# What are the possible ways to symbolically represent entities, within formal logic?

What are the different solutions proposed in the academic literature to represents symbolically individual entities within formal logic expressions?

One solution I am aware of is to use Latin letters. For example, while the letter x in Fx is used to stand for any individual within a predetermined collection, the letter a in Fa is used to stand for one unique individual. For example, a may stand for Socrates so that Ha could mean that Socrates is human and Ma could mean that he is mortal.

Is there any other solution discussed in the academic literature?

Thank you for any scholarly references.

• In predicate logic usually x,y are used for individual variables and a,b for individual constants. Commented Feb 20 at 17:57
• There are modern extensions of Aristotle's term logic, called term functor logic, where individuals are expressed by terms, as in syllogistic, instead of constants and variables. This is worked out in Sommers-Englebretsen, An Invitation to Formal Reasoning, see also informal intro on Siris. Commented Feb 20 at 21:01

The sense of idea of ἄτομον, translated into Latin as individuum, that is, what we get by individuation is so primordial for us that it is uniformly an invariable constituent of thought and language (viz., the concept of individual is an indispensable ingredient of natural language metaphysics).

The grammatical devices to refer to individual entities (nouns, pronouns and the relevant phrases) may exhibit variations across natural languages, however, they all can be explained in one way or another around an ontological correspondence between name and entity.

Artificial languages follow the same metaphysical perspective. The main formal devices that fulfil the referential functions of natural language grammar are individual constants and individual variables (of which values are supposed to be individuals of some universe). Usually, we simply call them constants and variables (denoted by a and x, respectively, in the question). Their implementations differ (consider, for example, several computer programming languages), but they all boil down to the same ontological view.

Built upon them, there are variable-binding term operators, such as Russell's ℩-operator and Hilbert's ε-operator, that are comparable to particular phrases.

Just for the sake of broadening our horizon, we may furthermore desire to get out of the frame of the constant-variable pair, or let us say, the term frame. Then, it might be said, by somewhat forced interpretation, that individual entities can be referred to by their absence, since they cannot be done without. Such a formal language is that of lambda calculus, in which variables are essentially place-holders. Hence, very roughly, the language defines functions and leaves the place of individual entities open.

• That is a really good answer. Commented Mar 25 at 4:19
• "The grammatical devices to refer to individual entities (nouns, pronouns and the relevant phrases)" The grammatical devices to refer to individual entities are noun phrases, phrases with a noun or a pronoun as its head. - 2. "lambda calculus, in which variables are essentially place-holders. Hence, very roughly, the language defines functions and leaves the place of individual entities open." I don't see how that would be specific to lambda calculus. In any language, words are essentially place-holders. Commented Mar 25 at 7:46
• "In any language, words are essentially place-holders": Is this an ideational theory of meaning? I think you ought to clarify it, not a mainstream view. Commented Mar 25 at 8:40