Introduction to Philosophical Logic/Arguments

Definition of an argument
An argument (in the context of logic) is defined as a set of premises and a conclusion where the conclusion and premises are separated by some trigger word, phrase or mark known as a turnstile.

For example:

1 I think; therefore I am.

There is only one premise in this argument, I think. The conclusion is I am and the turnstile is therefore (although the semi-colon may be thought of as part of the turnstile).

2 ''All men are mortal. Socrates was a man. So, Socrates was mortal.''

In this example there are two premises and the turnstile is so.

In English (and all other natural languages), the conclusion need not come at the end of an argument:

3 ''Pigs can fly. For pigs have wings and all winged animals can fly.''

4 I am a safe driver: I have never had an accident.

Here the turnstiles (for and :) seem to indicate where the premises come as opposed to where the conclusion comes. Other examples of turnstiles (indicating either conclusion or premises) are: so, thus, hence, since, because, it follows that, for the reason that, from this it can be seen that.

Sound arguments
A sound argument is an argument that satisfies three conditions. -True premises -Unambiguous premises -Valid logic

If one of these conditions is unsatisfied then the argument is unsound, though in the case of ambiguous premises, not necessarily so.

The first, second and fourth arguments (depending of course, on who "I" is - it is being assumed it is the author!) are all sound. Here are some more:

5 ''Grass is green. The sky is blue. Snow is white. Therefore coal is black.''

6 ''Grass is green. The sky is blue. Snow is white. Therefore pigs can fly.''

7 2+2=4; hence 2+2=4

8 2+2=4; hence 2+2=5

Note that it is not necessary that the truth value of the conclusion play a role in determining whether an argument is sound. This can be determined by considering the truth of the premises and the validity of the argument. However, if the conclusion is not true, the argument is not sound.

A sound argument always has consistent premises. This must be the case, since there is a possible situation (namely reality) in which they are all true.

Valid arguments
Of greater interest to the logician are valid arguments. A valid argument is an argument for which there is no possible situation in which the premises are all true and the conclusion is false.

Of the above arguments 2, 3 and 7 are valid. The reader should consider whether argument 1 is valid (read Meditations on First Philosophy by Descartes, chapters 1, 2).

It does not matter whether the premises or conclusion are actually true for an argument to be valid. All that matters is that the premises could not all be true and the conclusion false. Indeed, this means that an argument with inconsistent premises is always valid. There is no situation under which such an argument has all premises true and so there is no situation under which such an argument has all premises true and conclusion false. Hence it is valid. Similarly, an argument with a necessary conclusion can in no situation have all true premises and a false conclusion, since there is no situation in which the conclusion is false.

An argument with the single premise 'The conclusion is true.' is valid (regardless of the conclusion). An argument with the conclusion 'The premises are all true.' is also valid.

According to the definition of truth given previously, if the conclusion is false, its negation is true. Hence a valid argument can also be defined as an argument for which there is no possible situation under which the premises and the negation of the conclusion are all true. Hence, a valid argument is an argument such that the set of its premises and the negation of the conclusion is inconsistent. Such a set (the union of the set of premises and the set of the negation of the conclusion) is known as the counter-example set. It is called the counter-example set for the following reason: if a possible situation is found in which the members of this set are all true (and so the set is found to be consistent), this situation provides a counter-example to the arguments being valid, i.e. the existence of such a situation proves that the argument is not valid.

Counter-examples do not exist only for arguments, but also for statements:

Prime numbers are always odd		2 provides a counter-example (a number) to this statement. All animals have four legs			Human beings provide a counter-example (a type of animal). Years are 365 days long			Leap years provide a counter-example (a type of year). Years designated by a number divisible by four are leap years			The year 1900 provides a counter-example (a particular year). It always rains in England			A singularly sunny day in September (today, when written - a 								particular interval of time) provides a counter-example.

Counter-examples to declarative sentences refute their truth and are classes of things (thing being understood very broadly here) or particular things. Counter-examples to arguments refute their validity and are possible situations designated by sets of sentences (the counter-example set). Some clarification of the situation is often needed.

For example, take argument 4. It will be modified slightly as follows:

John has never had an accident; therefore, John is a safe driver.

It will be assumed here that accident means car accident and driver means motorist and safe means not liable to cause an accident. The counter-example set is:

John has never had an accident. John is not a safe driver.

Clarification by example: it may be that John has never driven in his life (and so never had an accident) because he is blind (and so cannot be considered a safe driver).

As mentioned, an inconsistent counter-example set implies that a conclusion is valid because it means that there is no situation under which the premises are all true and the conclusion is false (the negation of the conclusion is true).

Take argument 2; the counter-example set is:

All men are mortal. Socrates was a man. Socrates was not mortal.

These sentences cannot all be true at once. If Socrates was a man and he was not mortal, it could not be that all men are mortal. If all men are indeed mortal and Socrates was not mortal, he could not have been a man. If all men are mortal and Socrates was a man, he must have been mortal. Hence the counter-example set is inconsistent and the argument is valid.

Use a similar approach to show that arguments 3 and 7 are valid (and use it to consider argument 1). This method is known as reductio ad absurdum (which translates literally from Latin as "reduction to absurdity"). The negation of the conclusion is absurd given the truth of the premises and so the conclusion must be true.