Introduction to Propositional Logic – 11/07/2016

One of the many sub-disciplines of philosophy is logic. Logic is concerned with the question: when is an argument valid? (Of course, ‘argument’ means ‘inference’ here, and not ‘fight’. Philosophers don’t care whether a fall-out between partners is valid; they care whether their own reasoning is valid.)

I will here discuss some valid types of argument as well as some invalid types of argument, and how you can distinguish the valid from the invalid types.

Valid arguments – modus ponens and modus tollens

A valid argument is one where the conclusion must follow from the premises. In other words, if you take the premises to be true, then you must also accept the conclusion, if the argument is valid. If an argument is valid, the conclusion is a logical consequence. If it is logically possible to accept the premises, but not the conclusion, then the argument is a non sequitur (latin for: does not follow).

A non sequitur is a type of fallacy. Fallacies are invalid types of arguments. There are many variations. I will not discuss them all, but it is a good idea to familiarise yourself with the most common fallacies. Being able to identify fallacies protects you from being lured into other people’s poor reasoning. I will say more about fallacies later.

There are several types of valid arguments. Here are some examples:

 

Example 1.

p1. If Molly drinks too much, she’ll get a headache.

p2. Molly drinks too much

C. Molly will get a headache.

 

Example 2.

p1. If Molly drinks too much, she’ll get a headache.

p2. Molly does not have a headache.

C. Molly didn’t drink too much.

 

Example 1 is an inference of the form modus ponens (latin for: the way that affirms), also known as confirming the consequence. We can write such an argument down schematically as follows:

p1. P → Q

p2. P

C. Q

The letters P and Q can be replaced by a proposition, such as ‘Molly drinks too much’ or ‘Molly will get a headache’. A modus ponens is an inference used in propositional logic. The arrow is a connector sign used in logic that means ‘if… then’. So P → Q means ‘if P, then Q’. Note that the arrow doesn’t point both ways, so ‘if P, then Q’ does not imply ‘if Q, than P’. In this case, not all Molly’s headaches are caused by drinking too much, but drinking too much will always cause headaches in Molly.

Example 2 is an inference of the form modus tollens (Latin for: the way that denies), also known as denying the consequence. This can be schematically written down as:

p1. P → Q

p2. ¬Q

C. ¬P

This, too, belongs to the field of propositional logic, since P and Q can be replaced by propositions. The sign ¬ means ‘it is not the case that’, or in short ‘not’. It denies whichever proposition follows it, or, when brackets are used, it can deny the relation between two or more propositions, like ¬(P→Q). In this case, Q means ‘Molly will have a headache’, so ¬Q means ‘not Q’ or ‘It is not the case that Molly will have a headache’. You can now see why the conclusion must follow from the premises: given that p1 means that whenever it is the case that Molly drinks too much, the consequence is that she’ll get a headache (so there is no case in which Molly can drink too much but get away without a headache), and p2 means that she won’t have a headache, it is impossible that she drank too much. There is no rational or valid way to accept p1 and p2, but not the conclusion.

Whichever propositions you choose to replace P and Q with, modus ponens and modus tollens are always valid arguments.

Invalid arguments – two related fallacies.

It is easy to make a mistake in reasoning. Common mistakes are two fallacies which relate to modus ponens and modus tollens, but it is important to understand why they are invalid.

Here’s an example of the fallacy known as affirming the consequent:

p1. If Molly drinks too much, she’ll get a headache.

p2. Molly has a headache

C. Molly drank too much.

 

Or schematically put:

p1. P → Q

p2. Q

C. P

As noted above, the arrow in P → Q goes only one way. Molly always gets headaches after drinking too much, but that doesn’t mean that she never gets headaches caused by other things. So if you know that Molly always gets headaches after drinking too much, and you find Molly having a headache, you’d be jumping to conclusions if you’d conclude solely on that information that she’s been drinking too much. This fallacy is all too common. It lies at the basis of a lot of prejudice. If this is used against you by your annoying uncle at the family dinner table, you can now point out that he’s committing a fallacy and that therefore his argument is invalid. You’re welcome.

Another common fallacy is known as denying the antecedent. An example:

p1. If Molly drinks too much, she’ll get a headache.

p2. Molly didn’t drink too much

C. Molly won’t get a headache.

In other words:

p1. P → Q

p2. ¬P

C. ¬Q

Again: Molly’s headaches can be caused by other things than her drinking. Heat stroke, for example, or stress or migraine. So if she thinks that she is guaranteed to not get any headaches if she just doesn’t drink, she is sadly mistaken. Again, whichever propositions you use instead of P and Q, any inference that follows these structures is invalid.

Another related valid argument.

Of course, things change when the arrow points two ways. If Molly is the sort of person who never (never ever ever) gets headaches, the only exceptions being when she’s been drinking too much, then you get this:

p1. If and only if (iff) Molly drinks too much, she’ll get a headache.

p2. Molly has a headache.

C. Molly has been drinking too much.

Schematically:

p1. P ↔ Q

p2. Q

C. P

 

This is a valid inference.

P ↔ Q is a more efficient way of saying (P→Q)˄(Q→P), which means ‘(if P, then Q) and (if Q, then P). ˄ is a symbol we use to say ‘and’, and ˅ means ‘and/or’. So P˄Q means ‘it is the case that P and Q’, or just ‘P and Q’, and P˅Q means ‘either P is the case, or Q is the case, or both P and Q are the case’. With these symbols, we can also construct valid (or invalid, for that matter) inferences, such as:

p1. P ˄ Q

C. P

or

p1. P ˄ Q

C. Q

or

p1. P ˅ Q

p2. ¬P

C. Q

and many more.

We have now discussed the symbols ˄,   ˅,   ¬,   → and ↔. Let’s see if we can use them correctly.

Exercise 1

Translate the following schematic notations in terms of Molly’s drinking, assuming that P means ‘Molly drinks too much’ and Q means ‘Molly has/will get a headache’:

¬(P→Q)       ¬P→Q        P↔¬Q      ¬P˄Q     ¬(P˅Q)

Exercise 2

Is this a valid argument? If so, why? If not, why not?

p1. P → ¬Q

p2. Q

C. ¬P

 

How about this one:

p1. ¬(P → Q)

p2. P

C. ¬Q

 

Validity and truth

Whilst doing the above exercises, you may have thought: “but it can’t be true that whenever Molly drinks too much, she doesn’t get a headache! She’s bound to get a headache if she really drinks too much, because that’s just how human physiology works!”

You may be quite right about that. However, it is important to note that there is a clear distinction between truth and validity, and logic is only concerned with validity. Truth will follow if the premises are indeed true, but it is not the job of the logician to establish whether they are or not. It is possible to have a perfectly valid inference with false premises, and it is equally possible to have an invalid inference in which all premises as well as the conclusion are true. I will illustrate the this with some more examples.

Example 3

p1. If Molly drinks lemonade, she will be able to fly.

p2. Molly drinks lemonade.

C. Molly will be able to fly.

This inference is clearly bonkers. Molly can’t fly, and drinking lemonade won’t change a thing about that. However, this inference is just an ordinary modus ponens and therefore valid as can be. Remember that an inference is valid if the conclusion must be accepted if the premises are accepted. And that is the case. If we accept that drinking lemonade will enable Molly to fly, and we then find her drinking lemonade, then we must also conclude that she’ll be able to fly.

Example 4

p1. If Moby Dick is a whale, then he is violent

p2. Moby Dick is violent

C. Moby Dick is a whale.

If you’ve read Moby Dick, then you know that all of this is true. But is it a valid argument? Not one bit. You can recognise this inference as an incident of the affirming the consequence fallacy. That fallacy remains an invalid inference, regardless of the truth of its premises and conclusion. The conclusion, in this case, may be true, but it doesn’t follow from the premises. And remember that logic is only concerned with whether the conclusion of an inference follows from its premises.

Afterword

We have discussed some basic valid and invalid forms of argument in propositional logic. There are other types of logic (predicate logic, for example), and there are other, much more complicated inferences possible. Inferences with many more than just two premises are possible, and each premise can itself involve more than just two propositions. There are countless possibilities.

But why is this important to philosophers? Philosophers work with complex arguments. To be able to assess the quality of those arguments, it is often necessary to formalise them. This counts for, for instance, philosophy of science, where philosophers discuss which scientific methods really merits certain scientific conclusions based on the data available. It also counts for political philosophy, to name another example, where stripping an argument of its emotive factors allows us to rationally assess whether an inference is valid or not. Fallacies are all too common, and logic allows us to identify them and protect ourselves from being led astray by them. For philosophers of all kinds, if rationality has any value at all, logic is of utmost importance.

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