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Notes by RBJ on

Begriffsschrift

a formalised Language of pure Thought modelled upon the Language of Arithmetic

by Gottlob Frege

RBJ's Introduction
The Begriffsschrift CONTENTS List
The Inconsistency of the Begriffsschrift

RBJ's Introduction

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?
I was not surprised to find the grounds extremely tenuous. I was surprised when I concluded that the system is inconsistent (not because its hard to spot, but because its easy to spot and I wasn't aware that anyone had).

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.)

CONTENTS
Chap.TitleSections
 PREFACE 
I.DEFINITION of the SYMBOLS  
 §1.Letters and other signs
Judgement
 §2.Possibility that a content become a judgement. Content stroke, judgement stroke.
 §3.Subject and Predicate. Conceptual content.
 §4.Universal, particular; negative; categoric, hypothetic, disjunctive; apodictic, assertory, problematic judgements.
Conditionality
 §5.If. Conditional stroke.
 §6.Inference. The Aristotelian mode of inference.
Negation
 §7.Negation stroke. Or, either-or, and, but, and not, neither-nor.
Identity of Content
 §8.Need for a sign of identity of content, introduction of such a sign.
Functions
 §9.Definition of the words "function" and "argument". Functions of several arguments. Argument places. Subject, object.
 §10.Use of letters as function signs. "A has the property ." "B has the relation to A." "B is a result of the application of the procedure to the object A." The function sign as argument.
Generality
 §11.German letters. The concavity in the content stroke. Replaceability of German letters. Their scope. Latin letters.
 §12.There are some objects that do not --. There is no --. There are some --. Every. All. Causal connections. None. Some do not. Some. It is possible that --. Square of logical opposition.
II.REPRESENTATION and DERIVATION of SOME JUDGEMENTS of PURE THOUGHT
 §13.Usefulness of the deductive mode of presentation.
 §14.The first two fundamental laws of conditionality.
 §15.Some of their consequences.
 §16.The third fundamental law of conditionality, consequences.
 §17.The first fundamental law of negation, consequences.
 §18.The second fundamental law of negation, consequences.
 §19.The third fundamental law of negation, consequences.
 §20.The first fundamental law of identity of content, consequences.
 §21.The second fundamental law of identity of content, consequences.
 §22.The fundamental law of generality, consequences.
III.SOME TOPICS from a GENERAL THEORY of SEQUENCES
 §23.Introductory remarks
 §24.Heredity. Doubling of the judgement stroke. Lower-case Greek letters.
 §25.Consequences
 §26.Succession in a sequence
 §27.Consequences
 §28.Further Consequences
 §29."z belongs to the f-sequence beginning with x." Definition and consequences.
 §30.Further Consequences
 §31.Single-valuedness of a procedure. Definition and consequences.

PREFACE

"... we divide all truths that require justification into two kinds, those for which the proof can be carried out purely by means of logic, and those for which it must be supported by facts of experience."

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."

DEFINITION of the SYMBOLS

§1. Letters and other signs

Frege distinguishes between letters, which are left indeterminate (what we now call "variables") and other signs, each of which has some particular meaning (and we would call "constants").

Judgement

§2. Possibility that a content become a judgement. Content stroke, judgement stroke.

A judgement is expressed by asserting a content. The assertion of a content is done using the symbol "" (now called a "turnstile") of which the horizontal part combines the symbols which follow it into a totality and the vertical part asserts that totality. Frege distinguishes "contents that can become a judgement from those that cannot".

§3. Subject and Predicate. Conceptual content.

The subject-predicate analysis of sentences used in traditional logic does not appear in the formula language of mathematics and Frege has therefore found it convenient to dispense with it (except in the nominal sense that a judgement sign might be considered to predicate facthood of the content it precedes; this insignificant sense in which the subject predicate form is preserved reappears in Wittgenstein's Tractatus as a profundity about the forms of judgement).

§4. Universal, particular; negative; ... judgements.

Different kinds of judgement are more properly considered (in Frege's system) as judgements of different kinds of content. Many of the fine distinctions found in previous logical work have no value in the present context.

Conditionality

§5. If. Conditional stroke.

The graphical layout of the conditional (what we now call "material implication") is presented and is explained by giving its truth conditions.

§6. Inference. The Aristotelian mode of inference.

The rule of modus ponens is introduced (though not by that name) together with some ways of abbreviating its occurrences in proofs. This is the only form of inference used by Frege which has more than one premise. Other forms corresponding to the various Aristotelian modes of inference could be introduced but have been omitted for the sake of simplicity (being, in this system, superfluous).

Negation

§7. Negation stroke. Or, either-or, and, but, and not, neither-nor.

A short vertical stroke attached below the content line is introduced as the "negation stroke" expressing "the circumstance that the content does not take place". Examples are provided of how the negation stroke can be used in combination with conditionality, frequently explained by explicit descriptions of the truth conditions of the resulting combinations. It is shown how to do "or" and "and" and Frege observes that "and" and negation could have been used instead of conditionality and negation, but the inferences would then have been less simple.

Identity of Content

§8. Need for a sign of identity of content, introduction of such a sign.

Frege talks of identity as differing from other signs in relating names rather than contents. "For it expresses the circumstance that the two names have the same content". So in one sentence he seems to say that equality is referentially opaque, and in the next to deny it. A symbol for equality is introduced, and is explained as meaning that its two operands "have the same content" and therefore "we can everywhere put B for A and conversely" (substitution rules are given no explicit definition). This symbol serves indifferently as equality and logical equivalence. Frege spends some time explaining why an equality can be significant.

Functions

§9. Definition of the words "function" and "argument". Functions of several arguments. Argument places. Subject, object.

"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.

§10. Use of letters as function signs. "A has the property ." "B has the relation to A." "B is a result of the application of the procedure to the object A." The function sign as argument.

Application of an indeterminate function to its arguments is shown using a letter (usually a Greek capital) for the function, followed by the argument or arguments enclosed in brackets and separated by commas. e.g.:
(A)

(A,B)

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).

Generality

§11. German letters. The concavity in the content stroke. Replaceability of German letters. Their scope. Latin letters.

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.

§12. There are some objects that do not --. There is no --. There are some --. Every. All. Causal connections. None. Some do not. Some. It is possible that --. Square of logical opposition.

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).

REPRESENTATION and DERIVATION of SOME JUDGEMENTS of PURE THOUGHT

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:

  1. implication associates to the right
  2. universal quantification binds loosely
  3. the details of substitutions to be performed before the various applications of MP are omitted, and the order of the arguments (which was intended to be "MP condition, conditional"), is mostly but not always incorrect.

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.

§13. Usefulness of the deductive mode of presentation.

§14. The first two fundamental laws of conditionality.

A a (b a) (1)
A (c (b a)) ((c b) (c a)) (2)

§15. Some of their consequences.

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)

§16. The third fundamental law of conditionality, consequences.

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)

§17. The first fundamental law of negation, consequences.

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)

§18. The second fundamental law of negation, consequences.

A b b(31)
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)

§19. The third fundamental law of negation, consequences.

A a a(41)
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)

§20. The first fundamental law of identity of content, consequences.

A (c d) f(c) f(d)(52)
MP 8,52 f(c) (c d) f(d)(53)

§21. The second fundamental law of identity of content, consequences.

A (c c)(54)
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)

§22. The fundamental law of generality, consequences.

A (a.f(a)) f(c)(58)
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)

SOME TOPICS from a GENERAL THEORY of SEQUENCES

The Inconsistency of Begriffsschrift

That Russell showed how to derive a contradiction in the system of Frege's Grundgesetze der Arithmetic [Frege1893] is well known. The blame has been placed, by Frege and others, with axiom V of that system.

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:
NSA() ()

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:

    kinds of content
    Frege distinguishes "contents that can become a judgement from those that cannot". However, he does not define this distinction.

    Mention of RANK
    In discussing functions Frege does talk of a difference of rank between "every positive integer" and "the number 20". However, the purpose of this discussion is to explain that he treats "every positive integer" not as a concept at all, but as (what Russell later called) an incomplete symbol, to be rendered using a universal quantifier. There is nothing here to suggest that the notion of rank has any significant role in the exposition.

    Distinguishing function and argument
    In his discussion of functions Frege does distinguish between function and argument, but only to that extent which is necessary in describing functional abstraction in a type-free context. To infer from this that he had some type system in mind is unwarranted, and may be thought contradicted by his emphasising that the distinction "has nothing to do with conceptual content".

    substitution
    "The meaning of a German letter is subject only to the obvious restriction that, if a combination of signs following a content stroke can become a judgement, this possibility remains unaffected by such a replacement and that, if the German letter occurs as a function sign, this circumstance be taken into account."

    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.


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