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
‘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.
Examples
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.
Consider:
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…
Passive
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.
Semantics
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
‘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.