Languages, Natural and Formal


Natural Languages


‘Natural language’ is a piece of jargon to refer to languages like English, French, German, etc.


They are called natural because we do not (consciously) invent them; we naturally acquire them.


A natural language is neither good nor bad considered in itself, for it is not designed or invented for a purpose.



Some properties of natural language


Excluding properties of sound (phonology), the properties of a language are generally understood to fall into three areas: syntax (or grammar), semantics, and pragmatics.




‘Syntax’ refers to the way in which words may be combined to form phrases and sentences. The syntax of every language exhibits certain general features.


(i) ‘Infinity’

Every natural language is boundless in the sense that there is no restriction on the length of a sentence or phrase. We may say that languages are ‘infinite’: if you were to live for ever, you would never run out of new sentences.



1. (a) It is the case that Bill loves Mary

    (b) It is not the case that it is the case that Bill loves Mary

    (c) It is the case that it is not the case that it is the case that Bill loves Mary   

    (d) It is not the case that it is the case that it is not the case that it is the case that Bill

          loves Mary


We can carry on indefinitely generating new sentences by alternating the strings ‘it is the case that’ and ‘it is not the case that’.

Note: (a) means the same as (d), and (b) means the same as (c). No-one would ever bother to utter (d), given (a), but that is not the point. Syntax is concerned with the form or structure of sentences independent of what any speaker would in fact say in normal circumstances. It suffices that we can recognise the sentences as meaningful or well-formed, even though we would never bother to utter them.


Q: Does this mean that the infinity of language is essentially a redundancy? No.


2. (a) The girl behind the boy is blonde

    (b) The girl behind the boy behind the girl is blonde

    (c) The girl behind the boy behind the girl behind the boy is blonde


3. (a) I think Mary is blonde

    (b) I think you think Mary is blonde

    (c) I think you think I think Mary is blonde


4. (a) Dogs fight

    (b) Dogs, who fight cats, fight

    (c) Dogs, who fight cats, who fight dogs, fight


Again, we can carry on indefinitely, but here each sentence of the respective classes has a distinct meaning.


(ii) Rules

The reason we can understand an infinity of sentences just on the basis of understanding a finite number of words is that we also understand how words can be combined. The permissible ways in which words can be combined is syntax. We may think of syntax as a set of rules telling us how to form sentences.


The rules of syntax do not apply to words in terms of what they mean. The rules are blind to meaning; rather, the rules apply in terms of the grammatical categories to which words belong, i.e., nouns, verbs, adjectives, etc. For example, take 3. The categories of the words are as follows:


(5)a. N(oun) = I, you, Mary

    b. V(erb) = think, is

    d. Adj(ective) = blonde


So, here are some (very rough) rules:


(6)a. Form a sentence from a noun phrase (NP) followed by a verb phrase (VP).

    b. Form an NP just from a noun.

    c. Form a VP from a verb followed an adjective or a sentence.


Taken together, we know six things. But this is enough to generate an infinity of sentences on the model of 3.


Step 1: Following (6)c., we can form the VP ‘is blonde’; from (6)b, we have the NP ‘Mary’; and from (6)a. we have the sentence ‘Mary is blonde’.


Step 2: From (6)c, and the result of Step 1, we form the VP ‘think Mary is blonde’; and, just as in Step 1, providing an NP, we have the sentence ‘I think Mary is blonde’.

Step 3: From (6)c, and the result of Step 2, we form the VP ‘think I think Mary is blonde’. As before, we provide an NP to give the sentence ‘You think I think Mary is blonde’.


Step 4: From (6)c, and the result of Step 3, we form the VP ‘think you think I think Mary is blonde’. As before, we provide an NP to give the sentence ‘I think you think I think Mary is blonde’.


Step 6:….


This kind of generation is called recursive. A recursive rule is one which applies to its own output indefinitely, without restriction. Recursion enables us to generate an infinite class of objects - in this case sentences - from the recombination of a finite number of elements.


Via recursion, we can generate the infinite class of natural numbers - {0,1,2,3,4,…}- from 0 and ‘successor’ (suc):


(a) 0 is a number.

(b) If x is a number, then suc(x) is a number.


Can you see how this works?


(iii) Irregularity

The syntax of English (or any other natural language) is irregular relative to the major categories (noun, verb, etc.). That is, not all nouns, verbs, etc. behave the same under syntactic rules. Just to take one example…



The passive contrasts with the active form:


(6)a. Bill kicked the ball (active)

    b. The ball was kicked by Bill (passive)


There is no change of meaning. But consider:


(7)a. Beavers create dams.

    b. Dams are created by beavers.


b. can be read as saying that only beavers create dams; a. cannot have this reading.


A more complex case:


(8)a. Bill expected [the doctor to examine Mary]

     b. Bill expected [Mary to be examined by the doctor]

      c. Bill persuaded [the doctor to examine Mary]

      d. Bill persuaded [Mary to be examined by the doctor]


a. and b. mean the same thing; c. and d. don’t.


Some verbs don’t admit a passive at all:


(8)a. The dress fits Mary.

     b. *Mary is fitted by the dress.

     c. Mary resembles Jane.

     d. *Jane is resembled by Mary.

     e. The car weighs a ton.

     f. *A ton is weighed by the car.


(iv) Syntax dissociates from meaning

Meaning is not predictive of syntactic well-formedness. Below are examples of synonymous pairs of words of the same grammatical category which deviate from each other syntactically.


(9)a. It is likely that Bill will leave.

    b. Bill is likely to leave.

    c. It is probable that Bill will leave.

    d. *Bill is probable to leave.


(10)a. Bill asked what the was.

      b. Bill asked the time.

      c. Bill inquired what the time was.

      c. *Bill inquired the time.


(11)a. Harry stowed the loot away.

       b. Harry stowed the loot.

       c. Harry put the loot away.

       d. *Harry put the loot..


(12)a. Harry told his secret to the police.

      b. Harry told the police his secret.

      c. Harry reported his secret to the police.

      d. *Harry reported the police his secret.


We can say that meaning in natural language does not track syntax. Otherwise put, syntax doesn’t look as if it was designed to express meaning - meaning doesn’t predict syntax.






The semantics of a natural language such as English is an exceptionally complex and confusing area. Just what is part of semantics as opposed to syntax or pragmatics is a question much disputed.


One common understanding is that semantics is concerned with the relation between language and the ‘world’. Hence, the meaning of a sentence determines the conditions under which it is true. Schematically:


TM: If S means that p, then S is true iff p.


If ‘Snow is white’ means that snow is white, then ‘Snow is white’ is true iff snow is white.

If ‘Der Schnee ist weiss’ means that snow is white, then ‘Der schnee ist weiss’ is true iff snow is white.


The meaning of a word will determine what contribution it makes to the truth conditions of the sentences in which it occurs. Thus:


‘snow’ is true of  x iff x is snow.

‘white’ is true of x iff x is white.

‘Snow is white’ is true iff whatever thing ‘snow’ is true of, ‘white’ is true of the same thing, i.e. just if snow is white.


Whatever story we tell about semantics/meaning, a number of key phenomena must be accommodated.


(i) Ambiguity


Natural language is riddled with ambiguity. This is where a single word, phrase of sentence (understood orthographically or phonologically) has multiple meanings independent of context. There are two broad types of ambiguity.


Lexical ambiguity

This is where a single word has multiple meanings. Practically every word of English is ambiguous in some sense; the most oft cited examples, though, are ‘bark’, ‘bank’, ‘import’, etc.


So, ‘Bob went to the bank’ can mean (i) Bob visited a financial institution or (ii) Bob visited the side of a river.


In normal conversation, such ambiguity is resovled by the speech context. Consider:


(7) Bob knew that the bank shuts at 4; so he left work early


We naturally disambiguate (7) as ‘bank’ referring to a financial institution. On the other hand, perhaps the bank where Bob goes fishing shuts at 4. 



Syntactic ambiguity

This is where a sentence or phrase is ambiguous, not because of the ambiguity of any particular word in it, such as with (7), but because the phrase or sentence is consistent with different ways of understanding the combination of the constituent words, where each way gives rise to a distinct meaning. Here is a classic example:


(8)a. Mad dogs and English men go out in the mid-day sun

    b. [[Mad dogs] and English men]…

    c. [Mad [dogs and English men]]…


Does this mean (i) Dogs that are mad and English men… or (ii) Dogs that are mad and English men that are mad…?


This is called a scope ambiguity: does the adjective ‘mad’ describe - have scope over - just dogs or English men as well?


Here is another famous scope ambiguity:


(9) You can fool some of the people all of the time


Does this mean (i) Some of the people (just the men) are such that you can fool them all of the time or (ii) All of the time, you can fool some people or other?


Not all ambiguity is to do with scope. Consider the following:


(10)a. Visiting relatives can be boring (2)

      b. Mary wants to have a car tomorrow (2)

      c. Mary shot an elephant in her shorts (3)

      d. Bob had the car stolen (3)

      e. It’s too hot too eat (3)

      f. Bill is the man I want to succeed (2)

      g. Bill is the man I wanna succed (1)

      h. Mary had a little lamb (5)


(ii) Polysemy


 Polysemy is close to ambiguity, but importantly distinct. With an ambiguity such as ‘bank’, we might sensibly say that there are (at least) two words - bank(1) and bank(2) - which are pronounced the same, but mean different things. A word is polysemous when it has distinct meanings, but we don’t wish to multiply the words. Why not? Because the range of meanings are connected.

      Consider the pair ‘from… to…’:


(11)a. The lights went from red to green. (Change of state/property)

      b. The train went from London to Manchester. (Change of location)

      c. The house went from Bill to his daughter. (Change of ownership) 


These are different meanings, but they are closely connected: in each case, the subject of the sentence is in a initial state, as indicated in the object of ‘from’, and undergoes a change which results in a final state, as indicated in the object of ‘to’.

      See how many different meanings you can find for ‘keep’ (verb). What is the general connection between them all?





‘Pragmatics’ is used as a general term for referring to all kinds of effects on what we say - the thought we express - given the context of our utterance. One way of understanding pragmatics is as an ‘extra element’ on top of syntax and semantics. The latter two determine the ‘literal meaning’ of a sentence; given a literal meaning, pragmatics covers how we can express a different or extended meaning, on the basis of the ‘literal meaning’ + context. Detailed are some common aspects of language use that fall under pragmatics.


(i) Implicature


An implicature is where a speaker says something with a literal meaning, but intends her audience to infer that she means something else on the basis of shared background information. Here are some examples.


(1) Master to servant: “It’s cold in here”

Implicature: Close the window.

Background: Both master and servant are in a room with a draught caused by an open window.

The servant’s reasoning: I’m here to follow orders, but ‘It’s cold in here’ is not an order, and so would be an irrelevant thing to say. The master means to be giving me an order; given the background, his order must be to do with making it warmer, which would be achived by closing the window. The master means me to close the window.


(2) X’s referee to X’s prospective employer: “The candidate has excellent handwriting”

Implicature: The candidate is no good.

Background: The job requires no handwriting.

The prospective employer’s reasoning: The referee intends to be offering me a helpful assessment of X, but X’s handwriting is irrelevant. The referee must mean something else. If the only thing helpful the referee can say is what he does say, he must mean that the candidate is no good.


(3) X to Y in a bar: Is anyone sitting here?

Background: the chair is empty.

X’s reasoning: ?


(ii) Missing constituents

(1)a. It’s raining (where?)

    b. It’s too hot (for what?)

    c. He’s leaving (from where?)


(2)a. Bill and Mary are married (to each other or not?)

    b. Fixing the fault will take time (Sure, but how long?)

    c. There is nothing on TV tonight (worth watching?)


(ii) Narrowing

(1)a. The path is uneven. (How uneven? Not Euclidean?)

    b. Bill is tired. (How tired?)

    c. Bill wants a woman. (Any woman? Mother Theresa?)


(iii) Broadening

(1)a. France is hexagonal. (France is not really hexagonal)

    b. This steak is raw. (Surely the steak is not raw)

    c. The room was silent. (Really silent?)



Formal Languages


A formal language is one we invent for some purpose or other. A programming language such as LISP, for instance, is a formal language invented for the purpose of programming computers. Our concern in this course is with logic. ‘Logic’ is used ambiguously to refer both to the study of ‘good inference’ and to the various languages we invent in order to reveal or discover the nature of good inference, or at least a style of such inference.


It is easy to invent a language, where ‘language’ is understood to be a set of ‘symbol strings’.


Language L

(i) ab is a sentence.

(ii) If X is a sentence, then aXb is a sentence.


This gives us the language {ab, aabb, aaabbb,…}, i.e. all and only strings of the form n occurrences of ‘a’ followed by n occurrences of ‘b’.


Language L*

(i) aa and bb are sentences.

(ii) If X is sentences, then aXa and bXb are sentences.


 This gives us the language {aa, bb, abba, baab, aaaa, bbbb, aabbaa,…}, i.e., all and only strings of occurrences of ‘a’ and ‘b’, where the left-hand side is the mirror image of the right-hand side.


A formal language consists of:

(i) an alphabet of symbols A and

(ii) rules (syntax) for their combination.


A string X belongs to L just if X consists of symbols from A and is formed in accordance with the rules.


Syntax: Not a matter of discovery (unlike with a natural language), but stipulation in order to provide a specified set of strings. Such formal syntax is regular and tracks meaning (see under semantics). So, if A and B belong to formal category X, then A and B behave alike; in particular, if A, B are members of X, then #-A-# is well-formed just if #-B-# is well-formed. As demonstrated above, this is not so for English.


Semantics: By interpretation. The strings of a formal language do not have fixed meanings. We can decide how to interpret the strings. Of course, we don’t want rubbish, so we can design the syntax of the language to produce only those strings consistent with some interpretation (the strings might well be consistent with lots of other interpretations). This is quite unlike the case of natural language: it is not so much that natural language syntax is not designed, but that meaning is not a guide to syntax.

     This kind of semantics differs from natural language semantics. There is no ambiguity or polysemy. Each symbol is decided to mean one thing and each string means whatever it does as a function of the meanings decided upon for its constituent symbols.


     We shall be interested in the extent to which semantic properties of English can be formalised, i.e., be rendered in a formal language.


Pragmatics: Not applicable.


Before we see how formal languages help us understand inferences conducted in natural language, we need to be clear about what inference is, and what a good inference is.