Ideas from 'Begriffsschrift' by Gottlob Frege [1879], by Theme Structure

[found in 'Translations from the Writings of Gottlob Frege' by Frege,Gottlob (ed/tr Geach,P/Black,M) [Blackwell 1980,0-631-12911-1]].

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1. Philosophy / F. Analytic Philosophy / 6. Logical Analysis
Frege changed philosophy by extending logic's ability to check the grounds of thinking
                        Full Idea: Frege's 1879 logic transformed philosophy because it greatly expanded logic's reach - what thought can achieve unaided - and hence compelled a re-examination of everything previously said about the grounds of thought when logic gives out.
                        From: comment on Gottlob Frege (Begriffsschrift [1879]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 Intro
                        A reaction: I loved the gloss on logic as 'what thought can achieve unaided'. I largely see logic in terms of what is mechanically computable.
2. Reason / B. Laws of Thought / 1. Laws of Thought
We should not describe human laws of thought, but how to correctly track truth
                        Full Idea: Frege disagree that logic should merely describe the laws of thought - how people actually did reason. Logic is essentially normative, not descriptive. We want the one logic which successfully tracks the truth.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Jennifer Fisher - On the Philosophy of Logic 1.III
                        A reaction: This explains Frege's sustained attack on psychologism, and it also explains we he ended up as a platonist about logic - because he wanted its laws to be valid independently of human thinking. A step too far, perhaps. Brains are truth machines.
4. Formal Logic / C. Predicate Calculus PC / 1. Predicate Calculus PC
I don't use 'subject' and 'predicate' in my way of representing a judgement
                        Full Idea: A distinction of subject and predicate finds no place in my way of representing a judgement.
                        From: Gottlob Frege (Begriffsschrift [1879], §03)
                        A reaction: Perhaps this sentence could be taken as the beginning of modern analytical philosophy. The old view doesn't seem to me entirely redundant - merely replaced by a much more detailed analysis of what makes a 'subject' and what makes a 'predicate'.
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
For Frege, 'All A's are B's' means that the concept A implies the concept B
                        Full Idea: 'All A's are B's' meant for Frege that the concept A implies the concept B, or that to be A implies also to be B. Moreover this applies to arbitrary x which happens to be A.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Michal Walicki - Introduction to Mathematical Logic History D.2
                        A reaction: This seems to hit the renate/cordate problem. If all creatures with hearts also have kidneys, does that mean that being enhearted logically implies being kidneyfied? If all chimps are hairy, is that a logical requirement? Is inclusion implication?
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing)
                        Full Idea: Frege distinguished between asserting a proposition and expressing it, and he introduced the judgement stroke (a small vertical line, assertion) and the content stroke (a long horizontal line, expression) to represent them.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege Ch.3
                        A reaction: There are also strokes for conditional and denial.
The laws of logic are boundless, so we want the few whose power contains the others
                        Full Idea: Since in view of the boundless multitude of laws that can be enunciated we cannot list them all, we cannot achieve completeness except by searching out those that, by their power, contain all of them.
                        From: Gottlob Frege (Begriffsschrift [1879], §13)
                        A reaction: He refers to these laws in the previous sentence as the 'core'. His talk of 'power' is music to my ears, since it implies a direction of explanation. Burge says the power is that of defining other concepts.
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
In 1879 Frege developed second order logic
                        Full Idea: By 1879 Frege had discovered an algorithm, a mechanical proof procedure, that embraces what is today standard 'second order logic'.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Hilary Putnam - Reason, Truth and History Ch.5
                        A reaction: Note that Frege did more than introduce quantifiers, and the logic of predicates.
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
Frege replaced Aristotle's subject/predicate form with function/argument form
                        Full Idea: Frege's regimentation is based on the view of the simplest sort of statement as having, not subject/predicate form (as in Aristotle), but function/argument form.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege
                        A reaction: This looks like being a crucial move into the modern world, where one piece of information is taken in and dealt with, as in computer procedures. Have educated people reorganised their minds along Fregean lines?
5. Theory of Logic / G. Quantification / 1. Quantification
A quantifier is a second-level predicate (which explains how it contributes to truth-conditions)
                        Full Idea: The contribution of the quantifier to the truth conditions of sentences of which it is a part cannot be adequately explained if it is treated as other than a second-level predicate (for instance, if it is viewed as name).
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.2
                        A reaction: They suggest that this makes it something like a 'property of properties'. With this account it becomes plausible to think of numbers as quantifiers (since they do, after all, specify quantities).
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
For Frege the variable ranges over all objects
                        Full Idea: For Frege the variable ranges over all objects.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by William W. Tait - Frege versus Cantor and Dedekind XII
                        A reaction: The point is that Frege had not yet seen the necessity to define the domain of quantification, and this leads him into various difficulties.
Frege's domain for variables is all objects, but modern interpretations first fix the domain
                        Full Idea: For Frege there is no need to specify the domain of the individual variables, which is taken as the totality of all objects. This contrasts with the standard notion of an interpretation, which demands that we first fix the domain.
                        From: comment on Gottlob Frege (Begriffsschrift [1879]) by Michael Dummett - Frege Philosophy of Language (2nd ed) Ch.14
                        A reaction: What intrigues me is how domains of quantification shift according to context in ordinary usage, even in mid-sentence. I ought to go through every idea in this database, specifying its domain of quantification. Any volunteers?
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
Frege introduced quantifiers for generality
                        Full Idea: In order to express generality, Frege introduced quantifier notation.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege
                        A reaction: This is the birth of predicate logic, beloved of analytical philosophers (but of no apparent interest to phenomenalists, deconstructionists, existentialists?). Generality is what you get from induction (which is, of course, problematic).
Frege reduced most quantifiers to 'everything' combined with 'not'
                        Full Idea: Frege treated 'everything' as basic, and suggested ways of recasting propositions containing other quantifiers so that this was the only one remaining. He recast 'something' as 'at least one thing', and defined this in terms of 'everything' and 'not'.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Gregory McCullogh - The Game of the Name 1.6
                        A reaction: Extreme parsimony seems highly desirable in logic as well as ontology, but it can lead to frustrations, especially over the crucial question of the existence of things quantified over. See Idea 6068.
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
Proof theory began with Frege's definition of derivability
                        Full Idea: Frege's formal definition of derivability is perhaps the first investigation in general proof theory.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Dag Prawitz - Gentzen's Analysis of First-Order Proofs 2 n2
                        A reaction: In 'On General Proof Theory §1' Prawitz says "proof theory originated with Hilbert" in 1900. Presumably Frege offered a theory, and then Hilbert saw it as a general project.
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Frege produced axioms for logic, though that does not now seem the natural basis for logic
                        Full Idea: Frege's work supplied a set of axioms for logic itself, at least partly because it was a well-known way of presenting the foundations in other disciplines, especially mathematics, but it does not nowadays strike us as natural for logic.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by David Kaplan - Dthat 5.1
                        A reaction: What Bostock has in mind is the so-called 'natural' deduction systems, which base logic on rules of entailment, rather than on a set of truths. The axiomatic approach uses a set of truths, plus the idea of possible contradictions.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
It may be possible to define induction in terms of the ancestral relation
                        Full Idea: Frege's account of the ancestral has made it possible, in effect, to define the natural numbers as entities for which induction holds.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Crispin Wright - Frege's Concept of Numbers as Objects 4.xix
                        A reaction: This is the opposite of the approach in the Peano Axioms, where induction is used to define the natural numbers.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Frege's logic has a hierarchy of object, property, property-of-property etc.
                        Full Idea: Frege's general logical system involves a type hierarchy, distinguishing objects from properties from properties-of-properties etc., with every item belonging to a determinate level.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Peter Smith - Intro to Gödel's Theorems 14.1
                        A reaction: The Theory of Types went on to apply this hierarchy to classes, where Frege's disastrous Basic Law V flattens the hierarchy of classes, putting them on the same level (Smith p.119)
7. Existence / A. Nature of Existence / 1. Nature of Existence
Existence is not a first-order property, but the instantiation of a property
                        Full Idea: When Kant said that existence was not a property, what he meant was, according to Frege, that existence is not a first-order property - it is not a property of individuals but a property of properties, that the property has an instance.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Stephen Read - Thinking About Logic Ch.5
19. Language / C. Assigning Meanings / 4. Compositionality
Frege's account was top-down and decompositional, not bottom-up and compositional
                        Full Idea: Frege's account was top-down, not bottom-up: he aimed to decompose and discern function-argument structure in already existing sentences, not to explain how those sentences acquired their meanings in the first place.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 03 'Func'
                        A reaction: This goes with the holistic account of meaning, which leads to Quine's gavagai and Kuhn's obfuscation of science. I recommend compositionality for everthing.
28. God / B. Proving God / 2. Proofs of Reason / b. Ontological Proof critique
The predicate 'exists' is actually a natural language expression for a quantifier
                        Full Idea: On Frege's logical analysis, the predicate 'exists' is actually a natural language expression for a quantifier.
                        From: report of Gottlob Frege (Begriffsschrift [1879]) by Joan Weiner - Frege Ch.8
                        A reaction: However see Idea 6067, for McGinn's alternative view of quantifiers. In the normal conventions of predicate logic it may be that existence is treated as a quantifier, but that is not the same as saying that existence just IS a quantifier.