Logic is the science of necessary implication. By this, we mean that it is concerned, like mathematics, with the transformation of symbols, according to given rules, which result in the discovery of new truths.
A few logical statements in natural language are: If the shipment hasn't come by Friday, then it will be here on Monday.
Either we are going to McDonald's, or we are going to Burger King.
If you buy one, then you will get one free.
Logic is vital in fields as diverse as law, philosophy, and computer programming; it is almost impossible to persue these fields without some knowledge of formal logic, and the deeper one's understanding of logic, the more successful one will be.
This book is concerned with propositional logic, the construction and evaluation of dialectical arguments. The use of logic in mathematics and computer programming, are related and important; a programmer who knows the inferrences of a particular statement, will be able to understand their application to his craft.
There is another field of logic, called informal logic or the art of sophistry, which concerns itself with the discovery of argumentative fallacies and flaws. While informal logic is valuable to beginners in argumentation, as Hobbes says,
...the rules of Sophistry be needless for them that be perfect in logic. — The Art of Sophistry
if...thenthat are true.
either...orthat are true.
Before we begin the study of logic as such, it will be well to say a few words about the philosophy of logic, and especially its status as a science. That logic is a science is not in dispute; in Hegel's words, we
...take its derivation as a given. — The Science of Right
The questions before the philosopher of logic, then, concern what sort of science logic is, what other sciences depend on it and its place in the structure of those sciences. In order to approach these questions, we must first say what a science is.
Science is concerned with structured knowledge; that is to say, with the cataloguing and classification of facts. There are two chief kinds of science:
An empirical science, like geology or biology, is chiefly concerned with the cataloguing of facts; a formal science is concerned with the inferrence from one fact to another.
There are two main branches of formal science, mathematics and logic. Of empirical sciences there are many; examples are biology, geology, astronomy, &c.
Logic, as the science of necessary implication, is primarily a formal science; that is to say, its concern is with the form of argument rather than the content of it.
Make a vessel, and you have a vessel. Fill the vessel, and you have use of the vessel. — Tao Te Ching
For this reason, one of the earliest major theorems in the philosophy of logic is that, because the forms of logic are convincing even in the abstract symbolism of the propositional calculus; and because the forms are convincing regardless of the state of the external world, knowledge of logic constitutes eternal knowledge. If this be not so, then, because symbolic knowledge makes no reference to the external world, it is no kind of knowledge at all, for
...all our knowledge begins with experience. — Kant, Critique of Pure Reason
While logic, as such, has no need of such Platonic idealism to justify its existence, philosophically speaking if logic is to claim the title of science — and logic has every right to make such a claim, as does its sister science, mathematics — it must be taken for granted that, once learned, the methods of logic are eternally valid for the construction and criticism of arguments.
So, logic is a formal science, concerned with the cataloguing and classification of facts as regards the implications of an argument. What this means in practice is that, due to certain peculiarities of language, certain kinds of statement are identical to one another; they mean the same thing
. The transformation of one argument to an equivalent, by the methods of logic, is called an inference.
One of the major questions in the philosophy of logic is: is logic an empirical science?
The answer, as far as we are concerned, is yes. While the major work done in logic, and especially theoretical logic, concerns itself with the form of argument, the third criterion of truth in logic is soundness, which is an extra-logical consideration concerning the external truth of individual propositions.
The simplest logical argument is the bare proposition. Since bare propositions do not have a formal structure (so the criteria of validity and consistency do not apply), the only criterion of truth a bare proposition need supply is soundness, or the relation of the statement to the fact it is intended to represent.
This being the case, logic is an empirical science. The second consideration in this regard is, are the arguments that are necessarily true of logic, also necessarily true empirically?
This is somewhat more difficult to answer. It depends, essentially, on the truth of the argument itself. If it is not true that α implies β in the real world, an argument to that effect must necessarily be invalid.
We live in a universe that plays right into logic's hands. — Principles of Psychology]