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Ideas of Gottlob Frege, by Text
[German, 1848  1925, Led a quiet and studious life as Professor at the University of Jena.]
1874

Rechnungsmethoden (dissertation)

Ch.6

p.68

9831

Geometry appeals to intuition as the source of its axioms

p.2

p.279

18256

Quantity is inconceivable without the idea of addition


p.17

8939

We should not describe human laws of thought, but how to correctly track truth [Fisher]


p.19

7742

Frege reduced most quantifiers to 'everything' combined with 'not' [McCullogh]


p.23

17745

For Frege, 'All A's are B's' means that the concept A implies the concept B [Walicki]


p.23

9950

A quantifier is a secondlevel predicate (which explains how it contributes to truthconditions) [George/Velleman]


p.31

7728

Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) [Weiner]


p.37

7729

Frege replaced Aristotle's subject/predicate form with function/argument form [Weiner]


p.44

7730

Frege introduced quantifiers for generality [Weiner]


p.59

9991

For Frege the variable ranges over all objects [Tait]


p.118

10607

Frege's logic has a hierarchy of object, property, propertyofproperty etc. [Smith,P]


p.124

7622

In 1879 Frege developed second order logic [Putnam]


p.126

11008

Existence is not a firstorder property, but the instantiation of a property [Read]


p.133

7741

The predicate 'exists' is actually a natural language expression for a quantifier [Weiner]


p.161

17855

It may be possible to define induction in terms of the ancestral relation [Wright,C]


p.191

13609

Frege produced axioms for logic, though that does not now seem the natural basis for logic [Kaplan]


p.207

13824

Proof theory began with Frege's definition of derivability [Prawitz]


p.475

10536

Frege's domain for variables is all objects, but modern interpretations first fix the domain [Dummett]

§03

p.12

4971

I don't use 'subject' and 'predicate' in my way of representing a judgement

§13

p.29

16881

The laws of logic are boundless, so we want the few whose power contains the others

1881

Boole calculus and the Concept script

p.17

p.17

18265

We don't judge by combining subject and concept; we get a concept by splitting up a judgement

1884

Grundlagen der Arithmetik (Foundations)


p.15

2514

Frege tried to explain synthetic a priori truths by expanding the concept of analyticity


p.14

2515

Frege fails to give a concept of analyticity, so he fails to explain synthetic a priori truth that way [Katz]


p.1

13864

Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C]


p.3

8911

If abstracta are nonmental, quarks are abstracta, and yet chess and God's thoughts are mental [Rosen]


p.4

10803

Frege himself abstracts away from tone and color [Yablo]


p.7

10625

Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright]


p.7

13871

Frege claims that numbers are objects, as opposed to them being Fregean concepts [Wright,C]


p.8

16022

The idea of a criterion of identity was introduced by Frege [Noonan]


p.10

13872

Numbers are secondlevel, ascribing properties to concepts rather than to objects [Wright,C]


p.11

9154

Frege agreed with Euclid that the axioms of logic and mathematics are known through selfevidence [Burge]


p.11

10309

Frege says singular terms denote objects, numerals are singular terms, so numbers exist [Hale]


p.11

13874

Numbers seem to be objects because they exactly fit the inference patterns for identities


p.13

13875

Frege's platonism proposes that objects are what singular terms refer to [Wright,C]


p.13

13876

The syntactic category is primary, and the ontological category is derivative [Wright,C]


p.13

9816

For Frege, successor was a relation, not a function [Dummett]


p.15

13878

Concepts are, precisely, the references of predicates [Wright,C]


p.17

9945

Logicism shows that no empirical truths are needed to justify arithmetic [George/Velleman]


p.18

10642

Secondorder quantifiers are committed to concepts, as firstorder commits to objects [Linnebo]


p.23

9951

It appears that numbers are adjectives, but they don't apply to a single object [George/Velleman]


p.24

9952

Numerical adjectives are of the same secondlevel type as the existential quantifier [George/Velleman]


p.25

13879

For Frege, ontological questions are to be settled by reference to syntactic structures [Wright,C]


p.25

9953

Numbers are more than just 'secondlevel concepts', since existence is also one [George/Velleman]


p.25

13881

We need to grasp not numberobjects, but the states of affairs which make number statements true [Wright,C]


p.26

9157

The null set is only defensible if it is the extension of an empty concept [Burge]


p.26

9158

For Frege a priori knowledge derives from general principles, so numbers can't be primitive


p.27

9954

"Number of x's such that ..x.." is a functional expression, yielding a name when completed [George/Velleman]


p.29

10139

Frege gives an incoherent account of extensions resulting from abstraction [Fine,K]


p.30

10028

For Frege the number of F's is a collection of firstlevel concepts [George/Velleman]


p.30

9956

'The number of Fs' is the extension (a collection of firstlevel concepts) of the concept 'equinumerous with F' [George/Velleman]


p.33

10029

Numbers need to be objects, to define the extension of the concept of each successor to n [George/Velleman]


p.35

10030

'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [George/Velleman]


p.39

10033

Why should the existence of pure logic entail the existence of objects? [George/Velleman]


p.41

10034

The number of natural numbers is not a natural number [George/Velleman]


p.43

9973

The number of F's is the extension of the second level concept 'is equipollent with F' [Tait]


p.44

16500

Frege showed that numbers attach to concepts, not to objects [Wiggins]


p.44

9976

Frege accepts abstraction to the concept of all sets equipollent to a given one [Tait]


p.48

11030

The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt]


p.49

11031

'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt]


p.54

7731

How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Weiner]


p.55

15916

Frege's onetoone correspondence replaces wellordering, because infinities can't be counted [Lavine]


p.57

7736

A concept is a nonpsychological oneplace function asserting something of an object [Weiner]


p.59

7737

Identities refer to objects, so numbers must be objects [Weiner]


p.59

9990

Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Tait]


p.64

13527

Frege's cardinals (equivalences of oneone correspondences) is not permissible in ZFC


p.64

9631

Formalism fails to recognise types of symbols, and also metagames [Brown,JR]


p.66

10831

Frege only managed to prove that arithmetic was analytic with a logic that included settheory [Quine]


p.66

8690

From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Friend]


p.66

7738

Zero is defined using 'is not selfidentical', and one by using the concept of zero [Weiner]


p.76

11100

Frege's algorithm of identity is the law of putting equals for equals [Quine]


p.77

9832

Frege sees no 'intersubjective' category, between objective and subjective [Dummett]


p.78

10219

Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Shapiro]


p.87

10606

Frege treats properties as a kind of function, and maybe a property is its characteristic function [Smith,P]


p.91

12153

Geach denies Frege's view, that 'being the same F' splits into being the same and being F [Perry]


p.91

9834

A class is, for Frege, the extension of a concept [Dummett]


p.92

9835

It is because a concept can be empty that there is such a thing as the empty class [Dummett]


p.96

9838

Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Dummett]


p.111

13887

Frege started with contextual definition, but then switched to explicit extensional definition [Wright,C]


p.112

9841

Frege was the first to give linguistic answers to nonlinguistic questions


p.114

13889

Fregean numbers are numbers, and not 'Caesar', because they correlate 11 [Wright,C]


p.118

7739

Arithmetic is analytic [Weiner]


p.123

10010

Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes]


p.125

9844

Originally Frege liked contextual definitions, but later preferred them fully explicit [Dummett]


p.135

18104

Frege, unlike Russell, has infinite individuals because numbers are individuals


p.136

13897

Each number, except 0, is the number of the concept of all of its predecessors [Wright,C]


p.162

9853

Identity between objects is not a consequence of identity, but part of what 'identity' means [Dummett]


p.166

8782

Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Hale/Wright]


p.167

9855

Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects [Dummett]


p.167

9854

We can introduce new objects, as equivalence classes of objects already known [Dummett]


p.168

9856

Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett]


p.171

8785

For Frege, objects just are what singular terms refer to [Hale/Wright]


p.190

17442

Frege thinks number is fundamentally bound up with oneone correspondence [Heck]


p.191

13608

Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Bostock]


p.200

9870

Early Frege takes the extensions of concepts for granted [Dummett]


p.204

9564

For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Chihara]


p.223

9875

Frege was completing Bolzano's work, of expelling intuition from number theory and analysis


p.246

5658

Numbers are definable in terms of mapping items which fall under concepts [Scruton]


p.246

10802

Frege's 'parallel' and 'direction' don't have the same content, as we grasp 'parallel' first [Yablo]


p.252

15948

Frege developed formal systems to avoid unnoticed assumptions [Lavine]


p.257

10278

Without concepts we would not have any objects [Shapiro]


p.258

10804

Thoughts have a natural order, to which human thinking is drawn. [Yablo]


p.267

18142

Oneone correlations imply normal arithmetic, but don't explain our concept of a number


p.277

17636

A cardinal number may be defined as a class of similar classes [Russell]


p.281

9902

Frege's incorrect view is that a number is an equivalence class [Benacerraf]


p.305

10526

Fregean abstraction creates concepts which are equivalences between initial items [Fine,K]


p.305

10525

Frege put the idea of abstraction on a rigorous footing [Fine,K]


p.325

16883

Arithmetical statements can't be axioms, because they are provable [Burge]


p.354

17623

To understand a thought you must understand its logical structure [Burge]


p.355

17814

The natural number n is the set of nmembered sets [Yourgrau]


p.356

17816

Frege's logicism aimed at removing the reliance of arithmetic on intuition [Yourgrau]


p.357

17819

A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau]


p.358

17820

If you can subdivide objects many ways for counting, you can do that to setelements too [Yourgrau]


p.361

16891

Despite Gödel, Frege's epistemic ordering of all the truths is still plausible [Burge]


p.371

16896

If numbers can be derived from logic, then set theory is superfluous [Burge]


p.405

17427

Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]


p.407

17430

Fregean concepts have precise boundaries and universal applicability [Koslicki]


p.408

17431

Vagueness is incomplete definition [Koslicki]


p.409

17432

Frege's universe comes already divided into objects [Koslicki]


p.418

17437

Nonarbitrary division means that what falls under the concept cannot be divided into more of the same [Koslicki]


p.424

17438

Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Koslicki]


p.504

10551

If objects exist because they fall under a concept, 0 is the object under which no objects fall [Dummett]


p.504

10550

Frege establishes abstract objects independently from concrete ones, by falling under a concept [Dummett]


p.947

16905

Arithmetic must be based on logic, because of its total generality [Jeshion]


p.947

16906

The primitive simples of arithmetic are the essence, determining the subject, and its boundaries [Jeshion]

Intro

p.9

8620

Thought is the same everywhere, and the laws of thought do not vary

Intro

p.9

8619

To learn something, you must know that you don't know

Intro

p.7

8621

Mental states are irrelevant to mathematics, because they are vague and fluctuating

Intro

p.5

8622

Psychological accounts of concepts are subjective, and ultimately destroy truth

Intro p.x

p.3

8415

Never lose sight of the distinction between concept and object

Intro p.x

p.3

8414

Keep the psychological and subjective separate from the logical and objective

§005, 88

p.3

20295

All analytic truths can become logical truths, by substituting definitions or synonyms [Rey]

§02

p.18

17495

Proof aims to remove doubts, but also to show the interdependence of truths

§02

p.190

17443

Many of us find Frege's claim that truths depend on one another an obscure idea [Heck]

§02

p.944

16903

Justifications show the ordering of truths, and the foundation is what is selfevident [Jeshion]

§03

p.4

9352

An a priori truth is one derived from general laws which do not require proof

§03

p.5

9370

A statement is analytic if substitution of synonyms can make it a logical truth [Boghossian]

§03

p.108

8743

Frege considered analyticity to be an epistemic concept [Shapiro]

§03

p.359

16889

A truth is a priori if it can be proved entirely from general unproven laws

§03 n

p.4

8624

Induction is merely psychological, with a principle that it can actually establish laws

§10

p.16

8626

In science one observation can create high probability, while a thousand might prove nothing

§13

p.360

16890

Frege's problem is explaining the particularity of numbers by general laws [Burge]

§18

p.25

8630

Individual numbers are best derived from the number one, and increase by one

§24

p.31

8632

You can't transfer external properties unchanged to apply to ideas

§25

p.33

8633

There is no physical difference between two boots and one pair of boots

§26

p.35

8634

The equator is imaginary, but not fictitious; thought is needed to recognise it

§26

p.382

16900

Intuitions cannot be communicated [Burge]

§26,85

p.480

10539

Frege refers to 'concrete' objects, but they are no different in principle from abstract ones [Dummett]

§27

p.38

8635

Numbers are not physical, and not ideas  they are objective and nonsensible

§29

p.40

8636

We can say 'a and b are F' if F is 'wise', but not if it is 'one'

§30

p.41

8637

The number 'one' can't be a property, if any object can be viewed as one or not one

§34

p.58

9988

If we abstract 'from' two cats, the units are not black or white, or cats [Tait]

§41

p.53

8639

If numbers are supposed to be patterns, each number can have many patterns

§42

p.54

8640

We cannot define numbers from the idea of a series, because numbers must precede that

§44

p.57

8641

You can abstract concepts from the moon, but the number one is not among them

§46

p.41

17460

A statement of number contains a predication about a concept

§46

p.43

11029

'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt]

§46

p.125

14236

Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed [Oliver/Smiley]

§47

p.61

8642

Abstraction from things produces concepts, and numbers are in the concepts

§53

p.65

8643

Affirmation of existence is just denial of zero

§53

p.65

8644

Because existence is a property of concepts the ontological argument for God fails

§54

p.59

9989

Units can be equal without being identical [Tait]

§54

p.403

17426

A concept creating a unit must isolate and unify what falls under it

§54

p.405

17428

Frege says counting is determining what number belongs to a given concept [Koslicki]

§54

p.406

17429

Frege says only concepts which isolate and avoid arbitrary division can give units [Koslicki]

§55?

p.123

10013

Numerical statements have firstorder logical form, so must refer to objects [Hodes]

§56

p.68

9046

Our definition will not tell us whether or not Julius Caesar is a number

§57

p.69

8645

Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"

§57

p.69

9999

For science, we can translate adjectival numbers into noun form

§60

p.71

8646

Words in isolation seem to have ideas as meanings, but words have meaning in propositions

§60

p.126

9846

Defining 'direction' by parallelism doesn't tell you whether direction is a line [Dummett]

§61

p.72

8648

Ideas are not spatial, and don't have distances between them

§61

p.72

8647

Not all objects are spatial; 4 can still be an object, despite lacking spatial coordinates

§62

p.111

9840

Frege initiated linguistic philosophy, studying number through the sense of sentences [Dummett]

§64

p.33

9822

Nothing should be defined in terms of that to which it is conceptually prior [Dummett]

§64

p.75

10556

We create new abstract concepts by carving up the content in a different way

§64

p.193

17445

Parallelism is intuitive, so it is more fundamental than sameness of direction [Heck]

§6468

p.232

9881

From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements [Dummett]

§6468

p.232

9882

You can't simultaneously fix the truthconditions of a sentence and the domain of its variables [Dummett]

§6468

p.233

9883

Frege introduced the standard device, of defining logical objects with equivalence classes [Dummett]

§66 n

p.77

8651

A concept is a possible predicate of a singular judgement

§68

p.79

18181

The Number for F is the extension of 'equal to F' (or maybe just F itself)

§68 n

p.80

8652

Numbers are objects, because they can take the definite article, and can't be plurals

§74

p.87

8653

Nought is the number belonging to the concept 'not identical with itself'

§77

p.90

8654

One is the Number which belongs to the concept "identical with 0"

§79

p.36

10032

'Ancestral' relations are derived by iterating back from a given relation [George/Velleman]

§87

p.99

8655

Arithmetic is analytic and a priori, and thus it is part of logic

§87

p.99

8656

The laws of number are not laws of nature, but are laws of the laws of nature

§90

p.102

8657

Mathematicians just accept selfevidence, whether it is logical or intuitive

4

p.355

17624

To understand axioms you must grasp their logical power and priority [Burge]

5557

p.118

18103

Numbers are objects because they partake in identity statements [Bostock]

p.x

p.3

7732

Never ask for the meaning of a word in isolation, but only in the context of a proposition


p.15

7725

'P or notp' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate [Weiner]


p.21

3307

Frege put forward an ontological argument for the existence of numbers [Benardete,JA]


p.22

13455

Frege did not think of himself as working with sets [Hart,WD]


p.33

7307

A thought is not psychological, but a condition of the world that makes a sentence true [Miller,A]


p.44

13473

Frege thinks there is an independent logical order of the truths, which we must try to discover [Hart,WD]


p.55

3318

Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' [Benardete,JA]


p.56

7309

Frege's 'sense' is the strict and literal meaning, stripped of tone [Miller,A]


p.59

3319

Frege gives a functional account of predication so that we can dispense with predicates [Benardete,JA]


p.67

6076

For Frege, predicates are names of functions that map objects onto the True and False [McGinn]


p.67

7312

'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness [Miller,A]


p.91

3328

Frege proposed a realist concept of a set, as the extension of a predicate or concept or function [Benardete,JA]


p.104

3331

If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA]


p.116

7316

Analytic truths are those that can be demonstrated using only logic and definitions [Miller,A]


p.119

5816

Frege said concepts were abstract entities, not mental entities [Putnam]


p.207

9871

Frege always, and fatally, neglected the domain of quantification [Dummett]


p.246

5657

Frege's logic showed that there is no concept of being [Scruton]


p.317

16880

Frege aimed to discover the logical foundations which justify arithmetical judgements [Burge]


p.320

16882

The building blocks contain the whole contents of a discipline


p.337

16885

To understand a thought, understand its inferential connections to other thoughts [Burge]


p.337

16884

Basic truths of logic are not proved, but seen as true when they are understood [Burge]


p.351

16887

Frege's concept of 'selfevident' makes no reference to minds [Burge]


p.369

16894

An apriori truth is grounded in generality, which is universal quantification [Burge]


p.371

16895

The null set is indefensible, because it collects nothing [Burge]

3.4

p.66

8689

Eventually Frege tried to found arithmetic in geometry instead of in logic [Friend]

p.228

p.228

9179

Frege frequently expressed a contempt for language [Dummett]

1891

Function and Concept


p.4

4028

Frege allows either too few properties (as extensions) or too many (as predicates) [Mellor/Oliver]


p.17

18899

Frege takes the existence of horses to be part of their concept


p.18

18806

Frege thought traditional categories had psychological and linguistic impurities


p.20

9948

Unlike objects, concepts are inherently incomplete [George/Velleman]


p.20

9947

Concepts are the ontological counterparts of predicative expressions [George/Velleman]

Ch.2.II

p.35

10319

An assertion about the concept 'horse' must indirectly speak of an object [Hale]

p.14

p.29

4972

I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false

p.30

p.30

8488

A concept is a function whose value is always a truthvalue

p.30

p.30

8487

Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism

p.32

p.32

8489

The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place

p.38

p.38

8490

Firstlevel functions have objects as arguments; secondlevel functions take functions as arguments

p.38 n

p.38

8491

The Ontological Argument fallaciously treats existence as a firstlevel concept

p.39

p.39

8492

Relations are functions with two arguments

1892

On Concept and Object


p.16

18995

Frege mistakenly takes existence to be a property of concepts, instead of being about things [Yablo]


p.21

9949

There is the concept, the object falling under it, and the extension (a set, which is also an object) [George/Velleman]


p.33

10317

It is unclear whether Frege included qualities among his abstract objects [Hale]


p.150

9167

Frege felt that meanings must be public, so they are abstractions rather than mental entities [Putnam]


p.474

10535

Frege's 'objects' are both the referents of proper names, and what predicates are true or false of [Dummett]

p.193

p.43

4973

As I understand it, a concept is the meaning of a grammatical predicate

p.196n

p.46

4974

For all the multiplicity of languages, mankind has a common stock of thoughts

p.199

p.49

4975

A thought can be split in many ways, so that different parts appear as subject or predicate

p.201

p.98

9839

Frege equated the concepts under which an object falls with its properties [Dummett]

1892

On Sense and Reference


p.1

8187

Frege was strongly in favour of taking truth to attach to propositions [Dummett]


p.4

11126

'Sense' gives meaning to nonreferring names, and to two expressions for one referent [Margolis/Laurence]


p.6

9462

Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first [Jacquette]


p.10

8164

Frege was the first to construct a plausible theory of meaning [Dummett]


p.13

9817

Earlier Frege focuses on content itself; later he became interested in understanding content [Dummett]


p.17

15155

Expressions always give ways of thinking of referents, rather than the referents themselves [Soames]


p.34

15597

Frege's Puzzle: from different semantics we infer different reference for two names with the same reference [Fine,K]


p.36

18752

'The concept "horse"' denotes a concept, yet seems also to denote an object [McGee]


p.41

8171

Frege divided the meaning of a sentence into sense, force and tone [Dummett]


p.46

10510

Frege ascribes reference to incomplete expressions, as well as to singular terms [Hale]


p.59

4954

Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined [Kripke]


p.59

17002

Frege's 'sense' is ambiguous, between the meaning of a designator, and how it fixes reference


p.79

18772

We can treat designation by a few words as a proper name


p.91

18778

Every descriptive name has a sense, but may not have a reference


p.100

4893

Frege was asking how identities could be informative [Perry]


p.154

18937

If sentences have a 'sense', empty name sentences can be understood that way [Sawyer]


p.154

18936

Frege moved from extensional to intensional semantics when he added the idea of 'sense' [Sawyer]


p.203

14075

Proper name in modal contexts refer obliquely, to their usual sense [Gibbard]


p.248

9180

Holism says all language use is also a change in the rules of language


p.357

7805

Frege started as antirealist, but the sense/reference distinction led him to realism [Benardete,JA]


p.395

10424

A Fregean proper name has a sense determining an object, instead of a concept [Sainsbury]


p.472

10533

We can't get a semantics from nouns and predicates referring to the same thing [Dummett]

Pref

p.8

7304

Frege explained meaning as sense, semantic value, reference, force and tone [Miller,A]

note

p.79

18773

People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander'

p.27

p.57

4976

The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense

p.28

p.58

4977

In maths, there are phrases with a clear sense, but no actual reference

p.30

p.60

4978

The meaning of a proper name is the designated object

p.33

p.63

4979

We are driven from sense to reference by our desire for truth

p.34

p.63

4980

The meaning (reference) of a sentence is its truth value  the circumstance of it being true or false

p.35

p.65

4981

The reference of a word should be understood as part of the reference of the sentence

p.40

p.69

18940

It is a weakness of natural languages to contain nondenoting names

p.41

p.70

18939

In a logically perfect language every wellformed proper name designates an object

1893

Grundgesetze der Arithmetik 1 (Basic Laws)


p.3

10623

Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Hale/Wright]


p.44

9975

Frege ignored Cantor's warning that a cardinal set is not just a conceptextension [Tait]


p.147

9190

A concept is a function mapping objects onto truthvalues, if they fall under the concept [Dummett]


p.177

13665

Frege took the study of concepts to be part of logic [Shapiro]


p.250

13733

Frege considered definite descriptions to be genuine singular terms [Fitting/Mendelsohn]

§25

p.217

9874

Contradiction arises from Frege's substitutional account of secondorder quantification [Dummett]

III.1.73

p.269

18252

Real numbers are ratios of quantities, such as lengths or masses

p.2

p.122

18271

We can't prove everything, but we can spell out the unproved, so that foundations are clear

p.4

p.6

18165

My Basic Law V is a law of pure logic

1894

Review of Husserl's 'Phil of Arithmetic'


p.32

9821

A definition need not capture the sense of an expression  just get the reference right [Dummett]


p.193

17446

Counting rests on oneone correspondence, of numerals to objects

p.323

p.323

9577

The naïve view of number is that it is like a heap of things, or maybe a property of a heap

p.324

p.324

9580

Our concepts recognise existing relations, they don't change them

p.324

p.324

9578

If objects are just presentation, we get increasing abstraction by ignoring their properties

p.324

p.324

9579

Disregarding properties of two cats still leaves different objects, but what is now the difference?

p.325

p.325

9581

Many people have the same thought, which is the component, not the private presentation

p.326

p.326

9583

Psychological logicians are concerned with sense of words, but mathematicians study the reference

p.326

p.326

9582

Husserl rests sameness of number on oneone correlation, forgetting the correlation with numbers themselves

p.327

p.327

9585

Since every definition is an equation, one cannot define equality itself

p.327

p.327

9584

Identity baffles psychologists, since A and B must be presented differently to identify them

p.328

p.328

9586

In a numberstatement, something is predicated of a concept

p.330

p.330

9587

How do you find the right level of inattention; you eliminate too many or too few characteristics

p.332

p.332

9588

Numberabstraction somehow makes things identical without changing them!

p.337

p.337

9589

Numbers are not real like the sea, but (crucially) they are still objective

1895

Elucidation of some points in E.Schröder

p.212

p.126

14238

A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class


p.147

11052

Psychological logic can't distinguish justification from causes of a belief

1900

On Euclidean Geometry

183/168

p.348

16886

The truth of an axiom must be independently recognisable

1902.06.22

p.127

18166

The loss of my Rule V seems to make foundations for arithmetic impossible

1902.07.28

p.113

18269

Logical objects are extensions of concepts, or ranges of values of functions

1903.05.21

p.270

18253

I wish to go straight from cardinals to reals (as ratios), leaving out the rationals

1903

Grundgesetze der Arithmetik 2 (Basic Laws)


p.109

13886

Later Frege held that definitions must fix a function's value for every possible argument [Wright,C]


p.139

10020

Frege's biggest error is in not accounting for the senses of number terms [Hodes]


p.261

9889

Real numbers are ratios of quantities [Dummett]


p.510

10553

A number is a class of classes of the same cardinality [Dummett]

§157

p.246

9886

Cardinals say how many, and reals give measurements compared to a unit quantity

§159

p.262

9890

The modern account of real numbers detaches a ratio from its geometrical origins

§160

p.277

9891

The first demand of logic is of a sharp boundary

§180

p.137

10019

Only what is logically complex can be defined; what is simple must be pointed to

§66

p.268

9845

We can't define a word by defining an expression containing it, as the remaining parts are a problem

§86137

p.252

9887

Formalism misunderstands applications, metatheory, and infinity [Dummett]

§91

p.147

8751

Only applicability raises arithmetic from a game to a science

§99

p.174

11846

If we abstract the difference between two houses, they don't become the same house

p.43

p.43

8447

In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference

p.43

p.43

8446

We understand new propositions by constructing their sense from the words

p.44

p.44

8449

Senses can't be subjective, because propositions would be private, and disagreement impossible

p.44

p.44

8448

Any object can have many different names, each with a distinct sense

1914

Logic in Mathematics


p.4

11219

Frege suggested that mathematics should only accept stipulative definitions [Gupta]

p.203

p.203

16862

The closest subject to logic is mathematics, which does little apart from drawing inferences

p.203

p.203

16863

Does some mathematical reasoning (such as mathematical induction) not belong to logic?

p.203

p.203

16864

If principles are provable, they are theorems; if not, they are axioms

p.204

p.204

16866

Tracing inference backwards closes in on a small set of axioms and postulates

p.204

p.204

16865

'Theorems' are both proved, and used in proofs

p.204

p.204

16867

Logic not only proves things, but also reveals logical relations between them

p.2045

p.204

16868

The essence of mathematics is the kernel of primitive truths on which it rests

p.205

p.205

16871

A truth can be an axiom in one system and not in another

p.205

p.205

16870

Axioms are truths which cannot be doubted, and for which no proof is needed

p.205

p.205

16869

To create order in mathematics we need a full system, guided by patterns of inference

p.206

p.206

16873

Thoughts are not subjective or psychological, because some thoughts are the same for us all

p.206

p.206

16872

A thought is the sense expressed by a sentence, and is what we prove

p.207

p.207

16874

The parts of a thought map onto the parts of a sentence

p.209

p.209

16876

We need definitions to cram retrievable sense into a signed receptacle

p.209

p.209

16875

We use signs to mark receptacles for complex senses

p.210

p.210

16877

A 'constructive' (as opposed to 'analytic') definition creates a new sign

p.212

p.212

16878

We must be clear about every premise and every law used in a proof

p.213

p.213

16879

A sign won't gain sense just from being used in sentences with familiar components

p.229

p.

9388

Every concept must have a sharp boundary; we cannot allow an indeterminate third case

1918

The Thought: a Logical Enquiry


p.5

8162

Thoughts have their own realm of reality  'sense' (as opposed to the realm of 'reference') [Dummett]


p.15

9818

A thought is distinguished from other things by a capacity to be true or false [Dummett]


p.129

7740

There exists a realm, beyond objects and ideas, of nonspatiotemporal thoughts [Weiner]


p.209

16379

Thoughts about myself are understood one way to me, and another when communicated


p.225

9877

Late Frege saw his nonactual objective objects as exclusively thoughts and senses [Dummett]

p.327 (60)

p.327

19465

There cannot be complete correspondence, because ideas and reality are quite different

p.327 (60)

p.327

19466

The word 'true' seems to be unique and indefinable

p.3278 (61)

p.328

19467

A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences

p.328 (61)

p.328

19468

The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets'

p.329 (62)

p.329

19469

We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion)

p.337(69)

p.337

19470

Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist

p.342(74)

p.342

19471

A fact is a thought that is true

p.343(76)

p.343

19472

A sentence is only a thought if it is complete, and has a timespecification

1922

Sources of Knowledge of Mathematics


p.3

9545

Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical [Chihara]
