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date: 24 January 2020

Logical symbols

Source:
The Oxford Dictionary of Philosophy

Logical symbols

Reading logical symbolism frightens many people more than it should. The very term symbolic logic sounds terrifying, and the presence of even a small amount of symbolism may deter many readers from otherwise perfectly intelligible texts. The following explanation introduces the symbolism used in this work, and lists some of the variations that may be encountered in other works. It should be noticed that technical terms used in this appendix are also explained under their own headwords in the body of the dictionary, and cross-references have been given where appropriate.

Lower-case italic letters from this part of the alphabet: p, q, r…, are used as propositional variables. This means that they stand for propositions or statements. Some logicians dislike these categories, and prefer to call them sentence letters, or sentential variables. In either event, they occur where a sentence can be substituted, just as the x and y of algebra stand where an expression for a number can be substituted. A statement like ‘If someone believes that p and q then he believes that p’ says that in any case in which someone believes a conjunction (such as ‘It is raining and it is windy’, then that person believes its individual parts (that it is raining). Variations encountered include capitals (P, Q,…), or italic capitals P, Q,….

Lower-case italic letters from the end of the alphabet: x, y, z…, are used as object variables. This means that they stand where reference to a person, or a thing, or a number might take place. Using such a variable, the example above could be phrased: ‘If x believes that p and q then x believes that p’, where x stands for any person. This notation is virtually universal, although the typographical appearance of the variables varies.

As in common mathematical usage, lower case roman letters, especially n, k, j…, are used in a context to refer to specific numbers. From the beginning of the alphabet, a, b, c…, are also individual constants, or terms used in a context to refer to specific things or people. Fa means that some specific thing, a, is F, and is therefore a self-standing sentence, true or false as the case may be. Fx by contrast is not, because nothing is picked out by the variable x.

Capital roman letters, F, G, R, stand for predicates and relational expressions. Particular instances of these are standard: for instance, identity (=), non-identity (≠), greater than and less than (>, <), and other mathematical relations. The usual convention is for predicate letters to stand before the terms to which they apply. Fn means that n is F; Rab means that a bears the relation R to b. In some works this would be written aRb.

The most simple relations between propositions studied in logic are the truth functions. These include:

Not. Not-p is the negation of p. Classically, it is the proposition that is false when p is true, and vice versa. In this work it is written not-p where the context is informal, and ¬p in more formal contexts. These mean exactly the same. Variations encountered include −p and ~p.

And. p and q is the conjunction of the two propositions. It is true if and only if they are both true. In this work it is written p & q. Variations encountered include p · q, and, more commonly, pq.

Or. p or q is the disjunction of the two propositions. It is true if and only if at least one of them is true. In this work it is written pq, and this is standard. Exclusive disjunction, meaning that one of p, q is true, but not both, is sometimes encountered, written pq.

Implication. Logic studies various kinds of implication. The most simple is called material implication. Here it is written pq. The most common variation is pq.

Equivalence. If pq and qp then p and q are said to be equivalent (they have the same truth value). Informally this is often expressed as p iff q. It is written pq. The most common alternative is pq.

This is the basic set of truth functions, in terms of which others are usually defined. In the predicate calculus the internal structure of propositions, as well as relations between them, is studied. The key notions are the two quantifiers:

The universal quantifier. In this work this is written ∀. (∀x)Fx means that everything is F. Variations that may be met include (Ax)Fx and (x)Fx.

The existential quantifier. In this work this is written ∃. (∃x)Fx means that something is F. The principal variation that may be met is (Ex)Fx.

In the predicate calculus numerical quantifiers can be defined, e.g. (∃nx)(Fx) means that there are n xs such that Fx. The principal variation is (∃!x)Fx (called E-shriek x), meaning that there is exactly one x such that x is F.

Terms may be defined from definite descriptions. The main examples encountered are (1x)Fx (the unique x such that x is F) and (µx)Fx (the least x such that x is F).

Modal logic studies the notion of propositions being necessary or possible. The basic notation is:

Necessarily p. Written □p. The main variation is Np.

Possibly p. Written ⋄p. The main variation is Mp.

In *metatheory, or the theory of logical systems, formulae and their relations become the topic. In this work capital roman A, B are variables for formulae, with A1…An referring to a sequence of formulae. In other works, Greek in various forms (α, β…) may be encountered. The principal relations that matter are:

There is a proof of B from A. This is standardly written A ⊦ B.

B is true in all interpretations in which A is true. This is standardly written A ⊧ B.

In traditional, or Aristotelian logic, there is not the same array of notions. Sentences are thought of as made up from terms, such as a subject and predicate, or the middle term of a syllogism. Capital roman letters (S, P, M) are used for these in this work. Set theory introduces a small new range of fundamental terms:

{x: Fx} refers to the set of things, x, that meet a condition F. This is now standard. A set may also be referred to by listing its members (‘extensionally’): {a, b, c} is the set whose members are a, b, and c.

The set with no members, or null set, is written ∅. An older variation is ∧.

Sets themselves are denoted by capital roman S, T, etc. There are many typographical variations possible.

∈ denotes set-membership. x ∈ S means that x is a member of the set S.

x ∈ {y: Gy} means that x is a member of the set of things that is G.

<…> refers to an ordered n-tuple.

The main notions used to construct sets include:

Intersection. S Union. S ∪ R is the set of things that belong either to S or to R. This too is standard.

Complement. S̄ is the set of things that do not belong to S.

Cartesian product. S×R is the set of ordered pairs whose first member belongs to S, and second belongs to R.

Relations between sets include:

Subset: S ⊆ R means that all members of S are members of R (notice that S ⊆ S).

Proper subset: S ⊂ R means that S is included in R (it is a subset, but not identical with R).

The main non-standard notation that may be encountered is Polish notation, which is explained in the body of the dictionary, as is substitutional quantification and its notation.