This paper studies the relationship between conjunctions in a natural language (Czech) and their logical counterparts. It shows that the process of transformation of a natural language expression into its logical representation is not straightforward. The paper concentrates on the most frequently used logical conjunctions, ∧ and ∨, and it analyzes the natural language phenomena which influence their transformation into logical conjunction and disjunction. The phenomena discussed in the paper are temporal sequence, expressions describing mutual relationship and the consequences of using plural.
The endeavor to express natural language sentences in the form of logical expressions is probably as old as logic itself. A very important role in this process is being played by natural language conjunctions and their transformation into logical connectives. The conjunctions are much more ambiguous than logical connectives and thus it is necessary to analyze their role in natural language sentences, in various contexts and types of texts. This paper presents a step towards such analysis for one particular language Czech.
Let us recall that the fundamental task of logic is to set
rules and methods for inferencing and referencing. On the
other hand, natural languages serve primarily for
communication. Speakers can reach and agreement or understand
each other even without a strict adherence to preset rules
(regardless whether they are morphological, grammatical
or stylistical). A human brain can obtain substantial
information also from ill-formed sentences what actually
makes them to fulfill their main goal, to serve as a tool
for communication. On the other hand, Noam Chomsky
introduced in [
Sentences in any natural languge are not isolated, their meaning typically depends on the context in which they appear, on the way how they are pronounced or even on some external factors as, e.g., gestures which accompany it. Natural languages also evolve in time according to the needs of the language community and although each natural language has a set of generally applicable rules (syntactic, stylistic, morphological etc.), there are many exceptions and irregularities which do not abide the rules as strictly as it is the case in logic.
Primarily due to this difference, the transformation of natural language sentences into their logical representation constitutes a complex issue. As we are going to show in the subsequent sections, there are no simple rules which would allow automation of the process – the majority of problematic cases requires an individual approach.
In the following text we are going to restrict our observations to the two most frequently used conjunctions, namely a (and) and nebo (or). 2
Sentences containing the conjunction a (and) The initial assumption about complex sentences containing the conjunction a (and) is inspired by the properties of the corresponding logical connective and – we suppose that the two clauses connected by the conjunction express two situations which are valid at the same time. This really holds in a number of complex sentences, as for example here:
(Lion is a feline and it lives in Africa.) (1) Jana (jJeavnea šiksoilneaasHchoonozlaalnedžíHnoemnzoacnliýesviplloisnteblie.d.) (2) In the sentence 1 we have used the so called gnomic present1. The truth value of the whole sentence is TRUE only in case that both clauses are TRUE, regardless of the context or current situation.
Complex sentences with gnomic present constitute probably the simplest case. It is not necessary to investigate whether the clauses are true or what are the conditions under which they might become true – they are simply true either always or never. Such sentences can be transformed into a logical representation in a simple and straightforward manner.2 In the logical representation of the example above we would of course use the construction A ∧ B for the conjunction a (and).
1Present tense can be used also for the so called extratensal processes which are valid always, regardless of the current situation. In our example we describe properties of an animal species and its habitat.
2Let us point out that mathematical theorems typically contain gnomic present.
The sentence 2 describes two situations being TRUE exactly in this moment3. The truth value of both sentences can be determined by a reference from the language to a real–world, where we will find out whether both clauses describe a valid situation.4
None of the two clauses from the sentence 2 is absolutely true (Jana does not spend every minute of her life in the school and Honza is not ill forever). However, when we utter any of these two statements, we do not mean that Jana should stay all the time in the school. The use of the present tense implicitly carries the information that she is there just now, in this moment. If we accept that in order to determine the truth value, we have to look into the real world and also take into the account the time when the sentence was uttered, we could paraphrase the sentence into a more unambiguous variant for example like this: Jana je práveˇ ted’ ve škole a Honza nyní leží
nemocný v posteli. (Jana is just now in the school and Honza is
now lying ill in bed.) – i.e. with the added information about time. Such sentence would then correspond to the logical scheme of the conjunction : A ∧ B.5
Natural languages use, of course, also other tenses – what if we would like to express the same content in the past, for example yesterday? (3) (4) (5)
(Jana was in a school yesterday and Honza
was lying ill in bed yesterday.) On the first sight, there seems to be no substantial problem. The only difference seems to be in the fact that we are not referring to a current moment, but to the moment in the past (in this case, yesterday). However, what if Honza will recover till the next day, will such sentence have the same truth value also tomorrow?
Regardless to what time the expressions refer to, we are interested in them only if they are TRUE in this current moment. We should thus simplify our sentence rather in the following way:
(Just now it is true that Jana is in the school, and, at the same time, Honza is lying ill in bed.) 3Of course, only if it is true that Jana is just now in the school and Honza lies ill in his bed.
4Determining the truth value of natural language expressions is
studied by epistemology, a simple explanation can be found for example in
[
5If we would like to consider tiniest details, we would have to
consider also the issue of proper names and singular terms – our sentence
does not specify which Jana and Honza we are talking about. More on
this topic can be found for example in [
Or, we could drop the initial part which we may consider to be implicitly present:
(Jana is in the school, and, at the same
time, Honza is lying ill in bed.)
Into such template it is possible to insert also the complex sentence introduced above:
(Jana was in a school yesterday and, at the same time, Honza was lying ill in bed yesterday.)
All the complex sentences mentioned above can schematically be described in the form A ∧ B. The fact that we can express the mutual relationship of clauses by means of a logical scheme actually means that we can work with them according to logical rules. For example, logical conjunction is commutative – and we really can swap the order of clauses in our complex sentence and still retain the original truth value. 2.1
Unfortunately, the conjunction a (and) doesn’t appear only in sentences describing actions which are happening in the same moment. All of the following sentences contain a (and) as its main conjunction:
(Honza fell and broke his arm.)
(Jana unlocked and entered the flat.) Šli jsme na výstavu a potom do kina.
(We have visited an exhibition and
then we went to a cinema.)
These sentences apparently aren’t commutative. The order of clauses cannot be swapped without affecting the truth value or meaning of the whole sentence. The reason is obvious – both clauses are ordered into a temporal sequence.
The conjunction a (and) isn’t a logical conjunction in these sentences, although it fulfills one fundamental basic condition – if the whole sentence is supposed to be true, then both clauses also have to be true.
The propositional logic nevertheless cannot cope with sentences of this kind. We might be tempted to attempt to solve this issue by means of the conditional construction Když..., (pak) ... (When... (then) ...):
Kd(yWž JhaennaJaondaemunklloac,kveedšl,athdeonbsyhteu.entered the flat.) (11)
and thus to find a certain scheme corresponding to an
implication. In natural languages, the modified sentence is
equivalent with the original one, but this is true only
because the construction Když..., (pak) ... (When... (then) ...)
not necessarily always means an implication. In this
particular case, its role is more temporal than conditional.6
Transforming the sentence into the scheme A → B is thus
incorrect. In [
The sentence:
Jana odemkla a soucˇasneˇ vešla do bytu.
(Jana unlocked and at the same time (12) entered the flat.) does not make much sense and thus we should not directly transform it into logical conjunction. However, the author itself admits that such simple test is not 100% reliable – the construction:
Gödel se narodil v roce 1906 a zemrˇel v roce 1978. (Gödel was born in 1906 and died in 1978.) (13) actually has all required properties of a conjunction: both clauses must be true if the whole sentence should be true; their order can be changed7. However, when we try to replace a (and) by the construction a soucˇasneˇ, (and at the same time), we won’t get a meaningful sentence:
(Gödel was born in 1906 and at the same
(14)
time died in 1978.)
Let us now return to the original sentence. We have
already mentioned that in predicate logic it is impossible to
describe it unless we loose an important information about
the order of events. What if we would use some other type
of logic? The type which seems to be ideally suited for
such kind of constructions is the temporal logic. It is in
fact the propositional logic enriched by the so called
temporal operators, by means of which we can express a
temporal sequence of actions. More information about this
kind of logic can be found, e.g., in [
So far, we have dealt with the conjunction a (and) in the cases in which it expressed conjunction. Let us now show that the same natural language conjunction may in some specific cases also serve as a logical disjunction. 6The conjunction když (when) is then ambiguous.
7Although it is more natural to use them in this order. Nevertheless, the variant with the reversed order does not violate neither linguistic rules nor the logical meaning of the sentence. (15) (16) (17) (18)
(Jana and Honza are students.) is actually a compound sentence:
(Jana is a student and Honza is a student.)
8More about the processes in the center of speech of a human brain
can be found for example in [
vcˇas a neprˇihlásí se ke státnicím, studia
letos nedokoncˇí. (If Honza won’t submit the thesis in time and doesn’t subscribe for the state exams,
he won’t finish his studies in this year.) If we would like to preserve the equivalence of a (and) and a logical conjunction, we could write this sentence schematically as (¬A ∧ ¬B) → ¬C. And indeed, the utterances corresponding to this scheme can be often heard from the Czech native speakers. If we look at the given sentence more closely, we will agree that in order to finish one’s studies it is indeed necessary to finish the thesis in time and at the same time to subscribe also for the state exams. If at least one of these two conditions is not fulfilled, Honza will not finish his studies. The scheme (¬A ∧ ¬B) → ¬C, on the other hand, requires both conditions to be invalid in order to obtain FALSE as the truth value of the whole sentence.
It would therefore be more correct to describe the complex sentence schematically as (¬A ∨ ¬B) → ¬C. The conjunction a (and) clearly substitutes logical disjunction in this context. Actually, even in the natural language it would be more correct to use the conjunction nebo (or) and to say:
vcˇas nebo neprˇihlásí se ke státnicím, studia
letos nedokoncˇí. (If Honza won’t submit the thesis in time or doesn’t subscribe for the state exams,
he won’t finish his studies in this year.)
The fact that this error is quite frequent in natural
language communication is documented for example in the
research of Vlastimil Chytrý [
i.e. it expresses two utterances.
In the sentence 18, the conjunction a (and) is equivalent to a conjunction in logic. However, let us investigate the following examples:
(Jana and Honza are friends.)
(Jana and Honza love each other.) (19) (20) (21) (22)
(Burma and Myanmar is the same thing)10
(Jana is a friend and Honza is a friend.) makes no sense, since we lose the information about a relationship between Jana and Honza.
In Czech, the second sentence could be rephrased as: Jana se miluje a Honza se miluje.
(Jana loves herself and Honza loves himself.) (23) but it’s meaning is not the same as in case of the original sentence, which is ambiguous in Czech. It contains a reflexive verb milovat se (to love someone), which expresses a relationship either between two subjects or of each of them to him/herself particularly. However, we usually use this verb in situations in which we want to express a relationship between two people. Anyway, this example is to illustrate that the same utterance can be formally represented using two different logical schemes. In case we would want to express the second meaning, we would write it down using the means of predicate logic as love_onesel f (Jana) ∧ love_onesel f (Honza) To catch the first meaning, we would have to use not the unary relation, but a binary one
which would be in this case symmetrical. Therefore, we would have to abandon the propositional logic to describe this type of sentences.
The last sentence from the list cannot be rephrased as:
(Burma is the same thing and Myanmar (24) is the same thing.)
This sentence makes no sense, since the phrase to be the same thing again implies a relationship between the entities. These examples actually clearly document the fact that the conjunction a (and) used in utterances which express a relationship cannot be used as a conjunction in logic. The following sentences represent other cases in which the conjunction a (and) refers to a relationship between the subjects:
Cˇ erné a bílé ponožky se pomíchaly.
(Black and white socks got mixed.)
(Jana and Honza got married.) (25) (26)
In the above examples we have shown that if the conjunction a (and) is used in the utterance which expresses a relationship, it cannot be used as a conjunction in logic. 2.4
The method of connecting smaller pieces of text than the whole compound sentences which we have introduced above can be called a distributive method in the mathematical sense of that term. However, the method is not flawless and we have already shown the examples for which it cannot be applied. We will now demonstrate the imperfections of the method that are not related only to lexicon/semantics (i.e. to particular words which do not let us use the method due to their lexical meaning), but rather to syntax.
Let us consider the following sentence: Po(šAt’ápkopsrˇtmivaenzldveellikvýeraedteˇažkbýigbaalníkd. heavy package.) (27) It is natural to agree with the premise that a postman delivered only one package, which was big and heavy at the same time.11 However, if we divide the sentence into two propositions: Pošt’ák prˇivezl velký balík a pošt’ák prˇivezl teˇžký balík. (A postman delivered a big package and a postman delivered a heavy package) (28) the most natural interpretation would probably be that a postman delivered two packages, one of them big and the other one heavy.
The distinction is even more obvious in the following sentence:
(There was a blue and green car parking
Although the word car is used here in singular, we would probably say that there were two cars parking on the street, one of them blue and the other one green. In case the author would use plural:
Na ulici stála modrá a zelená auta. (There were blue and green cars parking (29) (30)
we would probably come into conclusion that there were even more than two cars parking on the street.
These examples demonstrate that when connecting two adjectives, the interpretation of the conjunction a (and) is not clear. Whereas in the first example it is a description of one object having two characteristics, in the second example we describe two different objects having two different characteristics. However, we assign the same activity (same predicate) to both of these objects. The type of the structure is given by the particular adjectives. It is not common in a real word that the car would be both blue and green at the same time.12 In the case when it would be a dirty and scratched car, it would probably be perceived as only one vehicle.
More syntactic problems are connected with a plural. While in the sentence:
(There was a dirty and scratched car parking we do not insist on assigning both characteristics to all of the vehicles.
The second sentence thus cannot be interpreted as a conjunction, but rather as a disjunction.13 In the case of a plural we cannot consider this feature as a specific property of particular adjectives. This phenomena is not related to a specific semantics of given lexemes, but concerns all the adjectives in their fullness.
Below we list some other sentences which should be considered:
Cˇ lánek se zabývá aktivními a pasivními prˇíjmy.
(The article discusses the active and passive
(All the local famous and rich people met up incomes.) at the event.) 12In this case, the car would probably rather be described as bluegreen.
13However, we still need to take into account the level of which we speak. Whereas in connection with the noun (or a noun phrase) the adjective is attributed to we talk about disjunction (it is not required for both objects to have both characteristics), in context of the whole sentence the conjunction a (and) behaves as a conjunction again. It means that if there would be only dirty cars parking on the street, we would be wondering where are the scratched ones mentioned in the sentence as well. (31) (32) (33) (34)
(As a traveler, he got to far more interesting and remarkable places.) (35) 3
Interpretation of the Sentences
Containing the Conjunction nebo (or) As we have shown in the previous section, the interpretation of the conjunction a (and) is not an easy task. Surprisingly, the conjunction nebo (or) behaves more systematically.
The conjunction nebo (nebo) can be interpreted in two ways: • as a disjunction, • as an exclusive disjunction.
Apart from English, Czech language has a rather strict rules distinguishing between these two cases.14 If there is nebo (or) following the comma, it is an exclusive disjunction. In all the other cases, it is considered a common disjunction:
Cˇ ertovi se také rˇíká d’ábel nebo satan. (The demon is also called a devil or a Satan.) – a disjunction
(Honza is coming on Wednesday, or on Thursday.) – exclusive disjunction Naturally, in the spoken language we do not have a chance to find out whether there is a comma in the sentence or not.15 Therefore, we have a lexical distinction at our disposal: the exclusive nebo (or) becomes a correlative conjunction, namely bud’–nebo (either–or):
(Honza is coming either on Wednesday or on Thursday.) – exclusive disjunction (38)
Conjunction nebo (or) can also be a part of the more complex connection which can be further expressed using other logical conjunction. For illustration, see the following sentence:
At’ už Honza prˇijede, nebo ne, oslava se bude konat. (Whether Honza is coming or not, we will (36) (37) throw the party.) (39) 14In English, we use a comma preceding the conjunction or when it connects two independent sentences, regardless the relationship between them.
15In Czech, we place commas based on structural rules, i.e. not in places where there is a natural break in spoken utterance.
rozhodneˇ navštíví také prarodicˇe. (Whether coming on Wednesday or Thursday, Honza will definitely drop by his grandparents.) (40)
As for the sentence 39, we can write down the
proposition using a special logical conjunction Maybe and (MA,
truth depends on second proposition) described for
example in [
The sentence 40 is however much more complex. Although it expresses the contrast of the two possibilities, one of them is not a negation of another. Therefore we cannot use the conjunction MA, since it is a binary conjunction and we need to connect three propositions.16 The second sentence can be transformed into a logical notation in the following way: (Honza prˇijede ve strˇedu ⊕ Honza prˇijede ve cˇtvrtek) ∧
Wednesday ⊕ Honza is coming on Thursday) ∧ Honza will drop by his grandparents.) (41) Therefore, we would have to use the exclusive disjunction again.17 4
Conclusion In this article, we have discussed the interpretation of natural language sentences using the means of logic. We have shown that although some of the logical conjunction names are motivated by the natural language conjunctions and they quite often have similar meaning, it is not possible to translate them from natural language to logic directly. Especially for the conjunction a (and) we have introduced more complex problems (i.e. the issue of relations, plurals or sequence of tenses) which prevent us from identifying a (and) with a logical conjunction.
Also, we have brought an important analysis of the possibilities (and problems) which have to be considered when working beyond sentential level. We have shown how to transform these structures so that they could be described using the means of the propositional logic (which takes only the propositions – or, in other words, sentences).
161. Honza is coming on Wednesday. 2. Honza is coming on Thursday. 3. Honza will drop by his grandparents.
17Conjunction MA can be also expressed using a set of conjunctions {∧, ∨, ¬}. It is also interesting to consider whether the construction At’ už (...), nebo (...) (Whether (...) or (...)) can be captured using a common disjunction or using the exclusive one. In case we are describing an indisputable system (such as propositional logic), we already know that the situation A ∧ ¬A is impossible, so both versions of the translation – with ∨ and with ⊕ – are equivalent if there is the same formula in the connection (or, more precisely, the formula and its negation). Finally, we have to mention that there were both variants (with and without a comma) found in the corpus. 5
Acknowledgments This research was supported by the grant GA15-06894S of the Grant Agency of the Czech Republic and by the SVV project number 260 224. This work has been using language resources stored and/or distributed by the LINDATClarin project of MŠMT (project LM2010013).