1. Introduction

Broadly construed, Craig’s Interpolation Theorem ⁽¹⁹⁵⁷⁾ states that for any pair of formulas ϕ and ψ such that ϕ implies ψ, an interpolant is a formula γ such that ϕ implies γ, γ implies ψ, and the non-logical symbols in γ occur both in ϕ and in ψ. In other words, if we let Relϕ stand for the set of relation symbols in ϕ (resp. Relψ), then Craig’s theorem would say what follows: if ϕ and ψ are sentences such that ⊢ϕ⇒ψ, then there exists a sentence γ (an interpolant) such that ⊢ϕ⇒γ, ⊢γ⇒ψ, and Rel(γ) ⊆Relϕ ∩Relψ.

Lyndon gave a generalization of Craig’s Interpolation Theorem in terms of the polarities of the relation symbols involved in a formula ⁽¹⁹⁵⁹⁾. His theorem states that if ϕ and ψ are first order formulas, and ϕ implies ψ, then there exists a formula γ (an interpolant) such that ϕ implies γ, γ implies ψ, and every relation symbol which occurs positively (resp. negatively) in γ occurs positively (resp. negatively) both in ϕ and ψ. In other words, if we let Rel +ϕ (resp. Rel -ϕ stand for the set of relation symbols with at least one positive (resp. at least one negative) occurrence in ϕ , then, if ⊢ϕ⇒ψ, then there is an interpolant γ with respect to Rel ⁺ and Rel ^- .

Now, Hodges has argued, rather convincingly, that Lyndon’s Interpolation Theorem gives a generalization of the laws of distribution for traditional, Aristotelian syllogistic (1998), which is a claim we find quite interesting because distribution a concept we use when we talk about terms that appear under the scope of a universal quantifier (^{Keynes 1906}; ^{Sommers 1975}; ^{Wilson 1987})- serves a fundamental purpose in Sommers and Englebretsen’s Term Functor Logic a novel logic that recovers some insights of the traditional, Aristotelian logic- in so far as it helps account for a notion of validity (^{Sommers 1982}; ^{Englebretsen 1987}; ^{Englebretsen 1996}; ^{Sommers and Englebretsen 2000}; ^{Englebretsen and Sayward 2011}).

Given these preliminaries, we cannot help but wonder: granted these superficial links between Lyndon’s theorem, distribution, and Sommers and Englebretsen’s logic, is there some sort of interpolation for said logic? In this contribution we explore this issue and we suggest there is a plausible sense in which we could account for some sort of term interpolation for Sommers and Englebretsen’s Term Functor Logic. In order to reach this goal we first expound the basics of Term Functor Logic, together with a sense of term distribution, and then we offer a proof of our main contribution; at the end, we close with a brief discussion.

2. Term Functor Logic

Assertoric syllogistic the logic at the core of traditional, Aristotelian logic- is a term logic that makes use of categorical statements in order to capture a basic notion of assertion. A categorical statement is a statement composed by two terms, a quantity, and a quality. Typically, we say a categorical statement is a statement of the form:

where Quantity = {All, Some}, Quality = {is (are), is not (are not)}, and S and P are term-schemes, so that we obtain four kinds of categorical statements: the universal affirmative, the universal negative, the particular affirmative, and the particular negative. The following are examples of categorical statements, in said order:

From the standpoint of Sommers and Englebretsen’s Term Functor Logic (TFL, for short), we say a categorical statement in TFL is a statement of the form:

where ± is shorthand for the + and - functors, and S and P are term-schemes. So, for example, we can model the four traditional, categorical statements in TFL as follows, where the term L stands for logicians, and S stands for smart:¹

Given this language (i.e., LTFL= <T, ±>, where T=A,B,C,… is a set of terms, and ± is shorthand for the + and - functors), TFL offers a basic sense of validity as follows (^{Englebretsen 1996, p. 167}): a syllogism is valid (in TFL) iff (1) the algebraic sum of the premises is equal to the conclusion, and (2) the number of particular conclusions (viz., zero or one) is equal to the number of particular premises. And so, with this logic we can model assertoric inferences like the one shown in Table 1.

In this example we can clearly see how the previous definition works: (1) if we add up the premises we obtain the algebraic expression (-P + S) + (-L + P) = -P + S -L + P = -L + S, so that the sum of the premises is algebraically equal to the conclusion, and the conclusion is -L + S, rather than + S -L, because (2) the number of conclusions with particular quantity (zero in this case) is the same as the number of premises with particular quantity (zero in this case).²

The previous definition of validity is an algebraic rendition of the dictum de omni et nullo (don, for short), the principle that states that everything that is affirmed (resp. denied) of a whole can be affirmed (resp. denied) of a part (Pr. An. A1, 24b26-30), and so, alternatively, using the concept of distribution, we can define the same notion of validity as follows (^{Sommers 1989}): let ϕ(+M) stand for a statement with a positive occurrence of the term M (i.e., M is undistributed) in a formula ϕ, and let ψ(-M) stand for a statement in which M occurs negatively (i.e., M is distributed) in a formula ψ, then the rule would state that if ϕ(-M) and ψ(+M), then ϕψ, since -M and +M cancel each other out. In the previous example we can clearly see the term P occurs distributed in premise 1 (i.e., - P), and undistributed in premise 2 (i.e., + P), and so both occurrences cancel each other out, thus allowing the conclusion (i.e., -L + S).

This second account of validity, although equivalent to the first definition, depends upon a sense of term distribution defined as follows (^{Sommers 1989}): given a term T in a categorical statement ϕ, we say:

Thus, formally, given any statement, its terms are distributed (resp. undistributed) if and only if said terms occur next to the minus functor (resp. next to the plus functor). Informally, given its quantity and its quality, we say a statement distributes its terms if and only if they have a universal quantity or a negative quality. For example, if we recall the previous categorical statements we can pinpoint which terms are distributed (-) and which terms are not (+):

Now, up to this point, given the previous exposition, one could think the notion of validity for this logic only covers monadic or syllogism-like inferences, but that would be a hasty conclusion. We can extend the previous notion of validity in order to cover relational, singular, and compound inferences by noticing that any categorical statement is a well-formed formula of TFL, and either by enlarging the rules of inference (as in ^{Englebretsen 1996}) or by implementing tableaux proof methods (as in ^{Castro-Manzano 2018}; ^{Castro-Manzano and Reyes-Cárdenas 2018}; ^{Castro-Manzano 2020}) without losing the notion of term distribution. So, we can easily extend the notion of validity for this logic as follows:

Definition 1 (Valid inference (in TFL)). Let ϕ and ψ stand for arbitrary TFL well-formed formulas. An inference is valid (in TFL), namely, ⊢-ϕ+ψ iff ψ is obtained from ϕ by applying some adequate rule of inference (or by following a tableaux procedure).³

In other words, ϕ implies ψ if and only ψ is obtained from ϕ by applying some adequate rule of inference (like DON) or by following a tableaux procedure. As an example, and in order to close this section, consider a relational inference à la De Morgan in Table 2.

To wrap this up, and as we mentioned previously, Hodges has argued that Lyndon’s Interpolation Theorem gives a generalization of the laws of distribution for syllogistic (1998), which is something we find interesting because distribution helps account for the notion of validity for TFL, which stems from syllogistic (^{Sommers 1982}; ^{Englebretsen 1987}; ^{Englebretsen 1996}; ^{Sommers and Englebretsen 2000}; ^{Englebretsen and Sayward 2011}). Our objective now is to formally explore a sense of term interpolation for TFL.

3. Interpolation in Term Logic

Before we offer a formal result regarding term interpolation, let us consider some preliminaries. First, consider that the expression -ϕ+​ϕ is a valid formula or a tautology, and that its negation, --ϕ+ϕ = +ϕ-ϕ, is a contradiction (^{Englebretsen 1996}). Second, let us define a function Distϕ (resp. unDistϕ) that returns the set of distributed terms (resp. undistributed terms) in a categorical statement ϕ. For example, if ϕ:=-L+S then Distϕ = {L} and unDistϕ = {S}. In particular, in case ϕ is a tautology or a contradiction, we assign Distϕ = ⌀, and unDist(ϕ) = ⌀, for any term cannot be both distributed and undistributed. Third, we use the following sort of diagrams as to indicate what follows from what in a given inference:

So, in this example we are saying that ϕ implies ψ, and that ϕ is the set of premises -A+B,-C+A, while ψ is the conclusion - C + B.

Fourth, besides the DON inference rule, TFL includes the simplification rule (Simp) as follows: either conjunct can be deduced from a conjunctive formula; from a particularly quantified formula with a conjunctive subject-term, we can deduce either the statement form of the subject-term or a new statement just like the original but without one of the conjuncts of the subject-term (i.e., from +(+ X+ Y) ± Z we can deduce any of the following: + X + Y, + X ± Z, or + Y ± Z), and from a universally quantified formula with a conjunctive predicate-term we can deduce a new statement just like the original but without one of the conjuncts of the predicate-term (i.e., from -X ± (+Y + Z) we can deduce either -X ± Y or -X ± Z).

Finally, there is the addition rule (Add): any two previous formulae in a sequence can be conjoined to yield a new formula, and from any pair of previous formulae that are both universal affirmations and share a common subject-term a new formula can be derived that is a universal affirmation, has the subject-term of the previous formulae, and has the conjunction of the predicate-terms of the previous formulae as its predicate-term (i.e., from -X + Y and -X + Z we can deduce -X + (+ Y + Z)).

Given these preliminaries, we now suggest a notion of term interpolation for TFL as follows:

Proposition 1. (Term interpolation). Letϕ,ψ, andγstand for arbitrary TFL well-formed formulas. Ifϕimpliesψ, i.e., if⊢-ϕ+ψ, then there is an interpolantγs.t.⊢-ϕ+γ,⊢-γ+ψ, and every term that occurs (un)distributed inγoccurs (un)distributed in bothϕandψ.

Proof. We proceed by cases. (Case 1) So, first, suppose ⊢-ϕ+ψ, and let Distϕ ∩Distψ = unDistϕ ∩unDistψ= ⌀. In this case, we have four alternatives:

1. Both ϕ and ψ are tautologies with no terms in common, i.e., ϕ:=-T+T, and ψ:=-T'+T' s.t. T=T' for arbitrary well-formed formulas T andT’. In this case, any other tautology γ with no common terms is an interpolant, for unDist(γ) ⊆unDistϕ ∩unDistψ=⌀, and Dist(γ) ⊆Distϕ ∩Distψ=⌀.

2. If ψ is a tautology and ϕ is empty, i.e., ϕ:=⌀, and ψ:=-T+T. In this case, any other tautology γ with no common terms is an interpolant, for unDist(γ) ⊆unDistϕ ∩unDistψ=⌀, and Dist(γ) ⊆Distϕ ∩Distψ=⌀.

3. If ψ is a tautology and ϕ is an arbitrary formula with no terms in common, i.e., ϕ:=±T, and ψ:=-T+T s.t. T=T'. In this case, any other tautology γ with no common terms is an interpolant, for unDist(γ) ⊆unDist(ϕ) ∩unDistψ=⌀, and Dist(γ) ⊆Distϕ ∩Distψ=⌀.

4. If ϕ is a contradiction and ψ is any formula with no common terms with ϕ, i.e., ϕ:=+T-T, and ψ:=±T' s.t. T=T'. In this case, any other contradiction γ with no common terms is an interpolant, for unDist(γ) ⊆unDistϕ ∩unDistψ=⌀, and Dist(γ) ⊆Distϕ ∩Distψ=⌀.

(Case 2) Second, in case Distϕ ∩Distψ=⌀ and unDistϕ ∩unDist(ψ) =⌀, we have to proceed by induction on the size of such sets. So, for the base case, consider Dist(ϕ) -Distψ=⌀ and unDistϕ - unDistψ=⌀. In this case, all terms of ϕ are terms of ψ, both distributed and undistributed. Then, ϕ = γ is a proper interpolant because ⊢-ϕ+ϕ and ⊢-ϕ+ψ (by assumption), so that ⊢-ϕ+ψ follows by applying DON. Now, for the inductive case, consider Distϕ - Distψ = j and unDistϕ - unDistψ = k. In this case, since ⊢-ϕ+ψ, there are some common (un)distributed terms between ϕ and ψ. Fix some γ in such a way that Dist(γ) ⊆Distϕ ∩Distψ and unDist(γ) ⊆unDistϕ ∩unDistψ. We have to show that ⊢-ϕ+γ. Let Tj1 stand for a sequence of terms ±T1,…,±Tj. Then we have three general alternatives in which this condition holds by Simp:

But in all these alternatives, we can deduce ψ from γ, i.e., ⊢-γ+ψ, in such a way that every term (un)distributed in γ is also (un)distributed in ψ:

In alternative 1, ψ follows from γ by reductio, for suppose ψ is not the case. Then we would get +-Tj1+-Tkj+1, but then by Simp we obtain -Tj1, which contradicts γ. In alternative 2, ψ follows from γ by a similar procedure. Last, in alternative 3 ψ follows from γ as in the base case. But in all these alternatives ψ follows from γ and γ is such that Dist(γ) ⊆Dist(ϕ) ∩Distψ and unDist(γ) ⊆unDistϕ ∩unDistψ.

Now, for the inductive case in which Distϕ - Distψ = j+1 and unDist(ϕ) - unDistψ = k+1, consider the next alternatives:

But again, in all these cases, ⊢-γ+ψ , as follows:

(Case 3) Third, in case Distϕ ∩Distψ=⌀ and unDistϕ ∩unDist(ψ) =⌀ , let us use induction on the size of such sets. For the base case, consider Dist(ϕ) - Distψ=⌀ and unDistϕ - unDistψ=⌀ . In this case, all terms of ϕ are terms of ψ, both distributed and undistributed. For the inductive case, consider Dist(ϕ) - Distψ=⌀ and unDistϕ - unDistψ = k. In this case, since ⊢-ϕ+ψ , there are some common undistributed terms between ϕ and ψ . Fix some γ in such a way that unDist(γ ) ⊆unDistϕ ∩unDistψ . We have to show that ⊢-ϕ+γ. Then we have one alternative:

That ψ follows from ϕ is clear by reductio, for suppose ψ is not the case. Then we obtain +-Tj1+- (±T)  kj+1 , and by Simp, we get ∓Tkj+1 . But by applying DON to ϕ we get ±Tkj+1, which is a contradiction.

But now, ⊢-ϕ+γ (by DON), and ⊢-γ+ψ by reductio as follows:

Then clearly, for unDist(ϕ) - unDistψ = k + 1, consider the next alternative:

And thus ⊢-ϕ+γ, and ⊢-γ+ψ as follows:

(Case 4) Finally, the remaining case is similar to the third case, mutatis mutandis. □

To further illustrate this result, let us consider some examples.

Example 1 (A propositional case).⁴ Let ϕ:={-p+q,+p} and ψ:={--r--q}:

Since ⊢-ϕ+ψ, there is an interpolant γ such that unDist(γ) ⊆unDist(ϕ) ∩unDist(ψ) = q, and Dist(γ) ⊆Dist(ϕ) ∩Distψ=⌀, as follows:

In plain English, this example could say what follows, and the interpolant would be the undistributed statement “You are Socrates’ friend”:

Example 2 (A syllogistic case). Let ϕ:={-A+B,-C+A} and ψ:={-C+B}:

Given that ⊢-ϕ+ψ, there is an interpolant γ such that unDist(γ) ⊆unDist(ϕ) ∩unDistψ={B} , and Dist(γ) ⊆Distϕ ∩Distψ={C}:

This example could say what follows, and the interpolant would be “Every logician ^- is mortal ⁺”:

Example 3 (A polisyllogistic case). Let ϕ:={-A+B,-B+C,-C+D,-D+E} and ψ:={-A+E}:

Given that ⊢-ϕ+ψ , there is an interpolant γ such that unDist(γ) ⊆unDistϕ ∩unDistψ={E} , and Dist(γ) ⊆Distϕ ∩Distψ={A}:

Following our pattern of examples, this one could say what follows, and an interpolant would be “Every philosopher^- is material⁺ and every material ^- is corruptible⁺”:

Example 4 (A relational case). Let ϕ:={-B+(+L+G),-G+(+A+C),-C+M,-(+A+M)+F} and ψ:={-B+(+L+F)}:

Given that ⊢-ϕ+ψ , then there is an interpolant γ such that unDist(γ) ⊆unDistϕ ∩unDistψ={L,F}, and Dist(γ ) ⊆Diϕst∩Distψ={B}:

This example could say what follows, and an interpolant would be “Every logician^- likes⁺ some rules⁺ and every rule^- is useful⁺”:

Example 5 (Another relational case). Let ϕ:={+(-A+(-B))+C} and ψ:={-(-D+A)+(-D+(-B))}:

Since ⊢-ϕ+ψ , there is an interpolant γ such that unDist(γ) ⊆ unDistϕ ∩ unDistψ=⌀ , and Dist(γ) ⊆ Distϕ ∩ Distψ={A,B}:

Last, this example could say what follows, and an interpolant would be “Every circle^- is unsquared^-”:

4. Final Remarks

Given some superficial links between Lyndon’s theorem, term distribution, and Sommers and Englebretsen’s logic, in this contribution we have explored some sort of interpolation for TFL. We think this is an interesting result, at least, for the following reason: a research on interpolation for TFL has not been attempted yet, as far as we are aware, and such research suggests TFL has logical properties that bona fide logics typically share, which contributes to show that term logics in general, and TFL in particular, far from being superseded, are in the process of revival (^{Sommers 1982}; ^{Englebretsen 1996}; ^{Wang 1997}; ^{Correia 2017}; ^{Simons 2020}) and that their death certificate is fake (contra ^{Carnap 1930}; ^{Russell 1937}; ^{Geach 1962}).

Finally, having reached our goal, we would like to close this contribution by quickly answering some potential objections.

Objection 1. The notion of interpolation is usually defined in relation to results on proof theory, but this proposal does not develop any proof theory. It is true that interpolation is usually understood in relation to proof theory, but it would be false to claim that our result is void of developments on proof theory, since the main theorem requires the notion of proof in TFL; it is true, however, that we rely on such results or notions without discussion.

Objection 2. The notion of interpolation is usually defined in relation to results on model theory, but this proposal does not develop any model theory. Similarly, it would also be false to claim that our result is unrelated to model theory, since the main theorem requires the notion of distribution, which provides the formal semantics for TFL (^{Englebretsen 2017}).

Objection 3. The notion of interpolation has some clear meaning in classical logic, but there is no clear meaning for this so-called term interpolation. Finally, it would be unsound to argue that there is no clear meaning for term interpolation. Term interpolation would formally explain why distribution has been so important within traditional logic, since it would account for the strength of the dictum de omni et nullo, as ^{Hodges (1998)} would put it.