INSTITUTE OF GENERAL 
LINGUISTICS 
Groningen, March 1969 . 
Abstract of a paper for the Third International 
Conference on Computational Linguistics to be held in 
Sanga Sgby I - 4 September 1969 . 
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2. 
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Hobbes ~ Calculus of Words 
Hobbes' (1588-1679) philosophy of language transpires from 
his much quoted adage: "Words are wise men's counters". 
(Leviathan ch. IV, p. 25, in the edition of 1885) 
His ideas meant the rather revolutionary initiation of the 
preliminaries of computational linguistics. 
Hobbes' pioneering work underwent two influences, namely: 
A) a doctrinary influence from Nominalism (e.g. Oocam 1300- 
1350), which was overlapped by 
B) the disciplinary impact of contemporary Physics, more 
precisely of the newly established Mechanics (Galileo 156 ~- 
1641). 
Leibniz (1646-1715) is Hobbes' main heir; he develops Hobbes' 
ideas, together with questions of artificial languages and 
symbolic systems, into the conception of Representation, which 
became the nuclear theorem of his philosophy. 
Frege~(1848-1925) and Russell (born 1872) are the main 
participants in the last relay that took Hobbes' initiative 
to our day.) 
Prof.Dr. Pieter A. Verburg 
Hagepr eekkamp 4 
Gronin~en - The Netherlands 
HOBBES' CALCULUS OF WORDS 
Prof. Dr. P. A. V~.,-bu:g 
}-\]agep:eekka ~,. p 4 
G~onin(',en Tel 05900-~<355~ 
I. Hobbes' philosophy of language transpires from his much quoted 
adage: Words are wise men's counters. The context gives more 
evidence of his deeper intentions; the subsequent text runs as 
follows: " .......... , they do but reckon by them; but they are 
the money of fools, that value them by the authority of an 
Aristotle, a Cicero, or a Thomas, or any other doctor whatsoever, 
if but a man. - 'Subject to names', is whatsoever can enter into 
or be considered in an account, and be added one to another to 
make a sum, or subtracted one from another and leave a remainder. 
The Latins called account of money rationes, and accounting 
ratiocinatio; and that which we in bills or books of account call 
'items', they call nomina, that is 'names'; and thence it seems 
to proceed, that they extended the word 'ratio' to the faculty Of 
reckoning in all other things. The Greeks have but one word, k6~os 
for both 'speech' and 'reason'; not that they thought there was no 
speech without reason, but no reasoning without speech: and the 
act of reasoning they called 'syllogism', which signifieth summing 
up of the consequences of one saying to another." 
Hobbes' philosophy allows for the functioning of natural words as 
counters - or, as he more often calls it: as marks (in his Latin 
works: notae) - only in one respect, namely in the cognition of 
reality. The opposition 'cognitive versus communicative use of 
words' is maintained consistently; the following diagram may serve 
to make this clear: 
DIAGRAM I: Hobbes (1588-1679) 
Invention , Demonstration 
(Words as ) (Reason) (Words as ) 
(N°t--~Marks) e'~ I or I ( Si~Signs ) Reality(--( (c~itive)) Mind )-'* (c~unicative)) P'@ Others 
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The inventive application of natural words puts them to an enno- 
etical - i.e. intelligence-integrated - and instrumental use, 
which accurately handled turns out to be a reliable procedure for 
attaining rational truth. In so doing Hobbes posited the most 
striking and consequential linguistic theory of his time. 
The sign-use of natural language is allowed to be weaker; the 
interhuman communicative application is not a philosopher's 
privilege, but the everyday practice of ordinary people. They 
operate with prudence, which is the common mind as it consists of 
memor~y_~ perception and imagination. Philosophers, on the other 
hand, operate with their calculative reason, which is the counting 
and reckoning mind. 
Hobbes elaborates on his calculus of words in the sense that he 
demonstrates that the syntactic procedure is of an arithmetical 
nature in so far as we add or subtract the notions of which the 
common words are urged to become the bearers. 
2. Regarding influences Hobbes underwent from earlier theories and 
philosophies, we must first remember that he was an excellent 
classicist, who had acquired a thorough knowledge of all the 
outstanding authors of Antiquity. We suspect that Hobbes' 
opposition 'inventio versus demonstratio', is related to Plato's 
concept of the dual function of language, namely diakrinein towards 
things, didaskein towards fellow men. 
There is little basis for thinking of Hobbes as an Aristotelian. 
In the domain of language-theory and philosophical grammar, the 
~iddle Ages had yielded a realist grammar, largely under the 
influence of Aristotle's theories. The so-called Grammatica 
Speculativa did not appeal to Hobbes and only the fact that Leibniz, 
who continued and improved on Hobbes' work, reintroduced this 
theory, makes it worth mentioning at all. 
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The main trends Hobbes' linguistic philosophy had to cope with, 
were more topical. We can distinguish a doctrinar~ and a 
disciplinary trend. 
A. Nominalism had been strong and dominating since Ockham 
(1300-1350). Nominalist epistemology had taken the following 
basic form: 
DIAGRAM II: Nominalism 
Reality (only) ( (( Intellectus, 
individua) ) (-- Words, nominal-- ( conceptus, intentio 
To the nominalist, the direct intellectual or conceptual 
approach to reality is primary. From this approach he learns 
that reality consists of individua. This immediate inventive 
intellection is fundamental and the insertion of words into it 
is in a way redundant and possibly even misleading since many 
words stand for nonexistent generalities; reliable nomina only 
refer to indivi~ua. What Hobbes evidently learned from 
nominalist doctrine, was the intermediary role language could 
play as a means, as a tool in inventive thought - an 
epistemological model, which, however, he did not take over 
along with the critical and suspicious attitude towards 
language inherent in the nominalist view. For that matter, a 
critical attitude towards common language also characterized 
the politician and philosopher whom Hobbes served as a 
secretary~ namely Francis Bacon (1561-1626). Well-known are 
the latter's warnings against the pitfalls forming a constant 
danger for those who make too negligent a use of words. His 
'idola fori' especially, focused attention on lineal errors 
and inconsistencies. 
B. A much stronger impact on Hobbes' p~ilosophy came from 
Mechanics, the newly established sub-discipline of Physics, 
founded by Galileo and olosely related to the new ideas on 
motion of contemporary astronomers like Kepler. The scientific 
nature of Mechanics is in this case of great importance. 
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Inventors and designers of all kinds of machines were numerous 
in the Quattrocento and Cinquecento - one of them the great 
artist Leonardo da Vinci (1452-1519). Some published their works, 
though in the vernacular, and Galileo was acquainted with such 
writings. In Galileo this tradition of practical oraftmanshi p 
and technical skill merges with the academic tradition of the 
Aristotelian philosophy of nature. In other words: The practical 
manipulation of arithmetical and geometrical knowledge as 
acquired in the scholastic quadrivium on the one hand, and 
theories about natural motion that were radically non-mathematical 
and linked up with the pseudo-explanation of a transition from 
potentiality to actuality on the other hand, united in Galileo. 
His great achievement, which in his era places him in the 
category of outstanding scholars and inventive minds like Planck, 
de Broglie, Einstein and Heisenberg in our era, was his 
application of mathematics to physical research, more precisely, 
his analysis of macroph~sical motion by means of countin~ and 
measurin6 (As a verysimple but most significant example we 
point to the so-called parallelogram of forces). 
Galileo himself was highly conscious of the importance of his 
innovation and the perspectives it opened up. This is evidenced 
by statements like the following: "The book of nature is written 
in an alphabe~ of triangles, squares, parallelograms, circles, 
etc." Kepler had already said that the 'caelestic machina' was 
no__~t 'instar divini animalis 2 sed instar horolo~ii' and Galileo 
often expressed himself in similar terms, especially in his 
famous adage: 'universum horolo~ium est'. 
The following diagram represents Galileo's position: 
DIAGRAM III: Galileo (1564-1641) 
Physics ) ( Means: ) ( 'intelletto, 
(Motion in Astronomy ) ~--~ (Mathematical) (---q ( ~' 
and Mechanics) ) ( Symbols ) ((Reason) 
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The transition to a philosophical theory like that of Hobbes was 
facilitated because Galileo himself led the way by talking about 
the Universe as a whole instead of about Physics as the object 
of his science. 
The transition from artificial mathematical symbolism to a more 
general concept of natural language was also expedited by Galileo 
himself already comparing cyphers and figures to an alphabet. 
H.obbes' extrapolation from Physics to Universe and the 
Substitution of natural language words for mathematical s~mbcls 
can be represented as follows: 
Diagram IV: from Galileo to Hobbes 
Galileo : Physics4---- Symbols4---- Intellect 
gobbes : Reality4--- Words ~ Reason 
This part of Hobbes' conception then is the left half (Invention) 
of Diagram I. 
The inventive use of words takes the form of a calculus. In 
itself the embedding of words as tools in the cognitive process 
corresponds to the nominalist 'ennoesis', i.e. inclusion into 
thinking, of nomina. Yet there is a decisive difference in 
evaluation: for the nominalist the use of words is firstly 
optional and secondly unreliable and full of uncertainty. For 
Hobbes, however, the computational use of natural words is 
firstly necessary to obtain rational, i.e. true insights, and 
secondly, provided the calculation is performed accurately, 
fully certain and reliable. The calculation can only succeed if 
performed by philosophers, i.e. rationalist philosophers, - the 
"wise men" of his above quoted adage. 
t 
For the calculative use of words Hobbes introduces the Latin 
term ratiooinari, i.e. reasoning and reckoning in one! In his 
view it consists of addition and subtraction, in short, of 
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arithmetical operations. He adduces all manner of evidence, 
for example that the Laticn ratio - a~d come to that the 
Greek loges too - had alwaFs properly signified reokonlmg, 
counting, calculating, etc.; further, that 'syllogism' 
properly means addition, summing up, etc. 
So Hobbes' calculus with words corresponds to grammar or 
syntax in natural language, conceived of as an operation 
with words; it may be mentiomsd that in this view the 
d~namic and functional character of syntax as a kind of 
technical procedure is retained - this in contradistinction 
to its fate in Leibniz' philosophy. Natural words are, for 
philosophers, usable as counters, i.e. as premeditated 
artificial symbols. There is more to this mark/nora role 
imparted to natural signs. As a matter of fact, their 
symbolic character is, in Hobbes' opinion, their essential 
nature, right from the beginning of creation. Adam invented 
words ex arbitrio! This theorem subsumes the concept of 
natural language and its units under that of artificial 
symbolism. We must hold against this thesis that the 
drawing up of ar4y artificial symbolism is only possible 
because of our human endowedness with a lingual faculty. 
Any system of symbols is a posteriori to natural "linguality" 
and only possible because of the a priori and innate human 
language faculty. Hobbes' subsumption is like the statement: 
Look how that mother resembles her daughter! 
3. LEIBNIZ - and not Spinoza or Locke - accepted and digested 
the inheritance of Hobbes' philosophy of language. Leibniz 
was a character averse to any sort of quarrel, dispute, 
controversy, contrast, opposition, and conflict in any form. 
-7- 
In religion, politics, science and philosophy he endeav- 
cured to achieve harmony by stressing mutualities, agree- 
ments, similarities, gradualities, shading transition and 
so on. Let me examine the circumstances in which Leibniz 
conceived of his mathematical philosophy of representation. 
~. Hcbbes began publishing as early as 1642 and continued 
until the year of his death, 1679. In 1668 "An essay towards 
a real character and a philosophical language" by J.Wilkins 
appeared in London. It is not fortuitous that the seven- 
teenth century in particular was very fertile with regard 
to artificial language projects. The vital point is that 
these artefacts are not intended to be learned as an 
easier means of interhuman communication - Hobbes' demon- 
strative use of language! - but, quite positively~ as 
"the distinct expression of all things a~ notions" - i.e. 
as an inventive language "which may likewise be styled 
philosophical, rational and universal" as well. For Leibniz 
the numerous artificial language projects bridged the gap - 
see Diagram IV - between Hobbes' exact, symbolic-notative, 
computational calculative and artificial application of a 
natural but unfit material, namely natural words of a 
natural vernacular, and the establishment of an a priori, 
purely artificial system of symbols, i.e. mostly written 
characters or figures. In the inventory of problems which 
Leibniz encountered in the second half of the century, the 
numerous newly-constructed artificial languages proved to 
be very important for his elaboration of Hobbes' ini- 
tiatives. 
B. Leibniz found that on the side of the demonstrative 
natural signs in interhuman discourse it was possible to 
operate calculatively as well~ This insight rendered Hcbbes' 
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functional opposition of inventive mark versus demonstrative 
sign essentially invalid. We see this where Leibniz expects 
in the case of interhuman disputes about matters of ethics 
or religion to bring about a solution through mathemat- 
ically stric~ and rationally convincing arguments; in such 
cases, claims Leibniz, we can say: Calculons! 
The fact that Invention and Demonstration coincide is also 
significant in that the "others", i.e. fellow-men, (right 
hand side of Diagram I) now also belong to reality (left 
hand side of Diagram I). If with Leibniz we further assume 
that non-human plus human reality together form a huge 
collection of units - in his terminology 'monads' - it is 
clear that e~ch human, rational monad emits spontaneous 
signals to other monads. Consequently this process must be 
reciprocal and any human monad must operate at once as a 
transmitter and a receiver. This means that Leibniz mingles 
the originally nomina\]ist active mind theory with a passive 
mind coucept (tabula rasa) similar to that of the realist 
~rammatica speculativa, which, il~deed, he reintroduces. 
~. And he does not even stop here. Towards the end of his 
l~fe Leibniz taught that thei.e is no question of any 
impact on the monads from outside, for this would interfere 
with the pre-established progress of world history. Monads 
have no windows. Their existence is representation. The 
following image, naturally not Leibniz' own~ ma~ serve to 
illustrate this point. The monads resemble a number of 
cinemas, completely shut off from outside, where day and 
night the same endless film is being shown with perfect 
synchronization. The only variation possible is ~hen the 
monad is asleep and the light being, as it were, somewhat 
weaker, illumination and clarity are reduced. The Leibniz- 
Jan idea of representation reaches Von Humboldt via Wolff. 
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~. Hobbes based the certainty that he would obtain true 
and reliable results from his reasoning on the accuracy 
and correctness of the computational procedure. His 
calculus with words is analogous to the intralingaal 
operations with words that we call syntactic or grammatical 
procedure. IIobbes was only an arithmetician, and a poor 
one at that. Leibniz was a geometer as well, i.e. an all- 
round mathematician, and a brilliant one too. Hobbes 
continues to see grammar as a procedure, a computational, 
an arithmetical procedure, but nevertheless a process. The 
older Leibniz eventually arrives at a static view of the 
world and a geometrically conceived model of grammar. Any 
language, whether natural or artificial, any mathematical 
symbolism, mirrors in its own system the structure of the 
• universe, the order of the world; and this correspondence 
is fundamentum veritatis, ths very basis of truth. LsibrJiz' 
static parallelism replaces Hobbes' dynamic-calculative 
approach, which strives for the truth. For Leibniz, the 
truth is present a priori, in the deep structure, the 
"grammaire rationelle et universelle" of any sign- or 
symbolsystem, except that in a, rather irrational, natural 
language it may be somewhat obscured. To demonstrate this 
he uses a geometrical comparison: if for example, a circle 
with inscribed figures is viewed from a point not along 
its vertical axis, it will appear oval, but the inner 
order and proportions will be preserved in spite of the 
distortion. 
In the latter case there is only less "clarity" because of 
the shift in "point de rue" but the fundamental adequacy 
and truth is maintained. 
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4. We wanted to confine ourselves to describing and histor- 
ically locating Hobbes' arithmetical philosophy of language. 
And we must confine ourselves to merely sketching Leibniz' 
role as Hobbes' heir and successor. Just as it may happen 
in architecture that the initiator of a new style is out- 
stripped by a successor, so, in language philosophy, 
Leibniz overshadows Hobbes. In linguistics proper Leibniz 
influenced the founder of the discipline, Bopp, though 
indirectly, through Wolff; in mathematics he influenced 
Frege, who wanted to draft a Leibnizian "Begriffssohrift"; 
in philosophy and mathematical logic he influenced Russell 
in the farter's attempt to create an ideal language, which 
he recommended as the goal of the philosophy of language. 
"The'ideal language' would satisfy perfectly the intentions 
to make the relation of 'picturing' the sole essential 
basis of symbolism ..... Russell .... is unwilling to abandon 
the notion that language must "correspond" to the "facts", 
through one-to-one correlation of elements and identity of 
logical structure." (Black) 
It has been said that the failure of Machine Translation 
was due to the absence of an adequate philosophy of 
language; I feel inclined to agree. Such a philosophy is 
still in its infancy and it cannot be successful without 
scanning the past for its origins. 
Prof. Dr.Pieter A. Verburg 
Hagepreekkamp 4 - Gronin~en 
The Netherlands 
