Inference, truth and validity Introduction

The fundamental focus of logic is on inference or argument. An argument in this sense is not a discourse, but a piece of reasoning.

Example 1

All footballers are bipeds; Socrates is a footballer; Therefore Socrates is a biped.

This is made up of sentences which are used to make statements which express propositions. The differences between these three (sentence, statement, proposition) are important for the philosophy of language, and will raise their heads again in the later stages of this course, but for now we shall set them aside and just talk of sentences of a language, where by 'language' we mean a convenient set of potential utterances which can be used to describe something or other.

That is, here we only consider indicative sentences, of the sort used to make statements. In natural languages, there are sentences of many other sorts, such as questions, commands, exclamations, requests and so forth. Those are just as much genuine sentences as the fact-stating ones: it is simply that they are not the primary concern of logic.

A sentence, then, says that things stand in a certain way. 'Socrates is a footballer' says that a certain individual, Socrates, has a certain property - that he regularly plays the Beautiful Game. Such a sentence has a truth value: it is either true or false (but not, of course, both). If it happens that things in the world stand in the way the sentence says, then it is true; otherwise it is false. In this particular case, it happens to be false, but it is still a perfectly good example of a sentence. An argument like our example about Socrates and his feet connects sentences together, forming a claim that one sentence (Socrates is a biped) is true if the other two are.

Some authors define an argument to involve not merely the premises and conclusion but also a series of steps leading from the one to the other. That is, they use the word "argument" for roughly what I call a "derivation" (see Chapter 2 below). For the purposes of these notes, however, an argument is simply a pair consisting of a set of premises and a conclusion.

We call the sentences from which an argument starts its premises, and the one to which it leads the conclusion. The premises could be some known facts, but they do not have to be. They could be something assumed or conjectured, or just supposed for the sake of argument. In the world of computation, they could be seen as a database, or part of one, and the conclusion in that case as a query. The premises are supposed to lend rational support to the conclusion in the sense that anyone who accepts the premises and is faced with the argument must, on pain of irrationality, accept the conclusion. Conversely, anyone who denies the conclusion while being aware of the argument must be prepared to reject at least one of the premises or again face the charge of irrationality. Logic and logicians cannot compel anyone to be rational, any more than ethics and its students can compel people to be moral, but we can lay down and investigate the abstract conditions of rationality. We do this by classifying and studying forms of argument.

The query succeeds from the database, the theorem is a consequence of the axioms, the conclusion follows from the assumptions if and only if the argument is valid. Note the difference between validity and truth: sentences (statements, propositions, assumptions, etc) are true or false; arguments (inferences, passages of reasoning) are valid or invalid. Of course, there can be perfectly valid arguments with false conclusions:

Example 2

All cats are reptiles; Snoopy is a cat; Therefore Snoopy is a reptile.

just as there can be invalid arguments with true premises and conclusions

Example 3

All logicians are rational; Restall is rational; Therefore Restall is a logician.

Greg Restall, at the time of writing of these notes professor of logic at the University of Melbourne, author of the best-known book on substructural logic and sometime editor in chief of the Australasian Journal of Logic, is presumably a logician if anyone

What there cannot be is a valid argument with true premises and a false conclusion. In fact, we take this as the definition of validity: to say that an argument is valid means that it is impossible for its premises to be true and its conclusion false. Conversely, an argument is invalid if there is a way for its premises to be true and its conclusion false. To assert the premises of a valid argument while denying its conclusion is to contradict oneself, whereas the premises of an invalid argument are consistent with the denial of its conclusion.

Notice that the two valid arguments above, about Socrates and Snoopy, have a lot in common. They are valid for the same reason: they are of the same form. Clearly any argument of the same form as these two will likewise be valid. We can express the form using letters to stand in for predicates and names of the English language much as they stand in for numbers in algebra:

All As are Bs;
x is an A;
Therefore x is a B.

To force idiomatic statements of natural languages into such abstract moulds requires some chopping and squeezing. This paraphrasing process relies on native speakers' feeling for acceptability, for goodness of fit and the like. Hence at the interface with real arguments in real languages like English, logic is not an exact science.

Why isn't the third example about Greg Restall valid for the same reason? - because it is not of the same form. It is of the form

All As are Bs;
x is a B;
Therefore x is an A.

and this is no good. You might as well argue:

Example 4

All dogs are animals; Skippy is an animal; Therefore Skippy is a dog.

This is a counter-example to the suggested argument form: an argument which is clearly of that form but which has actually true premises and an actually false conclusion. Note that exhibiting an invalid form does not necessarily show that the particular argument is invalid, for it may also be of another, valid, form. Nonetheless, 'You might as well say...' is a very powerful tactic in the dialectic of reason.

In logic, we are interested in cataloguing valid argument forms in a systematic calculus, so that we can carry out proofs and other kinds of reasoning at an abstract level. We tend not to be interested (as logicians) in arguments that are valid for reasons other than their form, such as:

Example 5

The People's Flag is red; Therefore the People's emblems are not all blue.

It is not just an accident that red flags aren't blue. The fact that nothing can be both red and blue (all over, at the same time) is a necessary truth of some sort, but it does not seem to be a logical truth. It arises from the semantics of colour vocabulary, which we take to be a different field of study. There are borderline cases, of course, which are forms that could be subjected to logical analysis but which normally get left out of the catalogue, such as

x is bigger than y;
y is bigger than z;
Therefore x is bigger than z.

The logic of comparatives, for instance, is a perfectly sensible topic of study, but we usually choose not to pursue it as a central part of pure logic.

Having abstracted from natural language to give ourselves some sentence forms, we isolate the particles that seem to be systematically involved in the validity of arguments using those forms. Words such as 'all', 'some', 'and', 'not' and 'if' seem like prime candidates, for instance. We thus arrive at a formal language consisting of formulae built up using these particles from unanalysed basic atoms or variables or the like. We then devise a calculus of formally valid inference and proceed to carry out logical investigations in and about that. The theory which flows from such investigations is important for many foundational and practical concerns, including:

• deductive databases
• rationality of agents
• foundations of mathematics
• philosophy of science
• computation theory
• hardware and software verification
• problem solving (planning, scheduling, diagnosis, etc)
• legal reasoning
• reasoning in and about ontologies

In these notes, we do not explore the applications of logic, important though they be, but are concerned rather to examine the formal theory itself. In later sections, we consider some criticisms of the dominant paradigm of logic, and note some alternatives to it, but from the standpoint of theory rather than practice.