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Logical Connectives in Natural Languages

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Logical Connectives in Natural Languages 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 syntax 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 truth or take the form of conditional statements. Universals have also been defined for semantics 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 denotation 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 logic provides fruitful models to understand the meaning of sentences, intentional logic offers tools to analyze attitude verbs, 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 propositions 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 arguments of the same type and return an element of this type. I will focus 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 function from f0;1gn to f0;1g. 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 negation (that returns the opposite of its input) (we ignore the unary tautology and contradiction). All binary functions are given in table1. Boolean functions can be grouped together. Given a set 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 f0;1g. The set composed of all functions defined in this way is noted [B]. A set of functions B is said to be a closed class if and only if B = [B], i.e. if B is closed under composition. 1.2 Post’s lattice 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) 2 f0;1g , 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) Proposition 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) Conjunction – ^, 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) Converse 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 completeness 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 subset 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 f"g, f#g f>;9g, f>;8g, f?;!g, f?; g, f^;:g, f_;:g Sets of 2 functions f!;:g, f!;9g, f!;9g, f!;8g, f!;=g, f9;:g, f9;$g f ;:g, f ;9g, f ;8g, f ;=g, f8;=g Sets of 3 functions f>;_;=g, f>;^;=g, f?;_;$g, f?;^;$g, f_;$=g, f^;$=g 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.
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