|The Begriffsschrift CONTENTS List|
|The Inconsistency of the Begriffsschrift|
The Begriffsschrift was Frege's revolution in logic, bringing to an end 2000 years in which Aristotelian logic prevailed and ushering in the age of symbolic or mathematical logic. These are my notes on finally getting round to looking closer at this piece of history.
I came to this with one particular question which I wanted to know the answer to:
|What grounds are there for supposing this to be a "second order logic" in anything like the modern sense of that phrase?|
Having been disappointed by the standard of rigour with which the logic of Principia Mathematica is defined, I had been lead by reports of Gödel's opinion that in this respect the Principia was a regression over Frege's work to expect that the Begriffsschrift would be better defined. This isn't really the case. I do not doubt that it is the major advance in logic which is held to be. The two most significant features in this respect being the transformation of boolean algebra from an informally developed branch of mathematics into a formal logical system, and the introduction of the universal quantifier to provide with greater generality than in Aristotle for non propositional inference. He also introduces functional abstraction in a very general sense, but he does not provide the necessary detail on how substitution is to work, and has not yet come to understand the risks which abstraction poses for consistency or any of the techniques for protecting against that risk.
I'm thinking of getting the Begriffsschrift machine checked. When I do I will do the type free version first and show the derivation of a contradiction, and then show that the contradiction does not go through when a type system is introduced but that all the rest does. (not that there is any need to machine check it, the proofs are easily hand checked.)
|I.||DEFINITION of the SYMBOLS|
|Identity of Content|
|II.||REPRESENTATION and DERIVATION of SOME JUDGEMENTS of PURE THOUGHT|
|III.||SOME TOPICS from a GENERAL THEORY of SEQUENCES|
In attempting "to ascertain how far one can proceed in arithmetic by means of inferences alone" Frege found language "to be an obstacle", which lead him "to the idea of the present ideography".
Having decided to forgo anything which is "without significance for the inferential sequence" Frege adopted the term "conceptual content" for the rest.
The ideography is to ordinary language as the microscope is to the human eye. Not so good for ordinary purposes, but a great deal better for the special purposes for which it was devised.
"If it is one of the tasks of philosophy to break the domination of the word over the human spirit by laying bare the misconceptions that through the use of language often almost unavoidably arise concerning the relations between concepts and by freeing thought from that with which only the means of expression of ordinary language, constituted as they are, saddle it, then my ideography, further developed for these purposes, can become a useful tool for the philosopher."
"if in an expression, whose content need not be capable of becoming an judgement, a simple or compound sign has one or more occurrences and if we regard that sign as replaceable in all or some of these occurrences by something else (but everywhere by the same thing), then we call the part that remains invariant in the expression a function, and the replaceable part the argument of the function."
There is an observation here about why "every positive integer" should not for these purposes be regarded as a compound sign. This explanation is confused in that it appears at first to be suggesting that "every positive integer" is the wrong type of thing to be used in the same place as one would a specific integer (not the same rank [gleichen Ranges]), but then goes on to explain it as what Russell was later to call an incomplete symbol. This passage may be taken to imply that the logic of Begriffsschrift is some kind of type theory, but this interpretation seems to me doubtful. The purpose is to presage the use of universal quantification to give an account of sentences involving "every", which an important point of departure from Aristotelian logic and is not connected with type constraints. If Frege had a type theory in mind (or a "second order" logic) he would certainly have needed to be much more explicit in presenting that feature to have any expectation of understanding in his readers.
We can also regard (A) as a function of the argument (which we would now call a "higher-order" function, without prejudice to the question whether a type system is involved).
The construction which we now call "universal quantification" is rendered by Frege using a concavity in the content sign (a small semicircular dip) in which appears the name of the variable to be bound by the quantifier (for which Frege uses german letters). The scope is the entire content to the right of the content sign containing the quantifier. Constraints are mentioned on "the meaning" of the letter, which we should perhaps take to constrain the range of quantification, and which are explained as constraints on what can be substituted for the bound variable. The constraint is that any combination of signs which is capable of becoming a judgement before a substitution will still be so capable after the substitution. In particular, if the letter appears as a function symbol "this should be taken into account".
This again suggests that a type constraint might be intended, but is not explicit enought to exclude some weaker constraint on substitution of a more syntactic nature.
From a general judgement may be derived any number of particular judgements obtained by making such a substitution and removing the concavity from the content sign.
Frege then discusses the use of generalisations as constituents of complex judgements, explaining that the purpose of the concavity it to determine the scope of the generalisation.
An additional inference rule is described permitting the introduction of generalisations, which may be introduced on the main content stroke for the judgement or on the conclusion of a conditional or nested conditional provided that the variable thus bound does not occur in the hypotheses of the conditionals and the bound variable only occurs in argument places. This last constraint is hard to understand. Does this make begriffsschrift into a first order system? I think not. I think what he means here is the argument positions of the function over which generalisation is taking place. Which he has previously stated might be in function positions in the expression. But how could it be otherwise, surely the occurrences of the variable indicate the argument places? Well not necessarily, Frege has said previously that an expression can be regarded as a function in which only some occurrences of a name vary with the argument, which is unnecessary and confusing since he has introduced no notation which would enable us to indicate which occurrences were genuine argument places. He is fixing that problem here by withdrawing the option to have occurrences of the variable which are not argument markers.
In this section Frege shows how various other ideas can be expressed using generalisation. Relations between some of these are shown diagrammatically (I think the diagram may relate to one from Aristotle, but I haven't located where the aristotelian version is).
I present here transcriptions of the proofs into a more compact modern notation. I have in mind getting these machine checked, until they are they will most likely contain errors arising in the transcription. Note that:
The value of logical derivation of propositional theorems is now very doubtful, the advent of the computer having made this kind of proof quite redundant. Even without using a computer, once a reasonable level of understanding is achieved it becomes easier to check a propositional conjecture by eyeball and wit than to check a purported proof of the conjecture.
This is no comment on Frege's achievement, but rather a hint about how seriously a modern student should take this aspect of logic.
As soon as quantifiers are introducted life gets more complicated.
|A||a (b a)||(1)|
|A||(c (b a)) ((c b) (c a))||(2)|
|MP 2,1||(a b) (c b a) (c b) (c a)||(3)|
|MP 3,2||((b a) c b a) (b a) (c b) (c a)||(4)|
|MP 1,4||(b a) (c b) (c a)||(5)|
|MP 5,5||(c b a) c (d b) (d a)||(6)|
|MP 6,5||(b a) (d c b) (d c a)||(7)|
|A||(d b a) (b d a)||(8)|
|MP 5,8||(c b) (b a) (c a)||(9)|
|MP 8,9||((e d b) a) ((d e b) a)||(10)|
|MP 9,1||((c b) a) (b a)||(11)|
|MP 5,8||(d c b a) (d b c a)||(12)|
|MP 12,12||(d c b a) (b d c a)||(13)|
|MP 5,13||(e d c b a) (e b d c a)||(14)|
|MP 12,14||(e d c b a) (b e d c a)||(15)|
|MP 5,12||(e d c b a) (e d b c a)||(16)|
|MP 16,8||(d c b a) (c b d a)||(17)|
|MP 16,5||(c b a) (d c) (b d a)||(18)|
|MP 18,9||(d c b) (b a) (d c a)||(19)|
|MP 18,19||(e d c b) (b a) (e d c a)||(20)|
|MP 9,19||((d b) a) (d c) ((c b) a)||(21)|
|MP 5,16||(f e d c b a) (f e d b c a)||(22)|
|MP 22,18||(d c b a) (e d) (c b e a)||(23)|
|MP 12,1||(c a) (c b a)||(24)|
|MP 5,24||(d c a) (d c b a)||(25)|
|MP 8,1||b a a||(26)|
|MP 26,1||a a||(27)|
|A||(b a) (¬a ¬b)||(28)|
|MP 5,28||(c b a) c (¬a ¬b)||(29)|
|MP 10,29||(b c a) c (¬a ¬b)||(30)|
|MP 7,31||((¬b a) (¬a ¬¬b)) (¬b a) (¬a b)||(32)|
|MP 7,28||(¬b a) (¬a b)||(33)|
|MP 5,33||(c ¬b a) (c ¬a b)||(34)|
|MP 12,34||(c ¬b a) (¬a c b)||(35)|
|MP 34,1||a ¬a b||(36)|
|MP 9,36||((¬c b) a) (c a)||(37)|
|MP 8,36||¬a a b||(38)|
|MP 2,8||(¬a a) (¬a b)||(39)|
|MP 35,2||¬b (¬a a) a||(40)|
|MP 41,27||¬¬(a a)||(42)|
|MP 40,41||(¬a a) a||(43)|
|MP 21,43||(¬a c) ((c a) a)||(44)|
|MP 5,44||((¬c a) (¬a c)) (¬c a) ((c a) a)||(45)|
|MP 45,33||(¬c a) ((c a) a)||(46)|
|MP 21,46||(¬c b) (b a) ((c a) a)||(47)|
|MP 23,47||(d (¬c b)) (b a) (c a) (d a)||(48)|
|MP 12,47||(¬c b) (c a) ((b a) a)||(49)|
|MP 17,49||(c a) (b a) ((¬c b) a)||(50)|
|MP 18,50||(d c a) (b a) d ((¬c b) a)||(51)|
|A||(c d) f(c) f(d)||(52)|
|MP 8,52||f(c) (c d) f(d)||(53)|
|MP 53,54||(c d) (d c)||(55)|
|MP 9,55||((d c) f(d) f(c)) (c d) f(d) f(c)||(56)|
|MP 52,56||(c d) f(d) f(c)||(57)|
|MP 30,58||g(b) ¬f(b) ¬(a. g(a) f(a))||(59)|
|MP 12,58||(a. h(a) g(a) f(a)) (g(b) h(b) f(b))||(60)|
|MP 9,58||(f(c) a) ((a. f(a)) a)||(61)|
|MP 8,58||g(x) (a. g(a) f(a)) f(x)||(62)|
|MP 24,62||g(x) m (a. g(a) f(a)) f(x)||(63)|
|MP 18,62||(h(y) g(x)) (a. g(a) f(a)) (h(y) f(x))||(64)|
|MP 61,64||(a. h(a) g(a)) (a. g(a) f(a)) (h(x) f(x))||(65)|
|MP 8,65||(a. g(a) f(a)) (a. h(a) g(a)) (h(x) f(x))||(66)|
|MP 7,58||(((a. f(a)) b) b (a. f(a))) (((a. f(a)) b) b f(c))||(67)|
|MP 67,57||((a. f(a)) b) b f(c)||(68)|
What is not so widely publicised is that the logical system of the Begriffsschrift, if interpreted solely in terms of that document, is also inconsistent (without benefit of axiom V).
Frege's logic has in modern times been interpreted as a second order logic. However, the need for the type constraints built into second order logic was not appreciated until after the inconsistency was discovered in the Grundgesetze. There is in the Begriffsschrift no hint that constraints of this kind were invisaged, and the logic therefore benefits from unconstrained functional abstraction sufficient to reproduce Russell's paradox, by introducing a definition such as:
There are hints in Begriffsschrift that some kind of constraints are necessary, even though not spelled out in detail, but it seems to me to stretch credibility beyond reasonable bounds to suppose that Frege had in mind adequate measures to ensure the consistency of the Begriffsschrift but failed to communicate them clearly.
Some examples of the kind of issue which he thought worth mentioning are:
The above passage suggests minimal constraints. However another passage seems more draconian. An italian letter (used for a free variable) may be replaced by a (bound) german letter only if the italian letter occurs only in argument places. Even though Frege has earlier explicitly allowed german variables in the function places. (I think I have figured this out now, but the explanation doesn't fit in the margin)
From this last quote we may infer that any constraints on substitution which are not "obvious" are not intended. The constraints necessary to ensure the consistency of Begriffsschrift are not obvious even to someone well grounded in modern logic, let alone to an audience which we must presume knew little better than Aristotle.