This is a work in progress. It will be added to as I have time and inclination. If anyone wants to help, or especially if you see an error, please PM me.
A note on style: I will use italics for two reasons. Most instances of italics will be to introduce technical words used in logic, or technical usages of common words. It is a bad habit among philosophers to use common words in often narrow senses - senses specific to philosophy. Logicians are no different. I will also use italics for simple emphasis, where I think it might be helpful. Hopefully, the context will make the distinction clear.
I have also decided that I will sometimes provide links for these words. These links lead to material that may be of interest, but are not necessarily primarily explicative - they may just be interesting ideas connected to the words. If they confuse, I suggest you ignore them. In some cases, I have added a link primarily for amusement - perhaps only for my own.
A note on notation: Different writers use different systems of notation for the symbols used in logic. I am following the notation I first learnt, Irving Copi’s, as a matter of convenience. Much thanks to TheStumps and Carleas for their invaluable assistance in this regard.
Faust
[size=150]Introduction[/size]
Probably one of the most difficult aspects of discussing logic is to define it. First, we must limit our usage of the word, for it has many. Here, we are considering only what is called formal, deductive logic – but we will not necessarily use those qualifiers after this.
Logic Defined
We will define logic by its purpose – which is to distinguish good arguments from bad arguments. Of course, now we must say what we mean by good and bad. For this, we will introduce another word that can be used in different ways, but for which we will adopt a specific and therefore technical usage – validity.
Logic distinguishes valid arguments from those that are invalid, according to a peculiar set of rules. More on those rules later. Suffice it, for the moment, to say that valid arguments are those that if the premises are true, then the conclusion must also be true. This is called a valid inference, or a valid argument. It should be noted that valid logical arguments then produce only what is called analytic truth – which is to say that the truth of the conclusion is dependent on the truth of the preceding premises and not upon any matters of fact.
It should also be noted that logic is not concerned with the actual process of inference, but only with the statements used to make an inference and their relations to each other. These statements are also called propositions, claims or assertions, often indiscriminately. Some writers draw a distinction between statements and propositions, but I will not. None of these are usually called sentences, although they have been by some writers in the past. I will draw this last distinction and will not use “sentence” as a synonym for statement, etc.
But these assertions are conveyed by sentences. We can properly say that an assertion has in common with the sentence used to convey it that sentence’s meaning, and that such a sentence can be of only one kind: the declarative sentence. To illustrate, any declarative sentence may be translated into another language, but the assertion it makes remains the same in either language. Likewise, the same sentence may make different claims when used in different contexts.
Thus “All men are mortal” is the same proposition as “All men will someday die”, even though they are two different sentences. And “I went to jail” makes a different claim in a recitation of one’s criminal history than it does during a Monopoly game, or so it could, at least.
So logic has as its subject matter arguments and the statements used to make them. So, what is an argument? It’s a collection of statements (premises) of which the last (the conclusion) is claimed to follow from those that precede it, which is to say that those statements preceding the conclusion are to be considered grounds for the conclusion. While the word “argument” can mean many other things in common parlance, it has this technical meaning in logic.
It’s worthwhile to note that logic does not create this relationship between the premises and the conclusion, however – a valid logical argument only tests and affirms this relationship. Again, we are not, as logicians, concerned with the actual process of inference – but only with subjecting completed arguments to a method which will clarify and illustrate that process, by testing it for errors.
Truth
Propositions are either true or false. If we cannot determine whether a declarative sentence is either true or false, then it does not contain a statement, for the purposes of logic. Another way to say this is that, logic presupposes that truth or falsity can be assigned to any proposition – an assignable truth or falsity being part of the definition of a proposition. While indeterminacy may be fascinating to the philosopher, it’s useless to the logician. And while the issue of indeterminacy has been insinuated into the subject of logic, we will not consider it here.
Arguments are never said to be true or false. They are valid or invalid. While it may be of great interest to us that statements be either true or false, the word “truth” is not applicable to arguments themselves. The seat of truth (or falsity) in logic is then propositions and not arguments. Again - arguments can be valid or invalid, but not true or false.
Valid and invalid arguments alike may contain true premises as easily as false ones, and while it may seem odd at first glance, any argument, valid or invalid, can have a true proposition as its conclusion even if its premises are not true. These results would be accidental to the argument, but still possible.
There are two conditions that must be satisfied to establish the truth of any argument’s conclusion as a conclusion, however. The argument must be valid, and the premises must all be true (this is also called a sound argument). The logician is concerned only with the first of those conditions. Thus the logician, [i]qua logician[/i], is not concerned with truth per se. The truth of the premises used in an argument must be established, or merely accepted, prior to the argument. And the truth of the conclusion (as a conclusion of the argument at hand), being in part dependent upon the truth of the premises, is similarly of no importance to the logician. Logic is the study of validity, and not of truth.