**Proof.**


DARAPTI is one of those syllogisms whose validity is dependent on existential import of the subject term of the two premises: this is made explicit as the first premise.

In fact, every valid categorical syllogism has one of just three forms. We let \* be a toggle operator taking positive terms to negative terms and vice versa, that is, if *a* is positive *a*\* = *a*', while if *a* is negative, *a* = *b*', *a*\* = *b*. Then every syllogism has as its core one of the three valid inference forms

POSITIVE E*ab*, N*bc*\* E*ac* (cf. DARII) NEGATIVE N*ab*\*, N*bc* N*ac* (cf. CELARENT) IMPORT E*a*, N*ab*\*, N*bc*\* E*ac* (cf. BARBARI)

All can be derived from one of these by choosing *b* or *c* to be positive or negative, relabelling, swapping the order of premises, and applying commutativity (*ab* = *ba*) to obtain simple conversion. Furthermore, either POSITIVE or NEGATIVE is derivable from the other via partial contraposition and relabelling, so in the end Aristotelian categorical syllogistic owes its validity to just two forms of syllogistic inference, with a little propositional help.

Before concluding this section, a word about the ironic designation 'Heidegger's Law'. The basic non-existence predicate 'N' is best read as "Nothing is (a)", and the definitionally empty term 'Λ' can often be read as 'nothing'. The axiomatic formula 'NΛ' can then be read as 'Nothing is (a) nothing', or, with a little linguistic chivvying, 'Nothing noths' or *Das Nichts nichtet*. Of course, Heidegger did not intend to say anything so straightforward or trivial, but it does refute Carnap's claim that the sentence has to be nonsense. *Au contraire*: suitably understood, it is a logical law.

It may seem a little perverse to have based this logic on the negative idea of non-existence rather than the positive one of existence. Of course, it is possible to do it the other way around, but in general the axioms for N are more satisfyingly elegant than those for E.

## **5. Intension and Extension**

One of the standard principles of the Boolean algebra that emerged from Boole's and others' work on the algebra of terms in the nineteenth century is that all empty terms are identical: we have, e.g., that *aa*' = *bb*', N*a* ∧ N*b* → *a* = *b*, N*a* → *a* = Λ. These are *not* theorems of our system and it is important to see why. Their analogues with equivalence '≡' replacing identity '=' *are* theorems, and if we were to add an axiom of extensionality

#### EXT *a* ≡ *b* → *a* = *b*

they would be theorems, and there would be no distinction between identity and equivalence. Most nineteenth century algebraic logicians understood their logic extensionally, so would be happy with this simplification. However, Brentano was not, and nor am I. The axioms involving identity = and those involving non-existence N are distinct in intent. Existence and non-existence have to do, for the most part, with contingent facts: there are narwals; there are no unicorns. There are some non-contingent principles involving N, obviously, our axioms such as term non-contradiction, but the premises in syllogisms and the antecedents in NWK and NEXH are typically contingent in application to actual propositions and actual inferences.

As will be seen more clearly when we consider diagrammatic representation, the axioms governing identity have to do not with contingent propositions but with the framework of discourse within which propositions and inference are employed. In any of the syllogisms considered, we are looking at three terms, their negations and conjunctions. For three terms, there are eight maximally specific combinations of conjunction and negation, for example *ab'c*, and the question may then arise whether N or E is true of this term. The axioms governing identity (and conjunction and negation) are formal synonymies, there to tell us, in advance of any statements about what does or does not exist, when term expressions relate to the same possibilities. Of these, the most obvious perhaps is *a* = *a*", term double negation. No contingent facts have any bearing on these two expressions' relating to the same possibility of existence or non-existence. For this reason, I call the principles governing '=' *intensional* and those governing 'N' *extensional*. That does not mean I here endorse a modal logic or possible worlds, simply that the role played by framework description is di fferent from and prior to that played by questions of existence and non-existence.

## **6. Consistency**

The system is consistent. In the empty universe, every term is empty, and the extensional axioms are trivially true. Interpreting identity as equivalence, so are the intensional axioms. The empty universe is expressly not ruled out by the system: the dual to Heidegger's Law, namely.

EV There is something (rather than nothing) is not a theorem, because it is false for the empty universe.

## **7. Decidability**

It is well known that first-order monadic predicate logic is decidable [6]. We may interpret the term logic in monadic predicate logic by associating each term *a* with a monadic predicate *A*

> *a* → *Ax*

the term Λ with a necessarily empty predicate, for example

$$
\Lambda \mapsto \neg(x = x)
$$

with complex terms as follows

$$a' \mapsto \neg(Ax)$$

$$ab \mapsto Ax \land Bx$$

and the predicates as follows

$$a = b \mapsto \forall x (Ax \leftrightarrow Bx)$$

$$\mathsf{N}a \mapsto \neg \exists x (Ax).$$

In this way, each formula of the term logic is correlated with a formula of monadic predicate logic. The interpretation validates extensionality. It can be seen that all the axioms of the term logic system are valid formulas of monadic first-order predicate logic, and the validity of a formula or inference with finitely many premises containing *n* term variables may be decided on a domain of no more than 2*n* individuals.

## **8. Tree Proof Techniques**

Formulas and inferences with finitely many premises may be tested for validity or invalidity using tree techniques. The basic ideas were presented earlier for a slightly simpler system, so we can be brief [7]. We assume that all rules for trees for propositional logic are available, and we confine attention to term formulas using only the basic vocabulary of term variables, Λ, =, ', conjunction, and N, as well as propositional connectives. Any defined constants are eliminated first as per their definitions. The counterexample set of an inference to be tested consists of the premises together with the negation of the conclusion, or the negation of a formula if that formula's validity is to be tested. A tree starts with the counterexample set. It may then be extended according to the following rules:


If all branches close, the formula or inference is valid; if any branch remains open, the formulas along it may all be true and constitute a counterexample.

## **9. Diagram Techniques**

Diagrams for deciding the validity of logical inferences go back centuries, but the first e ffective ones are due to John Venn [8]. The idea, as applied to term logic, is to start with a diagram consisting of as many areas, or *cells*, as there are conjunctions of all simple terms and their negations contained in an inference. If there are *n* simple terms, that will be 2*n* cells. Venn's own curvilinear diagrams are inferior to the rectilinear ones proposed by Lewis Carroll, who ingeniously constructed diagrams for up to eight di fferent simple terms, and indicated how to extend these further [9] (p. 245 ff.: "My Method of Diagrams". Carroll was incidentally the first to use trees as an aid for solving logic problems: *ibid*., 279 ff. Since one of his problems ("Froggy's Problem", *ibid*., 338 ff.) is a sorites in 18 terms, which would require a diagram with 262,144 cells, taxing human capacity to solve, further aids were clearly needed.) The method for term logic as for syllogistic is to shade out those cells corresponding to N propositions, and indicate by crosses those cells corresponding to E propositions. The chief di fficulty is that an E proposition whose term is not a maximal compound of simple terms and their negations must straddle several cells disjunctively, a problem compounded in any term-logical formula or inference employing disjunction or its equivalent. For this reason, diagrams are practicable only for relatively small and straightforward problems. Trees branch easily, but the only way to branch a diagram is to treat several diagrams disjunctively.

An unfilled diagram for *n* term variables, with its 2*n* cells, represents the framework within which N and E propositions employing these variables are to be represented, and is neutral with respect to such propositions. The axioms for identity are then to be understood as indicating different but formally equivalent ways in which cells or groups of cells are indicated. This is why they play a different role in the logic from the N and E propositions.

## **10. Quantifiers**

It is natural to extend the term logic employed to date with variable-binding quantifiers. One reason is simply to enhance the representative scope of the system. Quantifiers binding term variables do not affect the decidability of the resulting system, (Ackermann, *loc.cit.*) but they bring greater expressive power. We take the universal quantifier as primitive and add the following axiom schemes, where *A* and *B* are sentences:

$$\text{QDIST} \qquad \forall a (A \to B) \to (A \to \forall a (B))$$

where *a* is any term variable which is not free in *A*;

> QINST #x2200;*a*(*A*) → *A*[*t*/*a*]

where *t* is a term expression (variable, constant or compound), *A*[*t*/*a*] is the result of substituting *t* for all free occurrences of *a* in *A*, and no free occurrence of *a* in *A* is in a well-formed part of *A* of the form ∀*t*(*B*) [10] (p. 172).

The particular quantifier may then be introduced in the standard way as dual of the universal:

$$\text{FORS}\\ \text{OME} \qquad \exists a(A) \leftrightarrow \sim \forall a(\sim A).$$

It should be noted that the particular quantifier does not in this system carry existential import. Since it is a theorem that ~EΛ, it follows that ∃*a*(~E*a*), so the quantifier cannot very well mean 'there exists', but must mean, neutrally, 'for some'. If we wish to talk about existing things, we have the predicate 'E' to hand.

One of the ways in which quantifiers introduce greater expressive power is that they facilitate expressions of number. Hitherto, expressions of the form E*a* only said that there is some *a*. This is compatible with there being one, two, ... any number of *a*s, and this is why the cells of any diagram for finitely many propositions need only be finite in number. Indeed, the terms need not denote individuals or pluralities of individuals at all: they could denote numberless stuffs, as do mass terms in ordinary language. The syllogism in Darii

All morphine is highly addictive

Some pain medication is morphine:

*therefore* Some pain medication is highly addictive

is no less valid for being about stuffs ("substances") rather than individuals. If we wish to introduce numbers, to count individuals or also consignments of stuff, we need quantifiers. Here is how to define 'at least two':

$$\geq 2 \qquad \mathsf{E}\_{\geq 2}a \leftrightarrow \mathsf{Ex}(\mathsf{E}ax \land \mathsf{E}ax\prime)$$

So we can define 'exactly one' as

> =1 E=1*<sup>a</sup>* ↔ E*a* ∧ ~E≥2*<sup>a</sup>*

#### **11. Pegagogical Advantages of Term Logic**

For students coming to logic with little or no background except in propositional calculus, term logic is quite natural and easy to understand. It is close to natural language (no bound variables, quantifiers as phrases not operators, logical and grammatical form closely similar); by comparison with predicate logic, there is minimal paraphrasing required; it has a straightforward and intuitive denotational semantics requiring no set theory, and is easy to do without special symbols. It can be treated by methods building easily on those of propositional logic: semantic diagrams, natural deduction proofs, axioms and trees. It has an accessible metalogic: it is sound, complete, and decidable, uses finitistic methods, affords a variety of approaches and good illustration of basic concepts. In difficulty, it is only slightly more complex than truth-tables and natural deduction for propositional logic, and is readily scriptable should one wish to write suitable computer programs. For an introduction to the history of logic, it allows much greater scope for comparison than post-Fregean predicate logic. It allows a variety of bases, apart from the one we have chosen. Alternative bases are equational (Leibniz, Boole, Jevons), subsumptional (Leibniz, Peirce, Schröder), existential (Leibniz, Brentano, Carroll), traditional (Aristotle, Łukasiewicz), or based on the singular copula (Le´sniewski, Słupecki). It admits of various extensions, most obviously by introducing predicates, especially relational predicates, but also modal [11], towards Le´sniewskian logic [3,12], and introducing higher types, up to and including full simple type theory.

On the negative side, it delays students' encounter with ∀ and ∃, with relations and multiple generality, has rather few links to modern mathematics, and being now unorthodox, suffers from a modern textbook gap. Most textbooks highlighting term logic (Łukasiewicz excepted) are antiquated attempts to keep pre-Fregean logic alive, often for non-logical reasons.

Nevertheless, I hope enough has been shown in this paper to sugges<sup>t</sup> that term logic, whether done this way or in some other way, remains worthy of the attention of logicians and teachers of logic.

**Funding:** This research received no external funding.

**Conflicts of Interest:** No conflict of interest.
