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CHAPTER 1. SENSE, DENOTATION AND SEMANTICS |
So, summing up our opinion about this tradition, it is always in search of its fundamental concepts, which is to say, an operational distinction between sense and syntax. Or to put these things more concretely, it aims to nd deep geometrical invariants of syntax: therein is to be found the sense.
The tradition called \syntactic" | for want of a nobler title | never reached the level of its rival. In recent years, during which the algebraic tradition hasourished, the syntactic tradition was not of note and would without doubt have disappeared in one or two more decades, for want of any issue or methodology. The disaster was averted because of computer science | that great manipulator of syntax | which posed it some very important theoretical problems.
Some of these problems (such as questions of algorithmic complexity) seem to require more the letter than the spirit of logic. On the other hand all the problems concerning correctness and modularity of programs appeal in a deep way to the syntactic tradition, to proof theory. We are led, then, to a revision of proof theory, from the fundamental theorem of Herbrand which dates back to 1930. This revision sheds a new light on those areas which one had thought werexed forever, and where routine had prevailed for a long time.
In the exchange between the syntactic logical tradition and computer science one can wait for new languages and new machines on the computational side. But on the logical side (which is that of the principal author of this book) one can at last hope to draw on the conceptual basis which has always been so cruelly ignored.
1.2The two semantic traditions
1.2.1Tarski
This tradition is distinguished by an extreme platitude: the connector \_" is translated by \or", and so on. This interpretation tells us nothing particularly remarkable about the logical connectors: its apparent lack of ambition is the underlying reason for its operationality. We are only interested in the denotation, t or f, of a sentence (closed expression) of the syntax.
1.For atomic sentences, we assume that the denotation is known; for example:
3 + 2 = 5 has the denotation t.
3 + 3 = 5 has the denotation f.
1.2. THE TWO SEMANTIC TRADITIONS |
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2.The denotations of the expressions A ^ B, A _ B, A ) B and :A are obtained by means of a truth table:
A B |
A ^ B A _ B A ) B :A |
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t |
t |
t |
t |
t |
f |
f |
t |
f |
t |
t |
t |
t |
f |
f |
t |
f |
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f |
f |
f |
f |
t |
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3.The denotation of 8 : A is t i for every a in the domain of interpretation2, A[a= ] is t. Likewise 9 : A is t i A[a= ] is t for some a.
Once again, this de nition is ludicrous from the point of view of logic, but entirely adequate for its purpose. The development of Model Theory shows this.
1.2.2Heyting
Heyting's idea is less well known, but it is di cult to imagine a greater disparity between the brilliance of the original idea and the mediocrity of subsequent developments. The aim is extremely ambitious: to model not the denotation, but the proofs.
Instead of asking the question \when is a sentence A true?", we ask \what is a proof of A?". By proof we understand not the syntactic formal transcript, but the inherent object of which the written form gives only a shadowy re ection. We take the view that what we write as a proof is merely a description of something which is already a process in itself. So the reply to our extremely ambitious question (and an important one, if we read it computationally) cannot be a formal system.
1.For atomic sentences, we assume that we know intrinsically what a proof is; for example, pencil and paper calculation serves as a proof of \27 37 = 999".
2. A proof of A ^ B is a pair (p; q) consisting of a proof p of A and a proof q of B.
3. A proof of A _ B is a pair (i; p) with:
i = 0, and p is a proof of A, or
2A[a= ] is meta-notation for \A where all the (free) occurrences |
of have been replaced |
by a". In de ning this formally, we have to be careful about bound |
variables. |
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CHAPTER 1. SENSE, DENOTATION AND SEMANTICS |
i = 1, and p is a proof of B.
4.A proof of A ) B is a function f, which maps each proof p of A to a proof f(p) of B.
5.In general, the negation :A is treated as A ) ? where ? is a sentence with no possible proof.
6.A proof of 8 : A is a function f, which maps each point a of the domain of de nition to a proof f(a) of A[a= ].
7.A proof of 9 : A is a pair (a; p) where a is a point of the domain of de nition and p is a proof of A[a= ].
For example, the sentence A ) A is proved by the identity function, which associates to each proof p of A, the same proof. On the other hand, how can we prove A _ :A? We have to be able to nd either a proof of A or a proof of :A, and this is not possible in general. Heyting semantics, then, corresponds to another logic, the intuitionistic logic of Brouwer, which we shall meet later.
Undeniably, Heyting semantics is very original: it does not interpret the logical operations by themselves, but by abstract constructions. Now we can see that these constructions are nothing but typed (i.e. modular) programs. But the experts in the area have seen in this something very di erent: a functional approach to mathematics. In other words, the semantics of proofs would express the very essence of mathematics.
That was very fanciful: indeed, we have on the one hand the Tarskian tradition, which is commonplace but honest (\_" means \or", \8 " means \for all"), without the least pretension. Nor has it foundational prospects, since for foundations, one has to give an explanation in terms of something more primitive, which moreover itself needs its own foundation. The tradition of Heyting is original, but fundamentally has the same problems | G•odel's incompleteness theorem assures us, by the way, that it could not be otherwise. If we wish to explain A by the act of proving A, we come up against the fact that the de nition of a proof uses quanti ers twice (for ) and 8). Moreover in the ) case, one cannot say that the domain of de nition of f is particularly well understood!
Since the ) and 8 cases were problematic (from this absurd foundational point of view), it has been proposed to add to clauses 4 and 6 the codicil \together with a proof that f has this property". Of course that settles nothing, and the Byzantine discussions about the meaning which would have to be given to this
1.2. THE TWO SEMANTIC TRADITIONS |
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codicil | discussions without the least mathematical content | only serve to discredit an idea which, we repeat, is one of the cornerstones of Logic.
We shall come across Heyting's idea working in the Curry-Howard isomorphism. It occurs in Realisability too. In both these cases, the foundational pretensions have been removed. This allows us to make good use of an idea which may have spectacular applications in the future.