Formal semantics of language and the Richard-Berry paradox

F ormal seman tics of language and the Ric hard-Berry parado x Stefano Crespi Reghizzi P olitecnico di Milano DEI - Dipartimen to di Elettronica e Informazione Piazza Leonardo da Vinci, 32 Milano 20133 T el. +39 02239 93518 stefano.crespireghizzi@polimi.it Octob er 25, 2018 Abstract The classical logica l an tinomy kno wn as Ric hard- Berry paradox is combined with plausi- ble assumptions a b out the size i.e. the desc riptional complexity , of T ur ing machines forma l- izing certain sen tences, to show that formalization of language leads to contradiction. Keywor ds : formalizatio n o f la nguage, Ric hard- Berry pa r adox, Kolmog orov complexity Article typ e : letter F or mal semant ics of language has r o ots in the disciplines o f lo gic, philosophy and math- ematics with pioneering w ork b y F rege and Russell in the early 20th century and impor tant addition b y T ar ski and Carna p in particular . It emerged as an imp orta nt area of linguistics in the 1970s fr om seminal research by Ric ha rd Mon tague. In recent years fruitfu l a pplica- tions of a sp ects of formal semantics ha ve been made in co mputational linguistic, ar tificial int elligence, and natural language pro cessing. V ar ious fragments of Englis h and other langua ges hav e b een care fully formalized, but the ultimate ob jectiv e of complete formalization is still remote, and not ev eryb o dy , to say the least, believes it to b e realis tic. This corresp o ndence exhibits an internal con tra dictio n inherent in lang uage formalization, in the cla s sical form of a lo g ical pa radox. The Ric hard-Berry parado x. It is g o o d to star t by r ecalling the logical para dox describ ed in the Principia Mathematic a (Russell and Whitehead, 1917), where the author s say the paradox “was suggested to us by Mr. G.G. Ber ry of the Bo dleian Library” . The Richard-Berry Paradox is the definition of a num b er as “the least num b er that cannot b e defined in few er than t wen t y words.” (1) The a ntinom y is explained by (Li and Vit´ anyi 19 97) a s follows: If this num b er e x ists, we hav e just describ ed it in thirteen words, con tradict- ing its definitional statemen t. If such num ber do es not exist, then all natur al nu mbers can be des crib ed in few er than t wen ty words. 1 . . . F or malizing the notion of “definition” as the shor test program from whic h a num- ber can b e computed b y the reference [T uring ] machine U , it turns out that t he quoted statemen t (reformulated appro priately) is no t a n effe ctive desc ription . It is known that the pro of of the imp ossibility of calculating the num b er descr ib ed in (1 ) gives a w ay of rephras ing G¨ odels incompleteness theorem. W e obser ve in statement (1) the w ord “least” has the us ua l a rithmetic sense, a s w ell as the meaning that the num b er r eferred to is the first encoun tered in the enumeration 1 , 2 , 3 , . . . . The paradox still ho lds if w e replace the “lea st n umber” by the “first n umber ” (in the enumeration) or , if, instea d o f natur a l num be r s, we use other discrete enumerable structures, suc h as rational num b er s. W ould the paradox still hold if n umber s ar e replaced by texts? Moving from this, the pr esent no te a rgues that the statement that natural language can be co mpletely formalized leads to antinom y . The formal seman tics pa rado x. Let us fo cus on some natural language, say En- glish, a nd as s ume that any text can b e precisely formalize d b y a T uring machine (or for that matter by an y o ther computationally complete for malism). This means that given a text, a pro cedure exists transla ting it to a formal definition, a s the description or prog ram of a T uring machine. W e do not rule out the p ossibility that a text b e formalized in differe nt wa ys, cor resp onding to differ e nt T uring machines. A T uring machine description can b e enco ded into a binary string, and the str ings describing T uring machines can be enum era ted by increa sing size . Thus any machine has a p os ition in the en umeratio n, and it makes se ns e to sa y that a mac hine comes before or after another. The notion of mac hine size can be made rigoro us enough, as done in the theory of co mplexity of Kolmo g orov, Chaitin, and Solomonoff, for which we refer to the classica l b o o k (Li and Vit´ anyi 1997 ). Here w e take s iz e a s synonym y of the p osition of a ma chine in the ab ove enum era tion. Since a text may have more than one forma l definition, w e consider the first one in the enum era tion as the reference definition. Th us the reference machine is the one of lea s t size among the definitions o f a text. The s ize of the reference mac hine formalizing a cer tain text will b e called the formal c om- plexity of the text. Now we can imagine to sort the E nglish texts in a scending o rder of their fo rmal co mplex- it y . This mea ns tex t one pr ecedes text t wo, if their r esp ective formal definitions as T uring machines, which w e hav e assumed to b e computable, ar e in that order in the en umeratio n. Since texts are now ordered, it makes sense to c onsider “the first text suc h that its formal complex ity is no t less than tw ent y .” (2) Sent ence (2) will be deno ted b y t (20 ), to emphasize that it is parameterize d b y the num ber t wen ty . By changing the numerical parameter, we may obtain similar sentences denoted as t (21), and so on. Before w e pro ceed with the main argument, we hav e to ma ke explicit tw o in tuitively reasona ble assumptions on the complexit y of formal descriptions. Un b oundednes s assum ption. F or ea ch integer n , there exists a text having formal com- plexity greater than n . The idea is that texts may requir e arbitrarily complex fo rmal descriptions. Logarithmic comple xit y . Consider the family of sentences t (20) , t (21) , . . . . F or a ny in- teger n greater than tw ent y , the formal complexity of t ( n ) do es not exceed the formal complexity of t (20) b y more than a qua nt ity prop o rtional to log ( n ). 2 T o justify the assumption, consider tha t the formal description o f t ( n ) includes t wo parts: one is indep endent of n and therefore has a s ize less than the size of t (20 ); the other part has a logarithmic complexity , since it is well known that integers can b e enco ded b y a po sitional num b er repres entation having a log arithmic n umber of digits. Main argumen t. Two cases are po ssible. 1. First supp ose “the fir st text” referred to in (2) exists, a nd consider the complexity of the for mal description of sen tence t (20). Two s ubca ses are po ssible. (a) The forma l complexit y o f t (20) is less than t wen ty . Since t (20) pr ovides a definition of ‘the first text. . . ”, we hav e found a forma l descriptio n of it co nt ra dicting the definition. (b) The formal complexity of t (20 ) is k ≥ 20. Then, from the logar ithmic co mplexity assumption, one can find a sufficiently large integer K g r eater than k such that the formal co mplexity o f se nt ence t ( K ) is less than K , th us obtaining a contradiction for the “the firs t text such that its formal complexity is not less than K .” 2. Second, supp ose “the first text” refer red to in (2) do es not exists. Then any text would hav e for mal complexity less than tw ent y , contradicting the unboundedness a s sumption. T o conclude, we obser ve the classical Richard-Ber r y par adox r elies on an en umeration of English sent ences ordered by the num b er of w or ds they contain, i.e. essentially by their length. This version of the pa radox order s the sen tences accor ding to the length of their formalizations (say b y T ur ing machines), assumed to exist. The ensuing contradiction prov es that a complete computational formalization of natural language s entences is impos s ible. References Li Ming and Vitanyi P . (1997 ). An Introduction to Kolmogo rov Complexity and its Ap- plications. (New Y ork: Springer). Russell B. a nd Whitehead A.N. (1917 ). Principia Mathematica. (Cambridge: Univ ersity Press). 3