Valid and invalid arguments

When we give arguments, the reasons, or premises we give need to support the conclusion. Or in other words, we need to know that the conclusion is true. The way we can see if the conclusion is true, is to check to see if the argument is valid. If the argument is invalid, then we cannot know the conclusion.

An argument is valid when if all of the premises are true, then the conclusion has to be true as well. There cannot be any circumstance where all the premises are true, but the conclusion turns out to be false. It should be noted when testing an argument’s validity, we do not need to know whether the premises are actually true (that comes later!), we only need to know if the premises are true, then the conclusion has to be true as well.

Let’s consider the argument mentioned in the first post explaining arguments:

  • All humans will eventually die
  • I am a human
  • Therefore,
  • I’ll eventually die

This argument is valid. It is valid because there is no possible circumstance where if it is true that I am human, and if it is also true that all humans will eventually die, that there will be a case where I will not eventually die. If I claimed to be something immortal, like an angel, then the premise ‘I am a human’ would be false. If I claimed that I had some special gene stopping me from ever dying, then the premise ‘All humans will eventually die’ would be false.

There is another way of seeing whether an argument is valid, which is to put the argument into logical form. Logical form is when we remove the language used of things possessing certain properties in standard form and replace with a letter to represent what said property; similar to algebra but with language. This would be the previous argument in logical form:

  • All x are y
  • X
  • Therefore,
  • Y

X represents the property of being human. Y represents the property of eventually dying. Here it can be more clearly seen that if all things that are x are also y, then if that thing is an x, it must also be a y.

Since we can look at the logic of the argument alone to test validity, an argument can have obviously false premises, even a false conclusion, yet still be valid. Take a look at this argument:

  • All vampires are skateboards
  • I am a vampire
  • Therefore,
  • I am a skateboard

Clearly neither premise is true and the conclusion isn’t true either. Nonetheless, the argument is valid because it has the same logical form as the previous argument.

Now we know how to know whether an argument is vaild, we can also see how it can be invalid, which is by showing how even if all the premises are true, the conclusion could be false. Consider the first argument said slightly differently:

  • All humans will eventually die
  • I will eventually die
  • Therefore,
  • I am a human

Both premises are true, and the conclusion is true, but the argument is actually invalid. This is because there are things other than humans that will eventually die, such as animals and plants. And this how you can demonstrate an argument’s invalidity, which is called a counterexample. Logical form can also help, which would be:

  • All x are y
  • Y
  • Therefore,
  • X

As can be seen, all we can say from this that all things that are x are also y, not that all things y are x. So, this does not make conclusion x necessarily true. Test yourself on the arguments below to see whether they are valid or invalid:

1: All motorcycle riders are criminals. This guy is a criminal. So, he must be a motorcycle rider.

2: The dog across the street is vicious. So, some dogs are vicious. (In philosophy ‘some’ means ‘at least one’)

3: I can’t find my phone. Therefore, it must have been stolen!

4: Only cats can ride bicycles. Humans are not cats. Hence, humans cannot ride bicycles.

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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