Logical connectives in natural languages

Leo´ Picat

Introduction

Natural languages all have common points. These shared features define and constrain them. A key property of all natural languages is their learnability. As long as children are exposed to it, they can learn any spoken or signed language. This process takes place in environments in which children are not exposed to the full diversity of their mother tongue: it is with only few evidences that they master the ability to express themselves through language. As children are not exposed to negative evidences, learning a language would not be possible if con- straints were not present to limit the space of possible languages. Input-based theories predict that these constraints result from general cognition mechanisms. Knowledge-based theories predict that they are language specific and are part of a Universal Grammar [1].

Which approach is the right one is still a matter of debate and the implementation of these constraints within our brain is still to be determined. Yet, one can get a taste of them as they surface in linguistic universals. Universals have been formulated for most domains of linguistics. The first person to open the way was Greenberg [2] who described 50 statements about and morphology across languages. In the phono- logical domain, all languages have stops for example [3]. Though, few universals are this absolute; most of them are more statistical or take the form of conditional statements. Universals have also been defined for with more or less success [4]. Some of them shed light on non-linguistic abilities. For example, a cross-linguistic study of the color lexicon revealed that the number and meaning of color words follow a strict hierarchy: if a language has only two color words, it is systematically a word for light and a word for dark; if a language has a word for blue, it also has words for green and red. This may follow from the physiology of vision as our eyes are best suited to detect green and red. Yet, these findings and the methodology behind them are subject to debate and not uncontroversial [5]. Other universals can take a more formal form. For instance, content words follow a continuity constraint that ensures that their is continuous in some conceptual space of entities. For animal names, it means that a single word cannot denote both a specific mammalian and a specific reptile without denoting all other animals between them in the phylogeny of species. These examples highlight the potential benefits of studying semantic constraints. It is a way of under- standing the interplay between general and language specific cognitive abilities. This project has the ambition to make a first step towards the analysis in these terms of logical words.

Logic and semantics have been in a tight relationship during the past centuries: classical provides fruitful models to understand the meaning of sentences, intentional logic offers tools to analyze attitude , tense or modality. The space of what logic allows is yet not entirely used by natural languages. Constraints that restrict the list of what can be used can be formulated as formal properties that operators must have to be lexicalized. For example there is no non-conservative determiners and such words are harder to learn [6]. This project proposes to extend this knowledge to one of the basic syncategorematic elements of logic: logical connectives. It stems from a simple observation: there is no word connecting two that always outputs true. The goal pursued here is two-fold: characterize the possible connectives and

1 understand what constraints shape their lexicalization.

Section 1 gives a formal description of logical connectives. Section 2 dives in their use in natural lan- guages. Section 3 replaces them in the bigger picture of logical words.

1 Logical connectives – Formally

Logical connectives are semantic objects that take one or more of the same type and return an of this type. I will on unary and binary connectives as connectives of higher arity are more uncommon in natural languages (even though binary connectives such as AND can have more than two arguments). Unary connectives are of type < x,x > and binary connectives of type < x,< x,x >>. Yet, connectives are also mathematical objects and this section explores them as such to describe their formal properties and their relation with each other.

1.1 Boolean functions Logical connectives can be seen as n-ary Boolean functions: a from {0,1}n to {0,1}. There are two unary functions and sixteen binary functions. They can be described by specifying for each input the output. This will form the truth-table of each connective. The two unary functions are the identity (that returns its input) and the (that returns the opposite of its input) (we ignore the unary and ). All binary functions are given in table1.

Boolean functions can be grouped together. Given a of functions B, one can extend this set using composition by defining new functions: with two binary functions f and g, one can define the function h such that h = g( f (x1,x2), f (x3,x4)) where x1, x2, x3 and x4 are in {0,1}. The set composed of all functions defined in this way is noted [B]. A set of functions B is said to be a closed B = [B], i.e. if B is closed under composition.

1.2 Post’s Emil Post published in 1941 a complete description of Boolean functions: the Post’s lattice [7]. He classified all functions into 70 closed classes embedded in one another, each class being defined by a particular property. This subsection exposes a simplified version of the lattice focusing only on unary and binary Boolean functions. Classes’ names are adapted from [8]. The definitions are given for binary functions but the reader will easily extend them to unary functions:

• Constant C Constant functions (the tautology and the contradiction). • Et and Vel Functions composed with the tautology, the contradiction, and, respectively, the con- junction and the disjunction.

• Monotonic M Functions f such that f (x1,x2) ≤ f (x3,x4) for all x1, x2, x3 and x4 verifying x1 ≤ x3 and x2 ≤ x4.

• 0 and 1 reproducing R0 and R1 Functions such that if they are given only 0s (or 1s), they return 0 (or 1).

2 • Self-dual D Functions f such that for all (x1,x2) ∈ {0,1} , f (¬x1,¬x2) = ¬ f (x1,x2). • Affine A Functions for which each input either always or never changes the output (classical exam- ples are the tautology or the biconditional)

• 0 and 1 separating S0 and S1 Functions such that if the output is 0 (or 1), there is at least one 0 (or one 1) in the input.

2 Q Q 0 1 0 1 0 1 1 0 0 0 P P 1 1 1 1 0 0 (a) Tautology – > (b) Contradiction – ⊥

Q Q 0 1 0 1 0 0 0 0 1 1 P P 1 1 1 1 0 0 (c) P – p (d) Negation P – ¬p

Q Q 0 1 0 1 0 0 1 0 1 0 P P 1 0 1 1 1 0 (e) Proposition Q – q (f) Negation Q – ¬q

Q Q 0 1 0 1 0 0 0 0 1 1 P P 1 0 1 1 1 0 (g) – ∧, AND (h) Alternative denial – ↑, NAND

Q Q 0 1 0 1 0 0 1 0 1 0 P P 1 1 1 1 0 0 (i) Disjunction – ∨, OR (j) Joint denial – ↓, NOR

Q Q 0 1 0 1 0 1 1 0 0 0 P P 1 0 1 1 1 0

(k) Material implication – → (l) Material nonimplication – 9

Q Q 0 1 0 1 0 1 0 0 0 1 P P 1 1 1 1 0 0

(m) implication – ← (n) Converse nonimplication – 8

Q Q 0 1 0 1 0 1 0 0 0 1 P P 1 0 1 1 1 0

(o) Biconditional – ↔ (p) Exclusive disjunction – =, XOR Table 1: List of the sixteen Boolean binary functions or logical connectives

3 Some properties are not taken into account in Post’s lattice:

• Associativity In an expression containing several times the same connective, this connective is said to be associative if the order of the operations does not matter.

• Commutativity The arguments of a commutative connective may be switched without affecting the output.

1.3 Functional A set of logical connectives is said to be complete if it is

• Sufficient Its closure under composition contains all Boolean functions (if it can reproduce the sixteen truth-tables presented in table1).

• Non-redundant No function of the set can be defined using other functions from the set.

Post proved that a set is functionally complete if it is not a of the closed classes R0, R1, M, D or A. This result can be used to construct these sets (see table2)[9].

Sets of 1 function {↑}, {↓} {>,9}, {>,8}, {⊥,→}, {⊥,←}, {∧,¬}, {∨,¬} Sets of 2 functions {→,¬}, {→,9}, {→,9}, {→,8}, {→,=}, {9,¬}, {9,↔} {←,¬}, {←,9}, {←,8}, {←,=}, {8,=} Sets of 3 functions {>,∨,=}, {>,∧,=}, {⊥,∨,↔}, {⊥,∧,↔}, {∨,↔=}, {∧,↔=} Table 2: Functionally complete sets of Boolean functions

Some properties of connectives only make sense when they are considered as sets:

• Distributivity A connective + distributes over a connective · if P + (Q · R) = (P + Q) · (P + R) for all P, Q and R.

• Absorption A pair of connectives + and · satisfies absorption if P + (P · Q) = P for all P and Q.

1.4 Conclusion The formal description of logical connectives provides an overview of what can be possibly expressed in natural languages. A notion of importance is as it ensures that a set of connectives can be used to reconstruct the sixteen binary Boolean functions.

2 Logical connectives – Verbally

This section explores the use of logical connectives in natural languages (the discussion is based on En- glish and French). All connectives are not lexicalized (the tautology or the contradiction are not for instance). This observa- tion leads to the simple conclusion that choices must be made regarding the expressible connectives. The inquiry will start with the idea that the main constraint to satisfy is to reach functional completeness. Even though it will be refined later, it seems like a logical point to start if languages should possess maximal expressible power. Subsection 1 examines how functional completeness could be achieved. Subsection 2 examines how it is achieved. Finally subsection 3 proposes constraints on logical connectives.

4 2.1 Expressing connectives To allow maximal expressive power, a natural language should provide a way to its speakers to express the sixteen possible connectives. One can imagine three different ways of achieving it.

2.1.1 Single words for each connective Lexcalizing every single connective is an easy solution as there is no underlying constraint. Expressing all truth tables requires no complexity as all relations can be stated using a single word. Yet, this is not a simple system if one takes the perspective of the learner. Without constraints to reduce the possible space, learning gets more complex if not impossible.

A grammatical definition of what a single word connective is is needed. The semantic type of unary connectives was defined as < x,x > and binary connectives as < x,< x,x >>, with x = t for the simplest cases. Connectives could be qualified as type-reproducing operators. In French, it corresponds to the famous mais, ou, et, donc, or, ni, car. This reduces the space of how a connective can be realized as a single word by excluding verbs for instance.

2.1.2 A compositional system The idea is to select a set of connectives satisfying functional completeness and compose the remaining connectives using functional composition. This solution requires a trade-off between simplicity and easi- ness. I define simplicity as simple to learn, i.e. as constrained as possible and easiness as easy to express, i.e. each connectives is expressed as shortly as possible.

The system should be easy enough so that expressing connectives do not get fastidious. For example if one chooses the NAND connective, expressing the conjunction of P and Q requires the use of three operators: (P NAND Q) NAND (P NAND Q). At the same time, it should be simple enough so that the set of connectives is restricted and simple to acquire.

2.1.3 Paraphrases Such a system would use single words for some connectives and paraphrases for the others. The para- phrase would not make use of connectives per se; this differentiates this solution from a compositional system. It could combine simplicity and easiness. The restricted set of connectives embeds several constraints facilitating learning. The paraphrases even though longer than single words, avoids the caveats of compo- sitionality by being shorter.

Yet this system raises and concerns. First, how should the choice of the single word connectives be made? It could be driven by meta-pragmatic constraints: a single word will always be shorter than a paraphrase. Then if a choice should be made between a paraphrase and a single word, frequent relations should be expressed with a single word. More complex ones would not because of another constraint enforcing simplicity on single word connectives. But this only pushes the problem one step further: how frequent relations are defined? Second, how is the distinction between compositional formula and paraphrases made? It would be more precise to define paraphrase as compositional expressions using not only connectives. For instance, if a system uses the negation and has a word such as the English entail, one could express the material nonim- plication as entail not. But this raises another issue. If the non-connective words used in the paraphrase are not connectives, how can the system ensure that the result of composition actually expresses the targeted truth tables. For instance, it has long been known that the material implication is not a perfect description

5 for natural language implication.

With these three possible systems in mind, it is now time to turn to real natural languages connectives.

2.2 Expressed connectives A quick survey of French and English shows that almost all connectives can be expressed (see table3).

Tautology ? Contradiction ? Proposition P ? Negation P ? Proposition Q ? Negation Q ? Conjunction et and Alternative denial ? Disjunction ou or Joint denial ni nor Material implication donc, implique, si alors so, entails, if then Material nonimplication n’implique pas, mais pas doesn’t entail but not Converse implication car, est implique,´ seulement si because, is entailed Converse nonimplication n’est pas implique´ is not entailed Biconditional si et seulement si, juste si if and only if, juste in case Exclusive disjunction ou bien either

Table 3: Lexicalization of logical connectives in French and English

2.2.1 Chosen strategy The system used in English and French resembles the paraphrases system we described in section 2.1.3. Even though this is a very informal observation, common and known connectives are lexicalized with a single word (negation, conjunction, disjunction, NOR, material and converse implication). Besides the conjunction, the disjunction and NOR, other connectives can be expressed with a paraphrase or compositionally. This four connectives hence have a special status.

The most interesting observation is that the tautology, the contradiction, NAND and the 4 connectives that ignores one of their arguments are not expressible in French and in English, neither with single word, paraphrases or compositionally (though one could imagine expressing proposition Q as p. Wait... Never- mind what I just said! q. But this goes beyond the classical cases this paper explores).

The system seems not to use compositionality to express connectives. Except for the material nonimpli- cation, connectives that do not have a single word to express them are lexicalized using paraphrases.

2.2.2 NOT, AND, OR, NOR This subsubsection explores the properties of the four connectives only lexicalized as a single word. To begin with, let’s note that this set is not functionally complete in the sense defined in section 1.3 as it is redundant (AND and OR can be defined with the negation and, respectively, OR and AND). The disjunction and the conjunction satisfy absorption and distribute over one another. NOR only distributes over the conjunction. All of them are commutative and associative.

6 Functional completeness seems not to be a goal of this choice of connectives as the strategy does not rely on compositionality. Taking this two conclusions together, an immediate consequence is that connectives need not to be con- sidered as set.

2.3 A first step towards a constrained system The analysis of expressed connectives is neither easy nor simple. First, it uses a mixed strategy with both single word connectives and complex expressions involving other grammatical categories. Second, the list of single word connectives is not constant across languages as some lacks a NOR.

Formulating constraints as universal is thus not the most basic first step one could make. Nevertheless, among the sixteen connectives, we are left with seven forbidden ones: the tautology, the contradiction, proposition and negation P and proposition and negation Q. Their analysis could lead to genuine constraints. These seven connectives cannot be expressed using a single word or a paraphrase. Only NAND could possibly be expressed verbally using compositional expressions such as not P and Q with a long pause between not and P.

2.3.1 Dependence on the input The tautology and the contradiction share the property of not depending on their inputs. Such a connective would violate the principle manner: to express truth, one would have to say P BLA Q where BLA would be the tautology operator. Besides, the very utility to express a tautology using an operator is worth asking. This leads to formulate a first constraint. Constraint 1. The output of a binary connective should depend on its input. The four connectives ignoring one of their arguments share the property of depending on only one of their inputs. This may be an even bigger violation of the principle of manner: why say P BLOP Q when one could say just P? This leads to a stronger version of the previous constraint. Constraint 2. The output of a binary connective should depend on both of its input. Pragmatic violations could drive the choice of single word connectives.

2.3.2 The case of NAND NAND is almost a unique connective: it is functionally complete and has almost none of the properties we mentioned above. But NOR is in the same position, except that it can be easily lexicalized. NAND hence appears to be a mystery towards which we shall return in the next section.

2.4 Conclusion As far as English and French are concerned, logical connectives are expressed using a mixed strategy of single words and paraphrases. Compositionality is seldom used and seems to be restricted to scholars’ talks. This point deserves a few lines. French and English has been used by scholars, philosophers and mathematicians for a very long time. Sharing and thinking around these disciplines is made easier if the meta-language can offer ways of expressing logical connectives. Hence, one cannot exclude the fact that English and French are exceptions regarding their rich lexicon of logical connectives. The pressure science has exerted on language may constitute another constraint, not a cognitive constraint but a social constraint. Without a typological survey over languages to assess this hypothesis, caution is required. Finally, expressible connectives must respect a pragmatic constraint that specifies that the role of their arguments should not be vacuous.

7 3 Logical connectives as logical words

This section reviews an article by Katzir and Singh [10]. In this paper, they explore the constraints on the lexicalization of logical operators. Logical connectives belong to this class as do quantifiers, modals or temporal adverbials. The authors hold the position that all these categories share common properties between their members. They can all be defined in the same terms. This explains a common asymmetry regarding the elements that can be lexicalized as single words. The number of lexicalizable elements is limited to four and only three of them actually are. The challenge is then two-fold: which constraints reduce this number to four? What blocks the lexicalization of the other operators? Subsection 1 presents the analysis for quantifiers. Subsection 2 extends this analysis to other logical operators. Subsection 3 comes back to logical connectives.

3.1 Square of oppositions From back to Aristole, philosophers and logician have noted negative relationships between quantifiers. They can be represented in a square in which the four corners correspond to the universal quantifier (A), the existential quantifier (I), the existential negation (E) and the universal negation (O).

contrary A E implies contradictory implies

I O subcontrary

Table 4: The square of oppositions a.k.a Aristotle’s square

Horns [11] noticed that only the A, I and E corner can be lexicalized, leaving the O corner as an exception. The mechanism behind this ban is that the meaning of the existential negation can be obtained through scalar implicatures. If one utters a sentence with the I corner, it would have been more informative to use the A corner, hence we can reinforce the I corner by adding the negation the A corner, i.e. the O corner. This follows from classical accounts of scalar implicatures. A first concern about this explanation is that even if the meaning of the O corner surfaces, it is not present per se, it is mixed with the I corner. So the that a O corner need not be lexicalized because its meaning can be obtained by combining other corners is subject to debate. Taking this concern apart, the following constraints is proposed Constraint 3. Only the quantifier belonging to the square can be lexicalized. This is obviously unattested empirically as shows other quantifiers like most or few. Katzir and Singh reviews an account of Landmand 2004 [12] in which other quantifiers do not have the same type as the quantifiers from the square. They are treated as adjectives that combine with a silent existential quantifier operator. Yet all elements that appears to be quantifiers share a set of properties like conservativity. This suggests a common ontology to all quantifiers and is an unsolved challenge in this framework. The constraint is indeed too restrictive for quantifiers but takes more importance when one replaces quan- tifiers in the bigger picture of logical words. This is the purpose of the following subsection.

8 3.2 Aristotle’s square extended Katzir and Singh propose a generalization of the square of oppositions to modals, temporal adverbials, and logical connectives. Each corner represents an operation done on a different domain using logical connectives.

• Quantifiers are defined on the domain of individuals.

• Modals on possible worlds.

• Temporal adverbials on possible times.

• Logical connectives on propositions (if the discussion is limited to type < t,< t,t >>).

The A corner corresponds to a conjunction, the I corner to a disjunction and the E and O corner are their . This homology between these different classes of functional words can be attested by their lexical realization: there is only one word for both the conjunction and the universal quantifier in some languages. The absence of the O corner in the lexicon can be seen as a reflection of the absence of NAND.

Even from this view, constraint3 does not hold. Temporal adverbials are not restricted to always, some- times and never. Often and rarely mirrors the quantifiers most and few. As was exposed above, logical connectives are not restricted to AND, OR and NOR. If one constraint was to emerge from the square it would be:

Constraint 4. The O corner cannot be lexicalized.

3.3 Are connectives as special as quantifiers? Constraint3 predicts that only AND, OR and NOR can be lexicalized as single words. The lexicalization of other connectives as paraphrases is not a problem in this framework if one adopts the same strategy as for quantifiers and disregards them as being of a different type. The main issue concerns the single words expressing the material and the converse implication. To save constraint3, one may wonder if words like so or because actually lexicalized these connectives.

Two failures of reasoning suggest that implications are in fact interpreted as biconditional. Affirming the consequent and negating the antecedent are very attractive fallacies. However this is still not in line with constraint3 as the biconditional is not in the square of oppositions. Let’s look at paraphrases of implications. The sentence if it rains, John does not come to the party is judged true in both following situations:

• it rains and John does not come to the party

• it does not rain and John comes to the party

However, the sentence John does not come to the party because it rains is only true in the first situa- tion. More generally, implications and biconditionals are hard to accept when both arguments are . Roughly, it is hard to accept truth if it stems from falsity. Formally, non-1-separating connectives are hard to grasp. I hold the view that because is to be interpreted as a conjunction with a of causality between its arguments. In this sense because is similar to but which is a conjunction with a presupposition of opposition between the two conjuncts. If we set apart paraphrased connectives, logical connectives respect constraint3. However, the following arises: is it pertinent to come up with ad hoc solutions to exclude logical words that do no fit in the square of oppositions? Besides its conceptual clarity, the square is not able to capture the full diversity of logical words. What should come out of it is rather only constraint4, that remains unexplained...

9 Conclusion

The learnability of natural languages in normal environments implies a restriction on what is possible and what is not. Universals and constraints are two sides of the same coin. They can be formulated for every disciplines of linguistics. Semantic universals open the way to understand the cognitive limitations and abilities that allow us to understand meaning. The of these constraints to non-linguistic domain could provide clues to answer the question of the specificity of language within the cognitive system of humans.

Successful analyses of quantifiers have been performed and were confirmed by experimental learning tasks. In this project, I tried to make a first step to apply the same heuristics to logical connectives. The formal analysis of logical connectives and functional completeness has not been fruitful as natural languages seems not to use compositionality of connectives to express the sixteen truth tables. Some connectives have been excluded for pragmatic reasons, but the ban on NAND remains unexplained (even though it seems to be shared with other logical words).

Classical logic is intrinsically limited in what it can express. The question of its pertinence to model natural languages has been asked and new systems have been proposed. Analyzing in classical terms may then be a dead-end. A switch of framework could provide new insights on logical connectives.

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