Introduction to Philosophy/Logic/Truth and Validity
Logic can get us from statements to further statements. So, to go back to the syllogism:
All men are mortal.
Socrates is a man.Therefore, Socrates is mortal.
The first two statements, or claims, are called the premises, while claim below the horizontal rule is called the conclusion. In an argument, the premises are things which you hope your interlocutor has already accepted - they may be empirical observations, for example.
Notice the convention of separating the conclusion of an argument from the premises with a horizontal rule. An alternative is to use the symbol. A, B C means that C follows from A and B. Alternative locutions are 'A and B entail C', 'C is a consequence of A and B', 'A and B derive C'.
An argument is valid if the conclusion follows from the premises. In logic, truth is a property of statements, i.e. premises and conclusions, whereas validity is a property of the argument itself. If you talk of 'valid premises' or 'true arguments', then you are not using logical jargon correctly.
True premises and a valid argument guarantee a true conclusion. An argument which is valid and has true premises is said to be sound (adjective) or have the property of soundness (noun).
I suppose I ought to say what an argument is in this context. An argument is a progression from premises to conclusion. Each statement in the argument is either a premise, or else follows from the previous statements in the argument. So two kids shouting "'tis" and "'tisn't" at each other does not constitute an argument, neither do two teenagers swearing at each other. This book is here to help you behave like civilized adults. If you wish to behave like uncivilized adults, please consult our sister project, Wikipedia.
Now sometimes you may see two adults pointing out facts to each other, and making inferences from those facts. We might say that these two adults are 'having an argument'. To be technical, it is a dialectic in which each side advances an argument in the sense meant here.
A mathematical argument is called a proof, and the conclusion of the argument is called a theorem. Sometimes only the really interesting conclusions are known as theorems, and the less important ones are given another name like lemma. Compare how we use the words 'lady' or 'gentleman' - these words can be reserved to refer only to people of status, or they can be used to refer to everyone.
Euclid, in The Elements, starts off with a set of premises from which he derives several volumes of conclusions, all in a rigorous manner. Such premises are known as 'axioms'. Logicians are trying to do something similar with arguments. We are reasoning about reasoning.
In arguments, mathematical or otherwise, each statement should be intuitively obvious given what has been said before - this is what is meant when we say that one statement follows from its predecessors. Logicians have tried to replace this appeal to the 'intuitively obvious' with a set of rules, called rules of inference, which form a special class of axiom. Of course the rules of inference should themselves be 'intuitively obvious' - we can't eliminate intuition completely.
So to abstract from our example about Socrates, we can write down a rule of inference:
From the premises 'all x are y' and 'a is x', we can infer 'a is y'.
So now we have a project: can reason be reduced to a small list of simple rules like this? Compare this with Euclid's project of showing that the mathematics of his day can be derived from a set of rules.
In a while we shall see how far we can get with a simple set of rules called propositional calculus. This is a very simple system, in which it is not possible even to properly express the deduction made about Socrates. But in presenting the propositional calculus I shall introduce some concepts and procedures which are useful for talking about any kind of logical calculus, in particular the predicate calculus, which I'll talk about later on.