feeling rather gluttonous, there would be nothing to stop me from ordering both. So, this would be an inclusive treatment of the disjunction.

A way we use a disjunction in argument can be to come to a conclusion when we know the disjunction is true but one of the claims is false. For example:

  1. I am at work or I am at home
  2. I am not at work
  3. Therefore,
  4. I am at home

This is valid, because if it is true that I am at work or I am at home, and it is true that I am not at work, it would have to be the case that I am at home. This is due to that the if the disjunction is true, I cannot be both not at work and not at home. Hence, if it is true I am not at work, the only possibility left is that I am at home.

Soundness: False dichotomy

When checking for soundness, we can reject a disjunction by showing either that both claims are false (or both claims are true if using it exclusively), or, that there are more options available than what is in the premise. For instance, you could say that ‘I am at work or I am at home’ is false by saying that I could be out playing sport, or at the local Pub, and so on. In philosophy, this is given a special fallacy name called ‘false dichotomy’. However, I dislike using this name because very often someone yell it out when they hear a disjunctive premise without really establishing whether the dichotomy really is false, so I tend to not use it.

Test questions

  1. Men and women are humans. I am a man. So, I am human (sound or unsound?)
  2. You can eat at McDonalds or eat at Subway for lunch. You cannot eat at McDonalds. Thus, you must eat at Subway. (valid or invalid?)
  3. I lift heavier weights than Andrew and Andrew lifts heavier weights than Ken, and Ken lifts heavier weights than me. (true or false?)

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.

Published by

Andrew Tulloch

I have a Bachelor of Arts majoring in Philosophy and Sociology, with a Political Science minor. I am currently completing my honours degree in Philosophy.

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