*Article* **Term Logic**

## **Peter Simons**

Department of Philosophy, Trinity College Dublin, College Green, Dublin 2, Ireland; psimons@tcd.ie Received: 19 December 2019; Accepted: 6 February 2020; Published: 10 February 2020

**Abstract:** The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, a constant empty term, and term conjunction and negation. The idea of basing term logic on existence or non-existence, outlined by Brentano, is here carried through in modern guise. It is shown how categorical syllogistic reduces to just two forms of inference. Tree and diagram methods of testing validity are described. An obvious translation into monadic predicate logic shows the system is decidable, and additional expressive power brought by adding quantifiers enables numerical predicates to be defined. The system's advantages for pedagogy are indicated.

**Keywords:** term logic; Franz Brentano; Lewis Carroll; logic trees; logic diagrams

## **1. Terminology**

A *term logic* is one in which the only categorematic expressions are terms, that is to say, nominal expressions. Examples of terms from ordinary language are: singular terms, such as 'Socrates', 'the North Pole', 'Vulcan'; plural terms, such as 'the Beatles', 'the signatories to the Geneva Convention'; and general terms, such as 'planet', 'black dog', 'negatively charged particle'. From this, it will be seen that the presence or absence of a definite article makes no di fference to whether an expression is a term or not. It will further be seen that terms may be simple or complex. In term logic itself, we will employ mainly term variables: there will be only two constant terms, given below. All other expressions in a term logic are formal, or what were once called syncategorematic. They are the logical constants needed to form sentences using terms, and such operators on terms as may form complex terms from simpler ones, and the logical connectives of propositional logic. Quantifiers will be added later.

The syllogistic of Aristotle and his successors was a term logic, as was that of such logical algebraists as Leibniz, Boole, Jevons, Venn and Neville Keynes. Term logic was augmented by relational expressions in De Morgan, Peirce and Schröder, but terms, except for singular terms, disappeared altogether from the predicate logic of Frege, Russell and their successors. An exception was the logical system of Le´sniewski, who retained plural and general terms, though Le´sniewski's system was also a predicate logic rather than a purely term logic. Term logic is a very natural medium for representing many inferences of ordinary discourse, more natural indeed than standard predicate logic. Though it has much less expressive power than predicate logic, being in its elementary form equivalent to monadic predicate calculus, it has much to recommend it from a pedagogical point of view, a fact recognised by Łukasiewicz, whose university textbook *Elements of Mathematical Logic* [1] augmented propositional calculus not with predicate calculus but with Aristotelian syllogistic.

The version of term logic we shall present owes much in inspiration to the logical reforms of Franz Brentano [2–4] with some influence from the logical writings of Lewis Carroll.

## **2. Language**

## *2.1. Grammar*

The grammar of our language will be categorial, with two basic categories: sentence (s) and term (n). (It is standard in categorial grammars to notate the nominal category by 'n' for 'name' rather than 't' for 'term'. We are following this tradition notationally, though we call the category by the older expression 'term'.) A functor category, the category of functor expressions taking arguments of categories β1, ... , β*n* as arguments and forming an expression of category α, will be denoted as αβ1 ... β*n*.

## *2.2. Basic Vocabulary*

The Table 1 Basic Vocabulary below gives the basic expression used, together with their syntactic categories, categorial indices, and how we describe them.


**Table 1.** Basic Vocabulary.

The intended meanings of the term-logical constants are given in the Table 2 below:

**Table 2.** Meanings of Basic Term-Logical Constants.


## *2.3. Basic Syntax*

In the interest of simplicity and brevity of expression, we delicately abuse the use/mention distinction and do not introduce special metavariables.

## *2.4. Terms*

Any term variable or term constant is a term If *a* is a term, so is (*a*) If *a* and *b* are terms, so is (*ab*) Nothingelseisatermexceptasallowedbydefinitions.

## *2.5. Sentences*

If *a* and *b* are terms, *a* = *b* is a sentence If *a* is a term, N*a* is a sentence If *p* is a sentence, so is ~(*p*) If *p* and *q* are sentences, so are (*p* ∧ *q*), (*p* ∨ *q*), (*p* → *q*) and (*p* ↔ *q*) Nothing else is a sentence except as allowed by definitions.

We will omit parentheses where no ambiguity results. Propositional connectives are assumed to bind in the order negation, conjunction, disjunction, implication, equivalence.

## **3. Axioms**

## *3.1. Propositional Logic Background*

We presuppose without mention axioms sufficient for classical bivalent propositional logic, with substitution and modus ponens as inference rules.

## *3.2. Term-Logical Axioms*

3.2.1. Intensional

for = ID *a* = *a* (Identity) LEIB *a* = *b* → (*p*[*a*] → *p*[*b*]) (Leibniz) where *p*[*x*] is any sentential context containing the term *x*.
