The Notational Basics of Formal Languages, Deductive Systems and Logical Consequence

Formal Languages and Natural Languages

Formal languages are of vital importance in logic. Such formal languages correspond, if only roughly or approximately, to particular parts of a natural language. For example, take the symbols which follow:

i)                    formal language → &, ∨,   ⊃,   ¬,     ∃,      ∀

                         ↓  ↓   ↓    ↓     ↓       ↓

ii)                   natural language→ and, or,  if-then,  not, at least one, for every…  

The words and symbols in the above are logical terms. Formal languages can also include non-logical terms, such as the sign ‘’ in

3         4

which basically means ‘is less than’. That is, ‘3 is less than 4’. However, more often than not we will get a set of schematic letters. These letters stand in for names, predicates and functions. So we can have the expression:

 a = x

and

Px

In which the symbol ‘a’ can stand for the name ‘Tony Blair’, the variable ‘x’ for a unspecified person or object, and ‘P’ for the predicate ‘is a politician’.

Just as we can see the signs of a formal language as stand-ins for parts of a natural language, so a formula of a formal language stands in for the logical form found in a natural language:

i)                    formal language formula

                                   ↓

ii)                   natural language formula

Logical Form and Grammatical Form

In the early Russell and Wittgenstein, such formal language formulas express, characterise or explicate the logical forms of expressions to be found in a natural language. Clearly, then, the logical forms discovered may be hidden. They are hidden, according to Russell and Wittgenstein again, because the grammatical forms of their natural language expressions hide their logical forms. For example, take Russell’s well-known example:

The king of France is bald.

This has an acceptable grammatical form. However, it hides its correct logical form, thus:

i)                    There is at least one king of France.

ii)                   That king is bald.

iii)                 If y is also king of France and is also bald, then y = x.

(There are many other ways of characterising Russell’s ‘theory of definite descriptions’.)

We can get to grips with such formal languages by using only two schematic letters – viz., ‘y’ and ‘f’:

y = a set of formulas

            f = a single formula

Logical Consequence, Deductive Systems and Deductive Validity

Now we can deal with the notion of logical consequence. We can say that f follows from y. That is, f is a logical consequence of y. However, in order to make sense and legitimise the claim that ‘f follows from y’ we need to introduce the notion of a deductive system. We can use the letter ‘S’ as a symbol for such a deductive system. We may only have axioms and rules of inference in the deductive system. How does such a system relate to our f and y?

Firstly, we need to bring in two more notions: argument and deductive validity. We have already said that ‘y’ stands for a set of formulas within a deductive system. Now we can say that y ends with f. That is, each member of the set of formulas, y, is either a member of y, evidently, or an axiom of S (the deductive system), or that certain formulas may follow from our y. These are off-shoots from y because they abide by the rules of inference of their containing deductive system (our S). These rules of inference take us from y to f. We can formalise this argument thus:

< y. f

The schema above is deductively valid according to S (our deductive system). Because we have brought in the fact that ‘< y. f’ depends on S, we can rewrite the above thus:

y ├ s f

The sign ‘├’ is the symbol of deductive validity. Thus the argument from y to f is deductively valid according to deductive system S, or ‘s’ in the schema above.

Logical Consequence, Models, Logical Proof

In dealing with the notion of logical consequence we must also bring in the models or interpretations of the formal language concerned. A model, as in the vernacular, is basically a structure which fulfils various logical functions and explicates the notion of logical proof. Take the following model:

M = (d, I)

In the above:

d = a set

M = a domain

I = a function

More technically:

d ( a set) = the domain of M

M            = the model

I              = a function which assigns extensions to the non-logical terms in the deductive system  or formal language.

The domain (d) of the model (M), is a set or a class. That is, the set or class of objects within a particular domain. The function, I, assigns the extensions to the non-logical terms in the formal language or system. For example, we may assign the following:

P ® the property being a politician or the class of politicians.

x ® a particular object or group of objects within a specific domain.

a ® The ‘a’ stands for, say, the proper name ‘Tony Blair’. And this name is assigned to a particular object – in this case, the person Tony Blair.

Constants

Here we can bring in the notion of a constant. We can say that the constant, c, or the name ‘a’, must be assigned to a specific object or individual, unlike the variable ‘x’. We can now join the constant, c, to the function, I, to get Ic. More precisely, Ic is a member of the domain, d, mentioned earlier. That is, the constant, c, is operated upon by the function, I, when it is assigned to the previous domain, d. Thus if

c = ‘Tony Blair’

then the function I can assign this constant, or name, to the domain or to the class of politicians.

Satisfaction

Now we meet the notion of satisfaction. The relation of satisfaction is that of the relation between M (the model) and our earlier f (our earlier consequence). More precisely, we can ask: Does M satisfy f? That is, does the model, M, satisfy the consequent, f? Does it correctly notate the way in which f

is a consequent of our earlier y (the set of formulas from which we derive f)? We can symbolically notate this notion of satisfaction thus:

M ⊧ f

This can be paraphrased as:

f is true under the interpretation or model M.

In other words, M shows us exactly how it is true that f is a consequent of certain axioms or premises (or our earlier y).

Model-Theoretic Consequence

Now we have yet another notion to grasp: model-theoretic consequence. This kind of consequence can again involve our f and y. We can now ask: Is the consequence of y, f? That is, is the consequence model-theoretic in nature?

What is required for the consequence to be model-theoretic is that every model or interpretation of each member of y, our set of formulas, also satisfies our consequence or conclusion (i.e. f). Is f a model-theoretic consequence of y? Yes, if every interpretation or model of y also satisfies f. That is, is f a model-theoretic consequence of every possible interpretation or model of ‘< y. f’? If the last formula is model-theoretically valid, we can symbolise its validity thus:

y ⊧ f

Thus we have a proof, ⊧, that f is a model-theoretic consequence of set y. And it is so with all possible models or interpretations of y.