Complexlang: a COMPact Logical EXperimental LANGuage Kelvin M. Liu-Huang Carnegie Mellon University [email protected] Published in Proceedings of SIGBOVIK 2019 (15 March 2019) Informal revisions (27 March 2019, 29 April 2019) Abstract and learning barrier of natural languages, Natural languages trade compactness and mathematical expressions seem better in these ways. consistency for efficiency. Complexlang is an a priori, The tradeoff is, of course, that mathematical declarative, ideographic spoken/written language descriptions can be very elaborate or unwieldy. which attempts to construct/ground the semantic We attempt to address all these concerns by structure of both morphology and syntax from first constructing Complexlang. Language should ideally principles using tools provided by propositional logic, synchronize speech, writing, and comprehension in set theory, type theory, number theory, object- order to facilitate learning. Like aUI (Weilgart, 1979) oriented programming, metaphysics, linguistics, and and Arahau (Karasev, 2006), by infusing individual classical field theory. In doing so, we hypothesize letters with meaning and using phonemic that speakers may converse in Complexlang with little orthography, words have transparent and largely training and learn some math and science in the deterministic etymology; writing, speech, and process. meaning can all be inferred from each other, reducing ambiguity, speeding up learning, and even allowing efficient and deterministic creation of 1. Introduction neologisms. For simplicity, the orthography is simply Most previous constructed languages intended for the IPA symbols of the phonemes. Unlike aUI and human use set out to improve etymological integrity Arahau, Complexlang attempts to express semantics (Zamenhof, 1887), semantic clarity (Bliss, 1965; entirely through logic, specifically patterned after set Karasev, 2006; Weilgart, 1979; Quijada, 2004), theory, rather than metaphor, resulting in compact, consistency (Weilgart, 1979; Cowan, 1997; Quijada, transparent, and unambiguous expressions. 2004), or other academic merits. Not many Semantically, consonants represent the set of all (Weilgart, 1979; Cowan, 1997; Bourland & Johnston, objects of a certain subtype. Compared to 1991; Quijada, 2004; Lang, 2014) have addressed mathematical expressions, introducing these cognitive benefits. First, the arbitrary phonetics and axiomatic sets, instead of rigorously defining morphology of most natural languages creates everyday objects that people agree on, greatly cognitive dissonance, which can be easily averted. increases the efficiency of communication. Vowels Also consider how mathematical expressions can represent Boolean, set, scalar, vector algebraic precisely express a great deal using a very small functions (possibly multiple all at once because many number of definitions. Compared to the ambiguity operations have analogous operations for Boolean/set/scalar/vector subtypes, and the operator Quijada, 2004) have attempted to explore or utilize is overloaded). Expressions are formed by selecting this hypothesis to improve cognitive function, but subsets containing the desired objects. Morphology most achieve this by through increased complexity is derived from inorder traversal because it is (Quijada, 2004). Meanwhile, Lojban is largely impossible to pronounce preorder and postorder of grounded in logic, though word formation is still many trees due to consonant duplicates (vowel arbitrary because they are synthesized from existing clusters also pose a problem). languages (Cowan, 1997), contributing to the Not only do we shape language, the Sapir- aforementioned cognitive dissonance. Complexlang Whorf hypothesis suggests that language also attempts to ground both morphology and syntax in influences (and perhaps determines) our thoughts logic using tools provided by propositional logic, set and behavior. Some previous conlangs (Weilgart, theory, type theory, number theory, object-oriented 1979; Cowan, 1997; Bourland & Johnston, 1991; programming, metaphysics, linguistics, and classical Table 1. All Complexlang vowels, their phonemes (indicated with IPA), and the function they perform (if multiple available, the function is chosen based on the input type). IP Boolean A input set input quantity input function input i ◁ ≝ var x in most recent ancestral : or ¤ ◁(퐼) ≝ Ith previous ◁(푆) ≝ previous var of subtype S letter ◁(푆, 퐼) ≝ Ith previous var of subtype S a ∧ (푃, 푄, … ) ∩ (푆, 푇, … ) × (,−, -퐼, ,−, -퐽, … ) ≝ 퐼,−1-퐽,−1- … u # ≝ 푗(. ) #(푆) ≝ 푗(푆) #(푆, 퐼) ≝ *푥 ∈ 푆 푗(푥) = 퐼+ o : (푆, 푃) ≝ ∪ *푥 ∈ 푍(푆) 푃+ : (푆, 푃, 푋) ≝ ∪ *푋 푥 ∈ 푍(푆) ∧ 푃+ e ¤(푆) ≝ 푑푒푎푙 푍(푆) ¤(푂, 푆) ≝ 푂 .푑푒푎푙 푍(푆) / * + ¤(푆, 푋) ≝ 푑푒푎푙( 푋 푥 ∈ 푍(푆) ) ¤(푂, 푆) ≝ 푂 푑푒푎푙(*푋 푥 ∈ 푍(푆)+) ɛ → (푃, 푄) ⊂ (푆, 푇) < (퐼, 퐽) ɔ ← (푃, 푄) ⊃ (푆, 푇) > (퐼, 퐽) ə ¬(푃) ∁(푆) −(퐼) ɨ ⊡ (푆) ≝ force field created by S ɪ ↔ (푃, 푄, … ) = (푆, 푇, … ) = (퐼, 퐽, … ) ʊ ⊕ (푃, 푄, … ) ∆(푆, 푇, … ) ≠ (퐼, 퐽, … ) y ⊞ (푆) ≝ 휇(퐽(푆)) ⊞ (퐼) ≝ 퐼 푑퐼 ɯ 훿(퐼, 퐽) ≝ 푑퐽 ɑ ∨ (푃, 푄, … ) ∪ (푆, 푇, … ) +(,−, -퐼, ,−, -퐽, … ) ≝ ,−-퐼 + ,−-퐽 + ⋯ where 푍(푆) = *푥 푥 ∈ 푆, 푗(푥) = 푖+ 푖 ∈ 퐽(푆) , 퐽(푆) ⊆ 핂n is the index set of S using the recommended units, 푗: 푆 → 퐽(푆) is the index of the input element in S, and 푘: 푆 × 퐽(푆) → 2푆 is the subset of S with input index Table 2. All Complexlang consonants, their phonemes (indicated with IPA), and the recommended set they represent. set subtype, 0th element, IPA emoji 풕(푺) recommended elements in set, S units of 푱(푺) _(푺, ퟎ) m ℝ real number all real numbers integer 0.0 k concept all information and concepts concept – p ⌘ maps all functions, functionals, etc. map – n ℤ integer* all integers integer 0 s ☺ organism all objects belonging to living entities organism I t ⚡ charge all spacetime infinitesimals of charged particles coulomb – – – electric field electric field magnitudes (∈ℝ) <s,m,m,m> – b 픹 Boolean† {FALSE, TRUE} integer FALSE h ⊚ soul/mind all souls/minds soul/mind my mind g ♡ feeling all thoughts and feelings feeling – ŋ ⍾ body all particles body my body d position all spacetime infinitesimals of all objects <m,m,m> here ⌖ ʔ Ø null none – – f ☼ internal all spacetime infinitesimals of massive particles joule – energy ʃ time all spacetime infinitesimals of all objects second now r ＊ entropy all spacetime infinitesimals of massive particles joule – z ⊙ mass all spacetime infinitesimals of massive particles kilogram – – – gravitational gravitational field magnitudes (∈ℝ) <s,m,m,m > – field v 핍 vector‡ all vectors integer – θ ⎔ thing all particles thing – * Integers are formed by "m" followed by any number of vowels. Each vowel, in the order shown in Table 1, corresponds to the digits 0-9, respectively. † The two Booleans are “bi” for FALSE or “ba” for TRUE. ‡ Vectors are formed by applying the “v” function to any number of scalars. field theory. Consequently, Complexlang would also phonemes among languages worldwide (Moran & allow two individuals (human to human, or perhaps McCloy). However, some were discarded due to even human to machine) with sufficient similarity with previously selected phonemes. mathematical and scientific education to converse Most of the selected vowels, i, y, ɯ, u, e, o, ɛ, with very little additional learning. Similarly, ɔ, a, and ɑ (indicated hereafter using IPA), coincide Complexlang could be used for compact transfer and with the IPA vowel gridlines, which benefit from high storage of information. sound contrast. The other selected vowels, ɪ, ʊ, and ə, are also quite phonetically and spatially distinct. Additionally, functions which have a tendency to 2. Phonetics neighbor other vowels were assigned vowels which Phonemes for Complexlang (see Table 1 and Table 2) correspond to semivowels. were greedily selected in order of the most prevalent The each of the voiceless consonants, k, p, s, transcribed through inorder traversal into a t, h, f, ʃ, and θ, were selected before its voiced sequence. counterpart, g, b, z, d, ɦ, v, ʒ, and ð, due to ease of Because information is lost during inorder articulation. No approximants were selected because serialization, we cannot deterministically deduce the these are easily confused with vowels. original tree from the spoken sequence. Hopefully, the tree structure can be deduced through context. However, if it cannot, then the structure can be 3. Orthography clarified using pitches; the speaker can sing the The orthography of Complexlang is phonemic (each expression by selecting a pitch for each vowel that is written letter corresponds to a single phoneme and lower than the pitch for the vowels in higher vice-versa). As shown in Table 1 and Table 2, branches. Complexlang is simply written using the Alternatively, preorder or postorder traversal corresponding IPA symbols for their phonemes. would preserve enough information during Alternatively, the vowels can be written using their serialization such that the tree structure can be mathematical symbols and the consonants using recovered. However preorder traversal has a emojis. tendency to create consonant clusters which are difficult to articulate; in particular, consonant pairs 4. Morphology and Syntax are completely impossible to enunciate clearly. Vowel clusters are also common and more difficult to Like in most languages and mathematical articulate. Postorder traversal suffers from the same expressions, Complexlang contains objects and problem. Additionally, the vowels (functions) appear relationships, can represent the order of those functions/relationships in a tree, and produces declarative statements which the speaker