Essential Terms for Symbolic Logic

There are many definitions of logic as a field of study. One definition  is the study of argument. For the purposes of logic, an argument is not a quarrel or dispute, the wonderful people of Monty Python notwithstanding, but an example of reasoning in which one or more statements are offered as support, justification, grounds, reasons, or evidence for another statement. The statement being supported is the conclusion of the argument, and the statements that support it are the premises of the argument. If the premises of an argument support its conclusion, the conclusion follows logically from the premises.  If you haven't seen Monty Python's Argument Clinic, watch it.

Studying argument is important because argument is the way we support our claims to truth. It is tempting to say that arguments establish the truth of their conclusions. But the study of logic forces us to qualify this statement. Arguments establish the truth of conclusions relative to some premises and rules of inference. Logicians do not care whether arguments succeed psychologically in changing people's minds or convincing them. The kinks and twists of actual human reasoning are studied by psychology; the effectiveness of reasoning and its variations in persuading others are studied by rhetoric; but the correctness of reasoning (the validity of the inference) is studied by logic.

To assess the worth of an argument, only two aspects or properties of the argument need be considered: the truth of the premises and the validity of the reasoning from them to the conclusion. Of these, logicians study only the reasoning; they leave the question of the truth of the premises to empirical scientists and private detectives.

An argument is valid if the truth of its premises guarantees the truth of its conclusion; or if the conclusion would necessarily be true on the assumption that all the premises were true; or if it is impossible for the conclusion to be false and all the premises true at the same time; or if the conclusion can be deduced from the premises in accordance with certain valid rules. It turns out that all these formulations are equivalent. (The last formulation is that of syntactic validity, the rest are formulations of semantic validity.) If an argument is not valid, it is invalid.

Note that only arguments can be valid or invalid, not statements. Similarly, only statements can be true or false, not arguments. Validity pertains to reasoning, not propositions, while truth pertains to propositions, not reasoning. The first fundamental principle of logic is the independence of truth and validity.

When the reasoning in an argument is valid and all its premises are true, then it is called sound. Otherwise the argument is unsound. If an argument is sound, then its conclusion must be true and we would be illogical to disbelieve it.

An argument is deductive if the premises claim to give conclusive grounds for the truth of the conclusion, or if the premises claim to support the conclusion with necessity. An argument is inductive if it makes the milder claim that its premises support but do not guarantee its conclusion. The black and white categories of validity and invalidity apply only to deductive arguments; inductive arguments are strong or weak. In a valid deductive argument with all true premises, the truth of the conclusion is necessary and its falsehood is impossible. In a strong inductive argument with all true premises, the truth of the conclusion is merely probable and its falsehood merely improbable. The kind of support that valid deductions provide their conclusions is not a matter of degree; it is "all or nothing". But the kind of support that strong inductions provide their conclusions is a matter of degree; it is "more or less". The conclusion of a valid deduction never contains more information than was contained the premises; the conclusion of an induction always does. That is why deductions possess certainty (they never tell us anything new) and why inductions are always uncertain in some degree.

Perhaps the best way to understand the difference between deductive and inductive arguments is in terms of the probability that the conclusion is true given the truth of the premises.  In the best of all deductive arguments, those that  are valid, the probability of the conclusion given the premises is always 1, that is, it is absolutely guaranteed.  In the best of all inductive arguments, the probability of the conclusion given the premises is always less than 1, that is, there is always some room for doubt.