Logical Notation

The way in which logical concepts and their interpretations are expressed in natural languages is often very complicated. In order to reach an overview of logical truths and valid inferences, logicians have developed various streamlined notations. Such notations can be thought of as artificial languages when their nonlogical concepts are interpreted; in this respect they are comparable to computer languages, to some of which they are in fact closely related. The propositions (1)–(4) illustrate one such notation.

Logical languages differ from natural ones in several ways. The task of translating between the two, known as logic translation, is thus not a trivial one. The reasons for this difficulty are similar to the reasons why it is difficult to program a computer to interpret or express sentences in a natural language.

Consider, for example, the sentence

(5) If Peter owns a donkey, he beats it.

Arguably, the logical form of (5) is

(6) (∀x)[(D(x) & O(p,x) ⊃ B(p,x)]

where D(x) means “x is a donkey,” O(x,y) means “x owns y,” B(x,y) means “x beats y,” and “p” refers to Peter. Thus (6) can be read: “For all individuals x, if x is a donkey and Peter owns x, then Peter beats x. Yet theoretical linguists have found it extraordinarily difficult to formulate general translation rules that would yield a logical formula such as (6) from an English sentence such as (5).

Contemporary forms of logical notation are significantly different from those used before the 19th century. Until then, most logical inferences were expressed by means of natural language supplemented with a smattering of variables and, in some cases, by traditional mathematical concepts. One can in fact formulate rules for logical inferences in natural languages, but this task is made much easier by the use of a formal notation. Hence, from the 19th century on most serious research in logic has been conducted in what is known as symbolic, or formal, logic. The most commonly used type of formal logical language was invented by the German mathematician Gottlob Frege (1848–1925) and further developed by the British philosopher Bertrand Russell (1872–1970) and his collaborator Alfred North Whitehead (1861–1947) and the German mathematician David Hilbert (1862–1943) and his associates. One important feature of this language is that it distinguishes between multiple senses of natural-language verbs that express being, such as the English word “is.” From the vantage point of this language, words like “is” are ambiguous, because sentences containing them can be used to express existence (“There is a Santa Claus”), identity (“Superman is Clark Kent”), predication (“Venus is a planet”), or subsumption (“The wolf is a vertebrate”). In the logical language, each of these senses is expressed in a different way. Yet it is far from clear that the English word “is” really is ambiguous. It could be that it has a single sense that is differently interpreted, or used to convey different information, depending on the context in which the containing sentence is produced. Indeed, before Frege and Russell, no logician had ever claimed that natural-language verbs of being are ambiguous.

Another feature of contemporary logical languages is that in them some class of entities, sometimes called the “universe of discourse,” is assumed to exist. The members of this class are usually called “individuals.” The basic quantifiers of the logical language are said to “range over” the individuals in the universe of discourse, in the sense that the quantifiers are understood to refer to all (∀x) or to at least one (∃x) such individual. Quantifiers that range over individuals are said to be “first-order” quantifiers. But quantifiers may also range over other entities, such as sets, predicates, relations, and functions. Such quantifiers are called “second-order.” Quantifiers that range over sets of second-order entities are said to be “third-order,” and so on. It is possible to construct interpreted logical languages in which there are no basic individuals (known as “ur-individuals”) and thus no first-order quantifiers. For example, there are languages in which all the entities referred to are functions.

Depending upon whether one emphasizes inference and logical form on the one hand or logic translation on the other, one can conceive of the overarching aim of logic as either the study of different logical forms for the purpose of systematizing the study of inference patterns (logic as a calculus) or as the creation of a universal interpreted language for the representation of all logical forms (logic as language).