__Week
2 - Validity and Formalisation__

__Recap__

In
week 1, we looked at some of the complexities of natural language. Following
the standard tripartite distinction, we distinguished between syntax,
semantics, and pragmatics.

(i)
Syntax

About
syntax, we found that it has a *recursive*
property, i.e., the rules which determine how words should be put together to
form sentences (of English) apply to their own products to generate further
sentences. This feature explains how it is that we can be *competent* with an unbounded number (an ‘infinity’) of sentences,
even though, given we are finite creatures, we would never get around to using
an infinite number of sentences (our *performance*
is bounded). A finite number of words and a finite number of recursive rules
allow to us generate an infinity of sentences.

One way of thinking about this is to
consider *novelty*. With notable
exceptions - ‘I love you’, ‘Pass the salt’, etc. - the sentences we utter and
hear are new to us. It would be quite mysterious how our linguistic
understanding could be *continuously novel*
if we didn’t have the equipment to generate any of an infinity of sentences as
the conversational situation demands.

As we shall see, recursiveness is a
property shared with formal languages, such as logic.

On the other hand, natural language
syntax is irregular, ‘messy’. Rules don’t apply universally (e.g., the
passive), and words of the same grammatical category (verb, noun, etc.) don’t
behave alike. Further, even synonymies (e.g., likely/probable, told/reported)
don’t behave alike. A neat way of capturing this idea is to say that *syntactic well-formedness doesn’t generalise
over substitution*.* *This is a
mouth-full. All it means is that if Fa is a sentence of English, then it
doesn’t follow that Fb or Ga are also sentences, even if F and G and a and b
belong to the same respective categories, even if they are synonymous pairs.
This feature has the following consequence: syntactic relations don’t track
semantic (meaning) relations. Otherwise put: the rules of syntax don’t care
about what words mean (at least not up to substitution).

As we shall see, such irregularity is *not* a property of logic.

(ii)
Semantics

About
semantics, we reached the same conclusion as above, except from the opposite
direction, as it were. There are complex semantic relations which are not
enshrined in the syntax. Most clearly, sentences can be ambiguous, i.e., the
same syntactic structure can support distinct meanings. We also saw that
individual words have aspects of meaning that don’t show up in each sentence
they occur in. *Bill kept the car*
implies ownership. *Bill kept the crowd
happy*, doesn’t.

Again, we have a mismatch between syntax
and semantics.

(iii)
Pragmatics

Pragmatics
is perhaps the most confusing area of language. This is because pragmatics
concerns the *use* of language, and so
involves a whole range of factors: the context of the conversation, the
psychology of the speakers, what has been previously said, etc.

We saw that aspects of pragmatics, such as
metaphor, irony, and implicature, appear to involve a distinction between *sentence* (literal) *meaning* and *speaker meaning*.
That is, we often use a sentence to communicate some thought which is not the
literal meaning of the sentence used. In such cases, we rely on the hearer to
know the literal meaning and figure out what we mean (our speaker meaning) by
her reasoning why we should utter a sentence with such a literal meaning. For
example, if I say ‘Bill is a lion’, I’m saying that Bill is brave or something,
but the hearer must know what ‘Bill is a lion’ literally means to be able to
understand the metaphor.

We also saw, however, that pragmatic
effects are quite pervasive; that is, the very notion of literal meaning as
something determined independent of pragmatics is not so stable. It might be
that what we think of as the literal meaning itself involves pragmatic factors.
If I say, ‘I want a red car’, I only want a car whose body is red, not its
whole surface, still less its interior. If I say, ‘I want a red ball’, I only
want a ball whose exterior surface is red. If I say ‘I want red paint’ I want
something is wholly red.

As we shall see, logic wholly ignores
pragmatics. This might or might not be a problem.

__Formalisation__

A
formal language is often spoken of as a *formalisation*,
where what is formalised is a set of intuitive concepts (or perhaps just one).
This may or may not involve a departure from our commonsense understanding of
the concepts, more often than not, it does. We may think of the formalisation
is an idealisation of a certain core aspect of our understanding of the
concepts. For example, *probability theory*
(a branch of mathematics) is a formlisation of our notions of chance and
probability. Decision theory is a formalisation of our notion of arriving at a
good decision, given our preferences and beliefs about outcomes. *Logic is a formalisation our notion of a
valid argument*.

Two key points:

(i)
None of these formlaisations are (necessarily) intended to be accounts of how
we *in fact* reason about probability
or validity.

(ii)
Once we have a formalisation, we can ignore the relation it has to the
intuitive concepts and simply investigate the properties of the formal system.
That said, it remains an interesting question to ask what the relation is
between the formal and intuitive concepts.

A
formalisation is, in effect, an invented language designed to expresses the
formal analogues of the intuitive concepts. The syntactic rules of the language
are designed so that they mirror or track semantic relations which hold between
the targeted concepts. In other words, the language is designed so that
semantic/conceptual notions can be reasoned about in a rule governed,
mechanical way.

This might seem terribly obscure. The fog
will only clear when we look in some detail at logic.

__Useless
Formalisations__ __ __

As
we saw in week 1, it is easy to design a formal language. All one needs is an
alphabet of symbols - ‘a’s and ‘b’s, say - and rules to combine them. Thus:

L
= {ab, aabb, aaabbb,…}

(i)
ab is a sentence of L.

(ii)
If *X* is a sentence of L, then a*X*b is a sentence of L.

(iii)
These are all the sentences of L.

This
is called a recursive definition of ‘a sentence of L’; the rules generate all
and only the L sentences. Note that ‘*X*’
is a variable ranging over sentences of L, it is not a symbol of L. So, neither
‘*X*’ nor ‘a*X*b’ are sentences, but (ii) remains fine - it can be read as
saying, ‘If you have any sentence of L, then putting a at the front of the
sentence, and b at the end of the sentence, results in another sentence’. The
variable ‘*X*’ is just a way of saying
this more simply. For example, the equation *x*0
= 0 doesn’t say that the 24th letter of the English alphabet times 0 equals 0.
It says that any number times 0 equals 0. ‘*x*’
is not a number, just as ‘*X*’ is not a
symbol of L.

L
is not an interesting formal language. Why not? Well, because nothing is being
formalised: we have the simple syntactic rules, but there are no concepts which
are being expressed - formalised - by the rules. What we want is a formal
language that actually succeeds in capturing some concepts within its
mechanical rules. Logic seeks to capture validity.

__Arguments__

Logic
is about arguments. Arguments here are not disputes but lines of reasoning from
premises or assumptions to conclusions. We say that an argument is a *good* one, if its conclusion *follows* from its premises. Logic is
about formalising what ‘good’ means here.

There
are, of course, many ways in which an argument can be deemed good. Logic is
concerned with a special way in which an argument can be good. The three
arguments below might well be considered good ones.

(1)a.
All bachelors are unhappy.

Bill is an unmarried man.

Therefore, Bill is unhappy.

b. The Sun has risen every morning since
the dawn of time.

There is no reason to think tomorrow
will be different.

Therefore, the Sun will rise tomorrow.

c. One hundred ill people saw the healer.

After seeing the healer, 90% of them recovered.

Therefore, the healer has some power to
cure the sick.

These
arguments *might* be persuasive, but
they are not logically valid. On the other hand, this argument is logically
valid:

(2)
All Frenchmen are fish.

Tony Blair is a Frenchman.

Therefore, Tony Blair is a fish.

What’s
good about a logically valid argument doesn’t depend on either its premises or
conclusion being true. A logically valid argument need not persuade you of its
conclusion at all.

__Validity__

Logic
is concerned with validity: logically good arguments are valid ones. There are
a number of ways to express validity. Here is a general way:

__Validity__:
An argument is valid if and only it is not logically possible for all its
premises to be true, and for its conclusion to be false.

Three
immediate things to note:

(i)
The validity of an argument can’t be determined by just looking at its premises
or just at its conclusion; it is necessary to look at both.

(ii)
To criticise an argument in terms of its validity, it is not enough to
establish that either its premises or conclusion is false.

(iii)
An argument might be invalid, even if its premises and conclusion are true.

(iv)
An invalid argument can have *some*
true premises, and a false conclusion.

Consider
argument (2). Its two premises are false, as is its conclusion. Validity,
however, is a property not based upon what is true or false given the way the
world is, but on what is true or false given an assumption about some other
sentences being true or false. Thus, to test for validity, we ask, *if the premises were true, would it be
possible for the conclusion not to be
true*. Otherwise put, *Is it possible for the conclusion to be
false, and the premises true?* Okay, well:

(i)
*If* all Frenchmen are fish, then any
given Frenchmen is a fish.

(ii)
The second premise tells us that Tony Blair is a given Frenchman.

(iii)
Thus, Tony Blair must be a fish, if the premises are true.

Going
the other way:

(i)
Assume that it is false that Tony Blair is a fish.

(ii)
If Tony Blair isn’t a fish, then he can be a Frenchman (i.e., the second
premise can be true) only if not all Frenchmen are fish (i.e., the first
premise is false).

(iii)
Thus, it is not possible for the conclusion to be false and for both premises
to be true.

We
may say: an argument is valid just if the assumption that the conclusion is
false generates an inconsistency between its premises. If no inconsistency is
generated, then the argument isn’t valid.

Consider
argument (1)c. Assuming that the healer has no powers (i.e., the conclusion is
false) generates no inconsistency. It is perfectly possible for the premises to
be true and for the healer to have no powers. How? Simply by assuming that 90%
of the sick people recovered from their colds independent of the healer’s
intercession.

Consider
argument (1)b. Assuming that the conclusion is false (i.e., the Sun will not
rise tomorrow) doesn’t generate an inconsistency in the premises. Of course the
Sun has always risen in the past, and I have no reason to think tomorrow will
be different, but the Sun might still blow up at any moment. That is logically
possible, and it doesn’t contradict the premises.

Argument
(1)a is an interesting case. As so far presented, it looks to be valid. We
shall return to it. In the meantime, think about the difference between it and
(2).

Here
is yet another way to understand validity:

An
argument is valid if and only if the negation of its conclusion conjoined with
its premises creates a contradiction.

Here
is another, more picturesque way:

Imagine
a world (actual or not) which is described by the premises of an argument. Can
the negation of the conclusion also consistently describe this world? If it
can, the argument is invalid. If it can’t, the argument is valid.

__Glossary__

Sainsbury
defines his terms quite clearly, but here are a few things which might help
you.

*Possibility*:
Validity is about what is logically possible, not what is actual or physically
possible. We can think of these three notions as contained in each other. This
may be represented as below, where ‘*x*
Í
*y*’ means ‘*x *is included in *y*’:

Actual Í
Physically possible Í
logically possible

The
*actual* covers what is in fact true.
The physically possible covers what could be true, given the laws of physics
(whatever they might be). The logically possible covers that which is
non-contradictory. This is a vast category. Anything which isn’t self-contradictory
is logically possible. It doesn’t follow that it is straightforward to tell
what is logically possible. Consider this mathematical claim:

(3)
For all numbers n, n + 1 >
n.

Would
the negation of (3) be contradictory? If anyone explains to me why (3) is in
fact false, I’ll buy them a drink of their choice.

*Consistency*:
A set of sentences is consistent if they can be all true or false together. In
other words, a consistent set is one which is not contradictory. Note,
consistency doesn’t imply truth.

*Negation*:
The negation of a sentence is its contradictory. If P is a sentence, and not-P
is its negation, then P and not-P is a contradiction. Contradictories are such
that if one is true the other must (logically) be false. With this in hand, we
can say that a consistent set is one which doesn’t contain P and its negation.
Inconsistent sets need not include negations, but we must be able to derive a
contradiction.

*Soundness*:
A sound argument is one which is valid and has true premises. Thus, a sound
argument is one which *proves* that its
conclusion is true. Validity doesn’t imply soundness, as example (2) shows.
Here’s a sound argument: All Englishmen are Europeans; Tony Blair is an
Englishman. Therefore, Tony Blair is European.

*Truth*:
Quine has said that “Logic chases truth up the tree of grammar”. What this
means will be explored below. But what is truth? Er, well, ah, yes. Logic
doesn’t tell us what truth is. It tells us how not to deviate from the truth on
the assumption that we begin with truth. For our purposes, we can think of
truth as a simple property which all sentences either have or lack (here we
ignore questions, exclamations, and the like).

__Take
a Breath__

This
is a fair amount of jargon. If you find it confusing, try to apply each of the
notions to a given argument rather than try to figure out what the definitions
say in the abstract. Rest assured, you are not supposed to get this stuff in
one sitting, as it were. If you do, great; if you don’t, then great - you have
something to look forward to.

__Some
Notation and Some Properties of Validity__

All
notation is conventional; the only restriction is that we be consistent in our
use. So, let us adopt the following convention. When we are talking about an
argument, we shall use uppercase English ‘A’ for assumptions and uppercase ‘C’
for conclusions. The double turnstile ‘†’
is used to connect premises to conclusions. Sainsbury (p.24) reads ‘†’
as ‘is valid’. This is slightly odd. It is better to read it as:

‘A_{1},…
A_{n} †
C’ means ‘A_{1},… A_{n} entail C’ (or C follows from A_{1},…
A_{n}), i.e., it is not possible for C to be false and A_{1},…
A_{n }all to be true.

‘A_{1},…
A_{n }†
C’ means ‘A_{1},… A_{n} don’t entail C’ (or C doesn’t follow
from A_{1},… A_{n}), i.e., it is possible for C to be false and
A_{1},… A_{n }all to be true.

__Validity
is transitive (LF, Ex. 1.15, p.27)__

This
means:

(4)
If [A_{1},… A_{n} † C] and [B_{1},…
B_{k}, C †
D], then [A_{1},… A_{n}, B_{1},… B_{k} †
D]

Explanation:
For (4) to be false, [A_{1},… A_{n} †
C] and [B_{1},… B_{k}, C †
D] must be true, and [A_{1},… A_{n}, B_{1},… B_{k}
†
D] must be false. Assume, then, that [A_{1},… A_{n} †
C] and [B_{1},… B_{k}, C †
D]. It follows that it is not possible for C to be false and A_{1},… A_{n
}to be true, and it is not possible for D to be false and A_{1},…
A_{n}, C to be true. Is it possible for [A_{1},… A_{n},
B_{1},… B_{k} †
D] to be false, i.e., for D to be false and A_{1},… A_{n}, B_{1},…
B_{k} to be true? Let us assume so and then generate a contradiction.
The contradiction would show that (4) must be true.

(i)
Assume, then, that D is false, and A_{1},… A_{n}, B_{1},…
B_{k} are all true.

(ii)
By the assumption of [B_{1},… B_{k}, C †
D], if D is false, then some of B_{1},… B_{k}, C must also be
false.

(iii)
We have assumed in (i) that B_{1},… B_{k} are all true; so that
leaves C to be false.

(iv)
But if C is false, then by the assumption of [A_{1},… A_{n} †
C], some of A_{1},… A_{n} must also be false.

(v)
But this contradicts our assumption in (i) that, A_{1},… A_{n},
are all true.

(vi)
Therefore, it is not possible for D to be false, and A_{1},… A_{n},
B_{1},… B_{k }to be true. QED.

__Validity
is Reflexive (LF, Ex. 1.16, p.27)__

This
means:

(5)
If C is among A_{1},… A_{n}, then [A_{1},… A_{n}
†
C].

Explanation:
Let C be A_{1}. For purposes of argument, assume that [A_{1},…
A_{n} †
C] is false, i.e., it is possible for C to be false and A_{1},… A_{n}
to be true. Now let’s generate a contradiction.

(i)
Assume, then, that C is false and A_{1},… A_{n} are true.

(ii)
It follows that A_{1} is true, but A_{1} is C, and so C is
true.

(iii)
This contradicts the assumption that C is false and A_{1},… A_{n }are
true.

(iv)
Therefore, it is not possible for , A_{1},… A_{n} to be true
and C to be false. QED.

__If
A _{1},… A_{n} __

(i)
Let A_{1} be the negation of A_{2}. This suffices for A_{1},…
A_{n} †.

(ii)
It follows that any set S containing A_{1} and A_{2} will be
inconsistent: S †.

(iii)
Therefore, in particular, A_{1},… A_{n,} B †.
QED.

__Necessary
truths are redundant as premises (LF, Ex 1.18, p.29)__

If † B, then [A_{1},…
A_{n} †
C] if and only if [A_{1},… A_{n}, B †
C]

(i)
‘†
B’ means that B is logically true (necessary); it follows from any premises.

(ii)
‘*X* if and only if *Y*’ (abbreviated to ‘iff’) means that if *X*, then *Y* and if *Y*, then *X*. Thus, to prove anything of the form ‘*X* if and only if *Y*’, we prove left to right (*Y*
from *X*), then right to left (*X* from *Y*).

We
want to show that ‘†
B’ can’t be true, while ‘[A_{1},… A_{n} †
C] if and only if [A_{1},… A_{n}, B †
C]’ is false. Let us assume, then, that
‘†
B’ is true and ‘[A_{1},… A_{n} †
C] if and only if [A_{1},… A_{n}, B †
C]’ is false, and generate a contradiction.

Step
1

(A)
We prove that if [A_{1},… A_{n} †
C], then [A_{1},… A_{n}, B †
C].

(i)
For purposes of argument, assume [A_{1},…
A_{n}, B †
C] is false, i.e., it is possible that C is false and A_{1},… A_{n, }B_{
}are true.

(ii)
But if C is false, then at least one of A_{1},… A_{n} must also
be false, if [A_{1},… A_{n} †
C]. (iii) If one of A_{1},… A_{n} must be false, then this contradicts our assumption
(i).

(iv)Therefore,
it is not possible for [A_{1},…
A_{n}, B †
C] to be false and [A_{1},… A_{n} †
C] to be true.

(v)
Therefore, if [A_{1},… A_{n} †
C], then [A_{1},… A_{n}, B †
C], which was to be proved. QED. (Incidentally, this proves that validity is
monotonic; it doesn’t make any difference if B is logically true or not.)

(B)
We prove, on the assumption that †
B, that if [A_{1},… A_{n}, B †
C], then [A_{1},… A_{n} †
C].

(i)For
purposes of argument, assume that [A_{1},… A_{n} †
C] is false, i.e., it is possible that C is false and A_{1},… A_{n, }are
true.

(ii)
But if C is false, then at least one of A_{1},… A_{n}, B must
also be false, if [A_{1},… A_{n},
B †
C].

(iii)
If one of A_{1},… A_{n} must be
false, then this contradicts our assumption (i).

(iv)
Therefore, if C is false and A_{1},… A_{n} are true and [A_{1},…
A_{n}, B †
C], then B must be false.

(v)
But B, by assumption, is logically true.

(vi)
Therefore, it is not possible for [A_{1},… A_{n} †
C] to be false, and [A_{1},… A_{n}, B †
C] to be true.

(vii)
Therefore, on assumption that †
B, if [A_{1},… A_{n}, B †
C], then [A_{1},… A_{n} †
C], which was to be proved. QED.

Step
2

(i)
We have proved If [A_{1},… A_{n} †
C], then [A_{1},… A_{n}, B †
C].

(ii)
We have proved If †
B, then If [A_{1},… A_{n}, B †
C], then [A_{1},… A_{n} †
C].

(iii)
Therefore, we have proved If †
B, then If [A_{1},… A_{n}, B †
C], then [A_{1},… A_{n} †
C], and If [A_{1},… A_{n} †
C], then [A_{1},… A_{n}, B †
C].

(iv)
Therefore, we have proved If †
B, then [A_{1},… A_{n} †
C] if and only if [A_{1},… A_{n}, B †
C], which was to be proved. QED.

__LF,
Ex. 1.18(b) __

Because
validity is monotonic (see Step 1(A), above):

If
[B†],
then [A_{1},… A_{n}, B †
C], whatever A_{1},… A_{n}, C may be.

Now,
we are asked whether if [A_{1},… A_{n}, B †
C], then [B†].
The answer is yes.

(i)
Assume [A_{1},… A_{n}, B †
C] is true, for any choice of A_{1},…
A_{n}, C.

(ii)
If, in particular, [A_{1},… A_{n}, B †
C] is true, for any choice of C, then A_{1},… A_{n}, B †.

(ii)a
Sub-proof: If A_{1},… A_{n}, B were all true, then [A_{1},…
A_{n}, B †
C], for a choice of false C.

(ii)b
Therefore, it is not possible for A_{1},… A_{n}, B all to be
true.

(iii)
But, by assumption, A_{1},… A_{n} can all be true; in
particular, there is an instance where A_{1},… A_{n} are true
and C is false and [A_{1},… A_{n}, B †
C].

(iv)
Therefore, it is not possible for B to be true.

(v)
Therefore, B †.

(iv)
Therefore, if [A_{1},… A_{n}, B †
C], then [B†].
QED

(Try
the other exercises for yourself, using the above examples as models.)

__Notes
on Proof__

(i)
Always be clear and precise; make no jumps, even if they are obvious. Behave
like a machine, as it were.

(ii)
For each formula, consider what follows from assuming that it is true/false.

(iii)
All of the above proofs are via *reductio
ad absurdum*. If we want to prove P, we assume not-P, and then generate a
contradiction based on not-P as assumption. Truths don’t lead to contradiction;
so, not-P must be false. Therefore, P is true.

(iv)
If your head hurts, don’t worry, it is supposed to.

__Formal
Validity__

So
far, we have only been considering validity. Logic, however, is concerned with
a strict understanding of validity: formal validity. The two notions largely coincide.

An
argument is formally valid iff it is valid in terms of its form.

‘Form’
here means essentially a generalisation over the words which occur in the
sentences of an argument. We keep certain words constant, and generalise over
others. When we generalise over the words of a given sentence, we can replace
them with letters. Why? Because it doesn’t matter what the words are.

Once
we have replaced all of the words we can generalise over with letters, we then
have the logical form. This might seem quite obscure. Let us consider the
example from above.

(2)a
All Frenchmen are fish.

Tony Blair is a Frenchman.

Therefore, Tony Blair is a fish.

b.
All Gs are F.

a is G.

Therefore, a is F

Note
that replacing G and F with any predicate, and a with any name, then we still
have a valid argument. We can say, (2)a is formally valid *because* any substitution instance of (2)b ((2)b with its letters
uniformly replaced by words) is valid.

Think
of the letters in an argument form as essentially generalisations over all of
the words which, if inserted into the form, would result in a valid argument,
i.e., all the words which would make the same contribution to determining
whether the containing argument is valid.

All
formally valid arguments are instances of a valid argument form.

The
logical form of a sentence is essentially a generalisation of the sentence in
terms of its structure that is relevant to determining its truth, i.e., the contribution the
sentence may make to determining the validity of an argument in which the
sentence occurs - think about the relation between (2)a and b.

With
all this in mind, let us return to the argument which appeared to be valid:

(1)a.
All bachelors are unhappy.

Bill is an unmarried man.

Therefore, Bill is unhappy.

This
seems valid because we know that bachelors are unmarried men, and so if Bill is
an umarried man, then he is a bachelor, and so is unhappy. But the argument is
not formally valid. Consider:

All
As are B

a
is C

Therefore,
a is B

This
is not a formally valid argument form, for it has the instance:

All
whales are mammals.

Goldie
is a fish.

Therefore,
Goldie is a mammal.

This
immediately tells us that the generalisation involved in finding the logical
form of a sentence is not sensitive to what individual words mean. Logic doesn’t
tell us that bachelors are unmarried men. Logic is only sensitive to the
categories to which words belong - names, predicates, etc. - and we replace
each word of such categories by a type of letter.

When
Quine says that “logic chases truth up the tree of grammar”, he means that
logic tracks how validity relations between sentences are determined by
relations between sentences in terms of their logical form.

A
valid argument is one which *preserves
truth*. That is, if you start with truths, then, reasoning logically, one
can only conclude with truths. Logic doesn’t tell us what is true, still less does
it tell us what we ought to believe. But it does tell us, in a mechanical way,
how we can guarantee truth form truths, as it were. The way is mechanical
precisely because truth is preserved in terms of form, not in terms of what individual
words, or, indeed, sentences, mean. Logic is a way of chasing - preserving -
truth via manipulation of form - grammar.

__Natural
Language Syntax and Logical Form__

We
can think of logical form as the syntax of logic: logic is not concerned with
the truth or validity of natural language sentences directly; it is only
concerned with the logical form of such sentences, and it is the logical form
which decides validity.

We
saw that natural language syntax is ‘messy’, logical form isn’t. Logic is
designed not to be messy. Each logical form is a generalisation over a class of
sentences, such that substituting one word for another of the same ‘type’
produces another well-formed sentence with the same logical form (witness the
Tony Blair example above). Natural language syntax is not like this.

Further,
logical form is designed to express or track certain semantic or truth relevant
relations between sentences (validity); natural language syntax isn’t so
designed.

Logical
form, in virtue of being a generalisation over words, abstracts away from individual
differences in the meaning of words: there is lexical ambiguity or polysemy. As
we progress, we shall also see how logical forms are not structurally ambiguous.

Logic
is a formalisation of validity.

__Logic
as a Formal Language__

Logic
is a formal language which seeks to formalise validity in terms of logical
form. Logic turns sentences into their logical forms, as it were, then assesses
arguments for validity in terms of such forms.

But
if logical form is essentially a generalisation away from particular words,
then can we generalise away from all words in a sentence? No. There are some
words which remain constant; we cannot generalise away from what they mean.
These are the so-called *logical* *constants*, and the so-called *propositional calculus* is the logic
which is concerned with the role they play in determining validity. In the
propositional calculus, we generalise over whole sentences, not just words, but
we leave the logical constants intact as words which uniformly make the same
contribution to validity whatever the context. We’ll come to them next.