Aristotle's Formal Language

1985 by Ralph E. Kenyon, Jr.

May 20, 1985*

Aristotle's attempts to state a formal structure for language were severely hampered by his lacking the use-mention distinction. He did not have an explicit device, such as the modern use of quotation marks, available to him; he had only the implicit and rudimentary device of naming, and such indicators as 'word' and 'sentence'. Since he lacked an explicit use-mention indicator, we should expect to find his writings somewhat ambiguous where he is attempting to discuss primarily syntactic structures.

The use-mention distinction is important in distinguishing between syntactic and semantic features of a language. Modern specifications for a formal language can be constructed totally without regard for any use of that language; in such a construction, the tokens of the language are mentioned only. To approximate the problem that Aristotle had, we must include the semantic elements as well. Structures which include semantic as well as syntactic elements are found in model theory. The purely syntactic structures of a language are formally specified; also, an interpretation for the language is formally specified. Symbols of the language are assigned to objects and relations among these objects. Questions about what symbols are mentioned are about the syntactic structure of the model, that is, about the language. Questions about how symbols are used are about the semantic structure of the model, that is, about the domain of interpretation.

Aristotle seems to have been more aware of the use-mention distinction in some areas than in others. For example, today we speak of homonyms as words which sound the same but have different uses. Aristotle spoke of homonymous things, which is an obvious confusion of the use-mention distinction. In spite of this evidence to the contrary, he seems to have been implicitly aware of the distinction. His attempts to specify a structure for language, in which he uses the device of 'naming', and the primordial mention-indicators of 'word', and 'statement' are remarkably lucid. Let us assume that, even though he was severely hampered by the lack of an explicit device, the distinction was apparent to him. In order for us to approximate the difficulty that Aristotle had, we must remember to keep this distinction somewhat ambiguous.

Modern Formal Logic

What is meant by a formal language? The formal language which attempts to capture the portion of natural language with which Aristotle was interested is the predicate logic. The language 'PL' [predicate logic] is a structure consisting of

(a) primitive symbols

(1) a1, a2, . . . -- constants
(2) x1, x2, . . . -- variables
(3) Aij -- predicate letters
(4) fij -- function symbols
(5) ¬, ⇒ -- logical symbols
(6) '(',')',',' -- punctuation
Note, the additional logical symbols of '∧' (and) and '∨' (or) and '≡' (if and only if) can be defined in terms of '¬' and '⇒' as follows.
(7) (A ∨ B) =Df. (¬A ⇒ B)
(8) (A ∧ B) =Df. ¬(A ⇒ ¬B).
(9) (A ≡ B) =Df. (A ⇒ B) ∧ (B ⇒ A)
(b) Terms

A term is (recursively) :

(1) a variable or a constant.
(2) if t1, t2, . . . tn are terms and fnk is a function symbol, then fnk(t1, t2, . . . tn) is also a term.

(c) well formed formula (wff)

(1) if Ank is a predicate letter and t1, t2, . . . tn are terms, then Ank(t1, t2, . . . tn) is a wff.
(Such a wff is called an atomic wff).
(2) if A is a wff then (¬A) is a wff (not-A).
(3) if A and B are wffs then (A ⇒ B) is a wff.
(4) if A is a wff and v is a variable then ((v)A) is a wff.
(5) All wffs can be generated by finitely many application of (1) thru (4) (there are no other wffs).

Note: The existential quantifier can be defined in terms of these wffs by:

(6) ((Ev)A) =Df. ¬((v)¬A)

Formal Theory

An axiomatic theory consists of a language, a set of logical axioms (in the language), a set of proper axioms, and a set of rules of inference. <L, AL, AP, R>

(a) Predicate Calculus Logical Axioms

(1) A ⇒ (B ⇒ C)
(2) (A ⇒ (B ⇒ C)) ⇒ ((A ⇒ B) ⇒ (A ⇒ C))
(3) (¬B ⇒ ¬A) ⇒ ((¬B ⇒ A) ⇒ B)
(4) (v)A ⇒ Av[t] (where t is freely substitutable for v in A) -- Av[t] is the wff obtained by substituting t for all occurrences of v in A.
(5) (v)(A ⇒ B) ⇒ (A ⇒ (v)B) (v not free in A)

(b)Predicate Calculus Rules of Inference

(1) from wffs 'A' and '(A ⇒ B)' we may infer 'B' -- modus ponens.
(2) from wff A we may infer (v)A -- gen


Interpretation

An interpretation (for a language) is a structure, <D,I>, consisting of a domain of objects, D, and an assignment function, I, satisfying the following.

(a) D is non-empty.
(b) The domain of I consists of all constants, predicate letters, and function symbols in L.
(c) The range of I consists of the following.
(1) if c is a constant, then I(c) ∈ D.
(2) if Aij is a predicate letter, then I(Aij) ⊆ Di
(3) if fij is a function letter, then I(fij):Di → D.
Note: If 'a1', 'a2', . . . 'an' are constant symbols and 'fnk' is a function symbol, then fnk(a1, a2, . . . an) ∈ D. Therefore, a term, for which particular values of any variables therein are selected, picks out an object in D, the domain of interpretation.

Truth. A wff with no free variables is called a closed wff or a sentence and represents a proposition which is true or false (with respect to the interpretation). If 'Ank' is a predicate letter and 'a1', 'a2', . . . 'an' are constants, then 'Ank(a1, a2, . . . an)' is true iff <a1, a2, . . . an> ∈ Ank. A wff with free variables stands for a relation on the domain of the interpretation.

Use-Mention Convention

In a formal theory, we shall apply the use-mention convention to the symbols of the language and the objects of the interpretation in the following way. A constant symbol of the language denotes the object in the domain of the interpretation. A constant symbol is used to refer to the object assigned to it by the interpretation function. Similarly, predicate letters and function symbols shall be used to denote, or refer to, the appropriate objects assigned. This will prove cumbersome in the following way; we will be required to use quotes to mention the symbols. The expressions ''A'', ''v'', ''a'', and ''f'' refer to the symbols, while the expressions 'a', 'A', and 'f', refer to elements of D, subsets of D, and functions on D, while 'v' refers to elements ranging over D. By applying this convention in this way, 'A11(a1)' is a sentence in L, while A11(a1) is a proposition composed of objects in the range of the interpretation function I. In the case of constant letters, I('a') = a. To simplify matters somewhat, whenever I('a') = a, we shall say that 'a' is a name of a. The expression N(a) shall pick out a name of a, which may be 'a'; but if a = b then N(a) might be 'b' as well.

Aristotle's Formal Language

Aristotle begins his explication of a formal structure for language by saying what a name, a verb, a negation, an affirmation, a statement and a sentence are. Because of his lack of explicit indicators to distinguish the mention of terms from their use, his language is fraught with ambiguity. If we were to use names with the same restrictions as Aristotle had, we would have great difficulty whenever we wished to discuss properties and characteristics of the names; our language would often seem to be talking about the things referred to by the names. We can specify 'a' is a noun or name; Aristotle could not. Aristotle refers to the conventionality inherent in the act of naming in order to mention names. He talks about the distinction between words and their uses by noting that words are symbols for our thoughts and refer to things. (16 a. 3.) He also notes that names are significant by convention. (16 a. 19) The significance of a name is the object it refers to, a semantic notion, whereas the convention in a name is the actual word itself, a syntactic notion. Separating these is one way of indicating the use-mention distinction without explicit indicators. It is by convention that a particular name refers to something; by pointing at the convention behind a name, the emphasis is on the symbol used to do the referring, not the object, and is an attempt to specify the syntax. He clarifies that he intends the syntactic notion by noting that parts of names are not significant (16 a. 21.), and there are no "naturally occurring" names (16 a. 26). Since names refer to objects, that is, each name is assigned to a particular object, we may infer that Aristotle has selected names as the atomic constants of the formal language, and the objects that names refer to as the objects in the interpretation for the language.

Nouns are the atomic constants.

Greek nouns have cases. Aristotle selects only the nominative case as valid atomic constants.

I wonder if Aristotle's initial impulse is to consider any arbitrary combination of symbols for analysis. Immediately after he defines 'name' or 'noun', he considers the combination of 'not' with a noun. It seems that there is a possible equivocation in Aristotle's use of 'not'. In some cases it seems to be used in the manner of set theory as the set-complement. Such is the case in which he considers 'not' with a noun. Such a structure he designates as an indefinite noun or indefinite name.

The lack of the use-mention distinction shows up again in Aristotle's characterization of verb. He says verbs are signs of things said of something else. (16 b. 6.) The term 'signs' marks the mention side of the distinction, while the phrase 'things said of something else' clearly has semantic connotations. The things said of something else suggests that there is a way of deciding or a "fact of the matter" regarding whether the something else should be considered to be included among those of which the thing is said. In other words, the things of which something is said comprise a set of objects. This set of objects, which satisfy the thing said of them, is clearly a set of objects which have names. A verb is a sign for this set. Use of the verb refers to this set. Moreover, when something is said of something else, that something is predicated of that something else. Aristotle is selecting verbs as his predicate symbols. The objects these verbs refer to are sets of objects in the interpretation.

As in the case of names or nouns, Aristotle considers the combination of 'not' with a verb. It is apparent that, in this case, he is concerned with the interpretation for 'not' which corresponds to set-complement. He calls a verb with 'not' an indefinite verb. (16 b. 11.) Aristotle restricts further what qualifies as verbs by noting the difference between past and future tenses, or inflections of verbs. (16 b. 16.) The technical term 'verb' will be reserved for the present tense only. (Aristotle called the past and future tenses "tenses of a verb", but not the present tense.) That use of a verb refers to something is stated explicitly by Aristotle.

When uttered just by itself a verb is a name and signifies something . . . but it does not yet signify whether it is or not. . . . but it additionally signifies some combination, which cannot be thought of without the components. (16 b. 19-25)

Aristotle's referents are ambiguous here because the verb and its referent comprise a composite object whose parts are referred to equivocally. The object named by the verb is incomplete and requires another component to produce a "thinkable" combination. By choosing the singular pronoun, 'it', Aristotle has implied a selection process in completing a verb. "Whether it is or not" is signified when the additional missing component is supplied to the verb. A verb designates the things of which it may be said; but by virtue of its incompleteness nothing is selected for a judgment.

Verbs are the predicate symbols.

A sentence is composed of significant expressions. A statement is a sentence which can be true or false. It is well to note here that nouns and indefinite nouns, including their cases, and verbs and indefinite verbs, including the tenses or inflections all comprise significant expressions. By selecting statements as comprising only those sentences which can be true or false, Aristotle has limited the application to propositions.

A single statement-making sentence is either

(1) an affirmation or
(2) a negation, or
(3) a single sentence in virtue of a connective uniting single statement-making sentences. (17 a.8)
Notice that this structure allows recursion!

An affirmation or negation is a simple statement. (17 a. 8.)

Aristotle says that a simple statement specifies either that something does or does not hold of something. (17 a. 23.) In other words, a simple statement specifies that something is (or is not) predicated of some named thing, i.e., that a simple statement consists in a name to which something is predicated (or not). The corresponding structures in PL are the one-place predicate letters applied to constant symbols. A simple affirmation consists of a name and a verb. If 'A1i' is a predicate letter, and 'aj' is a constant symbol, then 'A1i(aj)' is a simple affirmation.

For Aristotle, statements comprise well formed formulas.

(1) A statement consists of a name and a verb or an inflection of a verb (past and future tenses).
(2) If S is a statement then not-S is also a statement, namely the denial of S (S and not-S comprise a contradiction). The use of not here is clearly that of negation and not of set-complement.
(3) If S1 and S2 are statements and C is a connective then (S1 C S2) is also a statement.

Aristotle makes a precise distinction between universals and speaking of something universally. 'Universal' as a noun refers to "things", for example, man; 'universally' (speaking) applies to universal nouns and is indicated by the terms 'all' and 'every'.

Aristotle divides things into universals and particulars. By making such a distinction, he is setting up a schema for variables and constants. I look to modern computer languages for motivation for this interpretation. In some computer languages there are strong type distinctions. For example BASIC allows only numerical and string variables; a variable must be of one type or the other and can only take values in its own type. The new DoD language, ADA, allows many other types. A discrete type is allowed in which the objects over which the variable ranges are explicitly named. First a type is declared, and then a variable of that type is declared. ADA also allows "overloading" names; names may be reused for different functions; the compiler knows from the context which function is meant. The following ADA language program fragments illustrate the technique. (In ADA, '--' indicated a comment.)

 

-- Declare a discrete variable type called MAN

type MAN is (PLATO, ARISTOTLE, SOCRATES, CICERO, TULLY, JOHN, MARY, LINDA, BETH);

-- variables of type MAN can only be assigned a value
-- from the list in the type declaration. A discrete type
-- declaration implicitly creates constant objects of
-- the declared type. Now declare subtypes

subtype WOMAN is MAN range (MARY .. LINDA);
subtype LOVER_OF_KNOWLEDGE is MAN range (PLATO .. SOCRATES);
-- now declare variables with the same name as the type.
MAN : MAN; -- so we can ask IF SOCRATES = MAN THEN ...
WOMAN : WOMAN;
PHILOSOPHER : LOVER_OF_KNOWLEDGE;
-- These variables can only be compared or assigned values -- from the list in the declaration.
Aristotle's usage closely follows this model; for example 'man' is a variable of which 'Aristotle' is a constant. Because of the lack of indicators to explicitly mark the use-mention distinction, the objects so named and their associated class type variable values are intimately associated with their names. We can think of man as a class which includes all men; we can also think of 'man' as the name for this class, or as the name of a variable which ranges over the set of members of the class. (As a name of the class, it corresponds to the ADA language name of a type.) When Aristotle speaks of things divided into universals and particulars, he, in effect, speaks of variables and constants in his formal language structure (mention) at the same time as he speaks of their associated objects (use). His language is strongly typed with many types arranged in a hierarchical system. 'Living beings' has 'man' as a subtype; 'man' has both 'philosopher' and 'woman' as subtypes. 'Aristotle' is a living being within the range of 'man' and also within the range of 'philosopher', but not within the range of 'woman'. The simple distinction between universals and individuals matches with the distinction between variables and constants (which may be the values over which the variable ranges).

In addition, things may be spoken of universally as well as spoken of not universally. Speaking universally of something involves use of the terms 'all' or 'every'; speaking not universally involves the terms 'a' or 'some'. (17 b. 16-25) 'All men are white' involves speaking universally ('all') about a universal ('men').

In addition to the simple statements, universal and not universal statements are also well formed. Also, Aristotle specifies that the relation between a universal affirmation and a not-universal denial is that of contradictories. (17 b. 16.) Such is precisely the relationship between the univeral and existential quantifiers. Aristotle's characterization of speaking universally and speaking not-universally about universals gives the specifications for the remaining criteria for well formed statements by defining universal (and existential) quantification.

Aristotle's Symbol Structure Summary

Aristotle's language [AL] is a structure consisting of

(a) primitive symbols

(1) a1, a2, ... -- names of individuals (constants)
(2) v1, v2, ... -- names of universals (variables)
(3) Ai -- verbs (predicates)
(4) ¬, ∧ -- logical symbols
(5) '(',')',',' -- punctuation
Note, the additional logical symbols of '∨' (or) and '⇒' (material conditional) and '≡' (if and only if) can be defined in terms of '¬' and '∧' as follows.
(6) (A ∨ B) =Df. ¬(¬A ∧ ¬B)
(7) (A ⇒ B) =Df. ¬(A ∧ ¬B).
(8) (A ≡ B) =Df. (¬(A ∧ ¬B)) ∧ (¬(¬A ∧ B))

(b) well formed formula (wff)

(1) if Ak is a verb and ai is an individual name, then aiAk is a wff.
(2) if Ak is a verb and vi is a universal name, then viAk is a wff.
(3) if A is a wff then (¬A) is a wff (not-A)
(4) if A and B are wffs then (A ∧ B) is a wff.
(5) if A is a wff and v is a universal then ((v)A) is a wff.
(6) if A is a wff and v is a universal then ((Ev)A) is a wff.
(7) All wffs can be generated by finitely many application of (1) thru (6) (there are no other wffs).

Aristotle's Interpretation

For Aristotle, each name is intimately connected with its referent. The set of objects which have names is the domain of the interpretation and the natural assignment of a name to its referent allows interpreting the language in a straight-forward manner. A verb names a set of objects to which its predicate applies. Naturally, a statement consisting of a name and a verb would be a true judgment just in case the object named is one of the objects in the predicate set.

Conclusion

A moments reflection reveals that Aristotle has explicitly or implicitly laid out all the specifications for predicate logic except four. His system lacks function terms, such as would be exemplified in English by 'taller of .. and ..'; multiple place predicates, for example 'Socrates loves Plato.'; logical axioms; and explicit rules of inference. With the completion of "De Interpretatione", Aristotle achieved remarkable progress toward realizing PL; what impeded his further progress?

Geach3 points out that Aristotle was lead astray while he was working on two of these problems, how to account for multiple place predicates and rules of inference. Today, we represent 'loves' as a two place predicate (A21 in PL). 'Socrates loves Plato' is rendered as A21(a1,a2). In terms of the basic structure of spoken language, with its infix notation, Aristotle must render this two place predicate as a1A21a2. Although the meaning of 'Plato loves Socrates' is not the same, Aristotle undoubtedly noticed that the form is the same, a2A21a1. The name in the second position has the same form as the name in the first position. To allow this function, Aristotle drops his name-verb requirement in favor of the two place structure exemplified by 'Cicero is Tully.' Both his major metaphysical beliefs, that things are said of things and that things are in things, are personified by two place, infix notation statements. That Aristotle would adopt this new form as primary shows that he was attending to the syntactic structure rather than semantic considerations. It is the syntactic form of a1A21a2 and a2A21a1 that is the same; their use is different. The phenomenal success of syllogisms in explicating valid arguments in this form apparently prevented his ever returning to the basic subject-verb form.4

References

  1. J. L. Ackrill, Aristotle's Categories and De Interpretatione Oxford, 1963.
  2. Richard McKeon, The Basic Works of Aristotle, Random House, New York, 1941.
  3. P. T. Geach, "History of the Corruptions of Logic", Logical Matters, Basil Blackwell, Oxford, 1972
  4. ibid. pp. 47-48
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