formal logic, the abstract study ofpropositions, statements, or assertively used sentences and ofdeductive arguments. Thediscipline abstracts from the content of these elements the structures or logicalforms that they embody. The logician customarily uses a symbolicnotation to express such structures clearly and unambiguously and to enable manipulations and tests ofvalidity to be more easily applied. Although the following discussion freely employs the technical notation of modern symbolic logic, its symbols are introduced gradually and with accompanying explanations so that the serious and attentive general reader should be able to follow the development of ideas.

Formal logic is ana priori, and not anempirical, study. In this respect it contrasts with the naturalsciences and with all otherdisciplines that depend on observation for their data. Its nearestanalogy is to puremathematics; indeed, many logicians and pure mathematicians would regard their respective subjects as indistinguishable, or as merely two stages of the same unified discipline. Formal logic, therefore, is not to be confused with the empirical study of the processes ofreasoning, which belongs topsychology. It must also be distinguished from the art of correct reasoning, which is the practical skill of applying logical principles to particular cases; and, even more sharply, it must be distinguished from the art of persuasion, in which invalidarguments are sometimes more effective than valid ones.

(Read Steven Pinker’s Britannica entry on rationality.)

General observations

Probably the most natural approach to formal logic is through the idea of thevalidity of anargument of the kind known as deductive. A deductive argument can be roughly characterized as one in which the claim is made that some proposition (theconclusion) follows with strictnecessity from some other proposition or propositions (thepremises)—i.e., that it would be inconsistent or self-contradictory to assert thepremises but deny the conclusion.

If a deductive argument is to succeed in establishing thetruth of its conclusion, two quite distinct conditions must be met: first, the conclusion must really follow from the premises—i.e., the deduction of the conclusion from the premises must be logically correct—and, second, the premises themselves must be true. An argument meeting both these conditions is calledsound. Of these two conditions, the logician as such is concerned only with the first; the second, the determination of the truth or falsity of the premises, is the task of some specialdiscipline or of common observation appropriate to the subject matter of the argument. When the conclusion of an argument is correctly deducible from its premises, theinference from the premises to the conclusion is said to be (deductively) valid, irrespective of whether the premises are true or false. Other ways of expressing the fact that aninference is deductively valid are to say that the truth of the premises gives (or would give) an absolute guarantee of the truth of the conclusion or that it would involve a logical inconsistency (as distinct from a mere mistake of fact) to suppose that the premises were true but the conclusion false.

The deductiveinferences with which formal logic is concerned are, as the name suggests, those for which validity depends not on any features of their subject matter but on their form or structure. Thus, the two inferences(1) Every dog is a mammal. Some quadrupeds are dogs. ∴ Some quadrupeds are mammals. and(2) Every anarchist is a believer in free love. Some members of the government party are anarchists. ∴ Some members of the government party are believers in free love. differ in subject matter and hence require different procedures to check the truth or falsity of their premises. But their validity is ensured by what they have in common—namely, that the argument in each is of the form(3) EveryX is aY. SomeZ’s areX’s. ∴ SomeZ’s areY’s.

Line (3) above may be called aninference form, and (1) and (2) are then instances of that inference form. The letters—X,Y, andZ—in (3) mark the places into which expressions of a certain type may be inserted. Symbols used for this purpose are known asvariables; their use isanalogous to that of thex inalgebra, which marks the place into which a numeral can be inserted. An instance of an inference form is produced by replacing all the variables in it by appropriate expressions (i.e., ones that make sense in the context) and by doing so uniformly (i.e., by substituting the same expression wherever the same variable recurs). The feature of (3) that guarantees that every instance of it will be valid is its construction in such a manner that every uniform way of replacing its variables to make the premises true automatically makes theconclusion true also, or, in other words, that no instance of it can have true premises but a false conclusion. In virtue of this feature, the form (3) is termed a valid inference form. In contrast,(4) EveryX is aY. SomeZ’s areY’s. ∴ SomeZ’s areX’s. is not a valid inference form, for, although instances of it can be produced in which premises and conclusion are all true, instances of it can also be produced in which the premises are true but the conclusion is false—e.g.,(5) Every dog is a mammal. Some winged creatures are mammals. ∴ Some winged creatures are dogs.

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Formal logic as a study is concerned with inference forms rather than with particular instances of them. One of its tasks is to discriminate between valid andinvalid inference forms and to explore and systematize the relations that hold among valid ones.

Closely related to the idea of a valid inference form is that of a validproposition form. A proposition form is an expression of which the instances (produced as before by appropriate and uniform replacements for variables) are not inferences from several propositions to a conclusion but rather propositions taken individually, and a valid proposition form is one for which all of the instances are true propositions. A simple example is(6) Nothing is both anX and a non-X. Formal logic is concerned with proposition forms as well as with inference forms. The study of proposition forms can, in fact, be made to include that of inference forms in the following way: let the premises of any given inference form (taken together) be abbreviated by alpha (α) and its conclusion bybeta (β). Then the condition stated above for the validity of the inference form “α, therefore β” amounts to saying that no instance of the proposition form “α and not-β” is true—i.e., that every instance of the proposition form(7) Not both: α and not-β is true—or that line (7), fully spelled out, of course, is a valid proposition form. The study of proposition forms, however, cannot be similarly accommodated under the study of inference forms, and so for reasons of comprehensiveness it is usual to regard formal logic as the study of proposition forms. Because a logician’s handling of proposition forms is in many waysanalogous to a mathematician’s handling of numerical formulas, the systems he constructs are often called calculi.

Much of the work of a logician proceeds at a more abstract level than that of the foregoing discussion. Even a formula such as (3) above, though not referring to any specific subject matter, contains expressions like “every” and “is a,” which are thought of as having a definite meaning, and the variables are intended to mark the places for expressions of one particular kind (roughly, common nouns or class names). It is possible, however—and for some purposes it is essential—to study formulas without attaching even this degree of meaningfulness to them. The construction of a system oflogic, in fact, involves two distinguishable processes: one consists in setting up asymbolic apparatus—a set of symbols, rules for stringing these together intoformulas, and rules formanipulating these formulas; the second consists in attaching certain meanings to these symbols and formulas. If only the former is done, the system is said to beuninterpreted, or purely formal; if the latter is done as well, the system is said to be interpreted. This distinction is important, because systems of logic turn out to have certain properties quite independently of any interpretations that may be placed upon them. Anaxiomatic system of logic can be taken as an example—i.e., a system in which certain unproved formulas, known asaxioms, are taken as starting points, and further formulas (theorems) are proved on the strength of these. As will appear later (see belowAxiomatization of PC), the question whether a sequence of formulas in anaxiomatic system is aproof or not depends solely on which formulas are taken as axioms and on what the rules are for deriving theorems from axioms, and not at all on what the theorems or axioms mean. Moreover, a given uninterpreted system is in general capable of being interpreted equally well in a number of different ways; hence, in studying an uninterpreted system, one is studying the structure that is common to a variety of interpreted systems. Normally a logician who constructs a purelyformal system does have a particular interpretation in mind, and his motive for constructing it is the belief that when this interpretation is given to it, the formulas of the system will be able to express true principles in some field of thought; but, for the above reasons among others, he will usually take care to describe the formulas and state the rules of the system without reference to interpretation and to indicate as a separate matter the interpretation that he has in mind.

Many of the ideas used in theexposition of formallogic, including some that are mentioned above, raise problems that belong tophilosophy rather than to logic itself. Examples are: What is the correct analysis of the notion oftruth? What is a proposition, and how is it related to the sentence by which it is expressed? Are there some kinds of sound reasoning that are neither deductive norinductive? Fortunately, it is possible to learn to do formal logic without having satisfactory answers to such questions, just as it is possible to do mathematics without answering questions belonging to thephilosophy of mathematics such as: Arenumbers real objects or mental constructs?