Ideas of Gottlob Frege, by Theme

[German, 1848 - 1925, Led a quiet and studious life as Professor at the University of Jena.]

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1. Philosophy / E. Nature of Metaphysics / 5. Metaphysics as Conceptual
Syntactic category which is primary and ontological category derivative.
1. Philosophy / F. Analytic Philosophy / 1. Analysis
Frege initiated linguistic philosophy, studying number through the sense of sentences
Frege developed formal systems to avoid unnoticed assumptions
1. Philosophy / F. Analytic Philosophy / 2. Conceptual Analysis
Never lose sight of the distinction between concept and object
2. Reason / A. Nature of Reason / 5. Objectivity
There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts
Frege sees no 'intersubjective' category, between objective and subjective
Keep the psychological and subjective separate from the logical and objective
2. Reason / B. Laws of Thought / 1. Laws of Thought
We should not describe human laws of thought, but how to correctly track truth
2. Reason / D. Definition / 2. Aims of Definition
A definition need not capture the sense of an expression - just get the reference right
Later Frege held that definitions must fix a function's value for every possible argument
2. Reason / D. Definition / 3. Types of Definition
A 'constructive' (as opposed to 'analytic') definition creates a new sign
2. Reason / D. Definition / 7. Contextual Definition
Nothing should be defined in terms of that to which it is conceptually prior
We can't define a word by defining an expression containing it, as the remaining parts are a problem
Originally Frege liked contextual definitions, but later preferred them fully explicit
2. Reason / D. Definition / 10. Stipulative Definition
Frege suggested that mathematics should only accept stipulative definitions
2. Reason / D. Definition / 11. Ostensive Definition
Only what is logically complex can be defined; what is simple must be pointed to
2. Reason / E. Argument / 6. Conclusive Proof
Proof reveals the interdependence of truths, as well as showing their certainty
Proof aims to remove doubts, but also to show the interdependence of truths
We must be clear about every premise and every law used in a proof
2. Reason / F. Fallacies / 8. Category Mistake / a. Category mistakes
You can't transfer external properties unchanged to apply to ideas
3. Truth / A. Truth Problems / 2. Defining Truth
The word 'true' seems to be unique and indefinable
3. Truth / A. Truth Problems / 5. Truth Bearers
Frege was strongly in favour of taking truth to attach to propositions
3. Truth / B. Truthmakers / 5. What Makes Truths / c. States of affairs make truths
We need to grasp not number-objects, but the states of affairs which make number statements true
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
There cannot be complete correspondence, because ideas and reality are quite different
3. Truth / H. Deflationary Truth / 1. Redundant Truth
The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets'
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence
Since every definition is an equation, one cannot define equality itself
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
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Frege did not think of himself as working with sets
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The null set is indefensible, because it collects nothing
The null set is only defensible if it is the extension of an empty concept
It is because a concept can be empty that there is such a thing as the empty class
A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
We can introduce new objects, as equivalence classes of objects already known
Frege introduced the standard device, of defining logical objects with equivalence classes
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
A class is, for Frege, the extension of a concept
Frege proposed a realist concept of a set, as the extension of a predicate or concept or function
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)
The laws of logic are boundless, so we want the few whose power contains the others
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
In 1879 Frege developed second order logic
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Frege frequently expressed a contempt for language
Logic not only proves things, but also reveals logical relations between them
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
Does some mathematical reasoning (such as mathematical induction) not belong to logic?
The closest subject to logic is mathematics, which does little apart from drawing inferences
5. Theory of Logic / C. Ontology of Logic / 2. Platonism in Logic
Frege thinks there is an independent logical order of the truths, which we must try to discover
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
I don't use 'subject' and 'predicate' in my way of representing a judgement
Frege replaced Aristotle's subject/predicate form with function/argument form
Convert "Jupiter has four moons" into "the number of Jupiter's moons is four"
A thought can be split in many ways, so that different parts appear as subject or predicate
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
First-level functions have objects as arguments; second-level functions take functions as arguments
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
Relations are functions with two arguments
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
For Frege, predicates are names of functions that map objects onto the True and False
Frege gives a functional account of predication so that we can dispense with predicates
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
'Theorems' are both proved, and used in proofs
Despite Gödel, Frege's epistemic ordering of all the truths is still plausible
The primitive simples of arithmetic are the essence, determining the subject, and its boundaries
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
We can treat designation by a few words as a proper name
In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference
5. Theory of Logic / F. Referring in Logic / 1. Naming / b. Names as descriptive
Proper name in modal contexts refer obliquely, to their usual sense
A Fregean proper name has a sense determining an object, instead of a concept
People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander'
Any object can have many different names, each with a distinct sense
5. Theory of Logic / F. Referring in Logic / 1. Naming / c. Names as referential
The meaning of a proper name is the designated object
5. Theory of Logic / F. Referring in Logic / 1. Naming / d. Singular terms
Frege ascribes reference to incomplete expressions, as well as to singular terms
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
It is a weakness of natural languages to contain non-denoting names
In a logically perfect language every well-formed proper name designates an object
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
Frege considered definite descriptions to be genuine singular terms
5. Theory of Logic / G. Quantification / 1. Quantification
A quantifier is a second-level predicate (which explains how it contributes to truth-conditions)
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
For Frege the variable ranges over all objects
Frege always, and fatally, neglected the domain of quantification
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
Frege introduced quantifiers for generality
Frege reduced most quantifiers to 'everything' combined with 'not'
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Contradiction arises from Frege's substitutional account of second-order quantification
5. Theory of Logic / G. Quantification / 6. Plural Quantification
Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
Proof theory began with Frege's definition of derivability
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
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Basic truths of logic are not proved, but seen as true when they are understood
5. Theory of Logic / I. Semantics of Logic / 6. Intensionalism
Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first
Frege moved from extensional to intensional semantics when he added the idea of 'sense'
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
To understand axioms you must grasp their logical power and priority
Tracing inference backwards closes in on a small set of axioms and postulates
The essence of mathematics is the kernel of primitive truths on which it rests
A truth can be an axiom in one system and not in another
Axioms are truths which cannot be doubted, and for which no proof is needed
The truth of an axiom must be independently recognisable
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
To create order in mathematics we need a full system, guided by patterns of inference
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
Cardinals say how many, and reals give measurements compared to a unit quantity
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Quantity is inconceivable without the idea of addition
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
We cannot define numbers from the idea of a series, because numbers must precede that
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
I wish to go straight from cardinals to reals (as ratios), leaving out the rationals
Real numbers are ratios of quantities
Real numbers are ratios of quantities, such as lengths or masses
6. Mathematics / A. Nature of Mathematics / 3. Numbers / l. Zero
Treating 0 as a number avoids antinomies involving treating 'nobody' as a person
For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined
If objects exist because they fall under a concept, 0 is the object under which no objects fall
Nought is the number belonging to the concept 'not identical with itself'
6. Mathematics / A. Nature of Mathematics / 3. Numbers / m. One
We can say 'a and b are F' if F is 'wise', but not if it is 'one'
One is the Number which belongs to the concept "identical with 0"
6. Mathematics / A. Nature of Mathematics / 3. Numbers / o. Units
You can abstract concepts from the moon, but the number one is not among them
Units can be equal without being identical
A concept creating a unit must isolate and unify what falls under it
Frege says only concepts which isolate and avoid arbitrary division can give units
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted
Non-arbitrary division means that what falls under the concept cannot be divided into more of the same
Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage
Frege says counting is determining what number belongs to a given concept
Counting rests on one-one correspondence, of numerals to objects
Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
The number of natural numbers is not a natural number
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
We can't prove everything, but we can spell out the unproved, so that foundations are clear
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
Arithmetical statements can't be axioms, because they are provable
If principles are provable, they are theorems; if not, they are axioms
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
Frege claims that numbers are objects, as opposed to them being Fregean concepts
Numbers are second-level, ascribing properties to concepts rather than to objects
For Frege, successor was a relation, not a function
Numbers are more than just 'second-level concepts', since existence is also one
"Number of x's such that ..x.." is a functional expression, yielding a name when completed
Frege gives an incoherent account of extensions resulting from abstraction
For Frege the number of F's is a collection of first-level concepts
Numbers need to be objects, to define the extension of the concept of each successor to n
The number of F's is the extension of the second level concept 'is equipollent with F'
Individual numbers are best derived from the number one, and increase by one
A statement of number contains a predication about a concept
'Exactly ten gallons' may not mean ten things instantiate 'gallon'
Each number, except 0, is the number of the concept of all of its predecessors
A cardinal number may be defined as a class of similar classes
Frege's account of cardinals fails in modern set theory, so they are now defined differently
Frege's incorrect view is that a number is an equivalence class
Frege had a motive to treat numbers as objects, but not a justification
Frege showed that numbers attach to concepts, not to objects
Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts
Zero is defined using 'is not self-identical', and one by using the concept of zero
Frege started with contextual definition, but then switched to explicit extensional definition
The natural number n is the set of n-membered sets
A set doesn't have a fixed number, because the elements can be seen in different ways
Numbers are objects because they partake in identity statements
In a number-statement, something is predicated of a concept
A number is a class of classes of the same cardinality
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
There is the concept, the object falling under it, and the extension (a set, which is also an object)
Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions
Frege ignored Cantor's warning that a cardinal set is not just a concept-extension
Frege's biggest error is in not accounting for the senses of number terms
If you can subdivide objects many ways for counting, you can do that to set-elements too
Frege's problem is explaining the particularity of numbers by general laws
Numerical statements have first-order logical form, so must refer to objects
The Number for F is the extension of 'equal to F' (or maybe just F itself)
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle
Frege thinks number is fundamentally bound up with one-one correspondence
'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F'
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
The words 'There are exactly Julius Caesar moons of Mars' are gibberish
Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1
'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor
From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number?
Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension)
Our definition will not tell us whether or not Julius Caesar is a number
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
If numbers can be derived from logic, then set theory is superfluous
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
If numbers are supposed to be patterns, each number can have many patterns
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Numbers are objects, because they can take the definite article, and can't be plurals
Our concepts recognise existing relations, they don't change them
Numbers are not real like the sea, but (crucially) they are still objective
Numbers seem to be objects because they exactly fit the inference patterns for identities
Frege's platonism proposes that objects are what singular terms refer to
How can numbers be external (one pair of boots is two boots), or subjective (and so relative)?
Identities refer to objects, so numbers must be objects
Numbers are not physical, and not ideas - they are objective and non-sensible
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Geometry appeals to intuition as the source of its axioms
Frege's logicism aimed at removing the reliance of arithmetic on intuition
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
There is no physical difference between two boots and one pair of boots
The naďve view of number is that it is like a heap of things, or maybe a property of a heap
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
It appears that numbers are adjectives, but they don't apply to a single object
Numerical adjectives are of the same second-level type as the existential quantifier
'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many'
The number 'one' can't be a property, if any object can be viewed as one or not one
For science, we can translate adjectival numbers into noun form
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
My Basic Law V is a law of pure logic
Arithmetic must be based on logic, because of its total generality
Arithmetic is analytic
Logicism shows that no empirical truths are needed to justify arithmetic
Eventually Frege tried to found arithmetic in geometry instead of in logic
Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism
Frege offered a Platonist version of logicism, committed to cardinal and real numbers
Numbers are definable in terms of mapping items which fall under concepts
Arithmetic is analytic and a priori, and thus it is part of logic
Frege aimed to discover the logical foundations which justify arithmetical judgements
The loss of my Rule V seems to make foundations for arithmetic impossible
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Frege's logic has a hierarchy of object, property, property-of-property etc.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Why should the existence of pure logic entail the existence of objects?
Frege only managed to prove that arithmetic was analytic with a logic that included set-theory
Frege's belief in logicism and in numerical objects seem uncomfortable together
Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism fails to recognise types of symbols, and also meta-games
Formalism misunderstands applications, metatheory, and infinity
Only applicability raises arithmetic from a game to a science
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Abstraction from things produces concepts, and numbers are in the concepts
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
Mental states are irrelevant to mathematics, because they are vague and fluctuating
7. Existence / A. Nature of Existence / 1. Nature of Existence
Existence is not a first-order property, but the instantiation of a property
Affirmation of existence is just denial of zero
7. Existence / A. Nature of Existence / 2. Types of Existence
Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist
7. Existence / A. Nature of Existence / 3. Being / i. Deflating being
Frege's logic showed that there is no concept of being
7. Existence / A. Nature of Existence / 6. Abstract Existence
The equator is imaginary, but not fictitious; thought is needed to recognise it
7. Existence / A. Nature of Existence / 8. Criterion for Existence
Frege mistakenly takes existence to be a property of concepts, instead of being about things
7. Existence / C. Structure of Existence / 4. Ontological Dependence
Many of us find Frege's claim that truths depend on one another an obscure idea
Parallelism is intuitive, so it is more fundamental than sameness of direction
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
Frege refers to 'concrete' objects, but they are no different in principle from abstract ones
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / b. Levels of abstraction
If objects are just presentation, we get increasing abstraction by ignoring their properties
7. Existence / D. Theories of Reality / 7. Facts / c. Facts and truths
A fact is a thought that is true
7. Existence / D. Theories of Reality / 9. Vagueness / c. Vagueness as semantic
Vagueness is incomplete definition
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
For Frege, ontological questions are to be settled by reference to syntactic structures
7. Existence / D. Theories of Reality / 10. Ontological Commitment / c. Commitment of predicates
Second-order quantifiers are committed to concepts, as first-order commits to objects
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
'Ancestral' relations are derived by iterating back from a given relation
8. Modes of Existence / B. Properties / 1. Nature of Properties
Frege treats properties as a kind of function, and maybe a property is its characteristic function
8. Modes of Existence / B. Properties / 10. Properties as Predicates
Frege allows either too few properties (as extensions) or too many (as predicates)
It is unclear whether Frege included qualities among his abstract objects
8. Modes of Existence / D. Universals / 1. Universals
We can't get a semantics from nouns and predicates referring to the same thing
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
Frege establishes abstract objects independently from concrete ones, by falling under a concept
Logical objects are extensions of concepts, or ranges of values of functions
Frege says singular terms denote objects, numerals are singular terms, so numbers exist
9. Objects / A. Existence of Objects / 3. Objects in Thought
For Frege, objects just are what singular terms refer to
Without concepts we would not have any objects
The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place
Frege's 'objects' are both the referents of proper names, and what predicates are true or false of
Late Frege saw his non-actual objective objects as exclusively thoughts and senses
9. Objects / A. Existence of Objects / 4. Individuation / a. Individuation
Frege's universe comes already divided into objects
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
The first demand of logic is of a sharp boundary
Every concept must have a sharp boundary; we cannot allow an indeterminate third case
9. Objects / F. Identity among Objects / 1. Concept of Identity
The idea of a criterion of identity was introduced by Frege
Frege's algorithm of identity is the law of putting equals for equals
Frege was asking how identities could be informative
9. Objects / F. Identity among Objects / 5. Self-Identity
Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x'
9. Objects / F. Identity among Objects / 6. Identity between Objects
Identity between objects is not a consequence of identity, but part of what 'identity' means
11. Knowledge Aims / A. Knowledge / 2. Understanding
To understand a thought you must understand its logical structure
To understand a thought, understand its inferential connections to other thoughts
12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
For Frege a priori knowledge derives from general principles, so numbers can't be primitive
12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence
Frege's concept of 'self-evident' makes no reference to minds
Mathematicians just accept self-evidence, whether it is logical or intuitive
12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities
An apriori truth is grounded in generality, which is universal quantification
An a priori truth is one derived from general laws which do not require proof
A truth is a priori if it can be proved entirely from general unproven laws
12. Knowledge Sources / E. Direct Knowledge / 1. Intuition
Intuitions cannot be communicated
13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / d. Rational foundations
Justifications show the ordering of truths, and the foundation is what is self-evident
13. Knowledge Criteria / C. External Justification / 2. Causal Justification
Psychological logic can't distinguish justification from causes of a belief
14. Science / B. Scientific Theories / 1. Scientific Theory
The building blocks contain the whole contents of a discipline
14. Science / C. Induction / 1. Induction
Induction is merely psychological, with a principle that it can actually establish laws
In science one observation can create high probability, while a thousand might prove nothing
15. Nature of Minds / A. Nature of Mind / 1. Mind / c. Features of mind
Ideas are not spatial, and don't have distances between them
18. Thought / A. Modes of Thought / 1. Thought
Thought is the same everywhere, and the laws of thought do not vary
We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion)
Many people have the same thought, which is the component, not the private presentation
Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference')
A thought is distinguished from other things by a capacity to be true or false
18. Thought / A. Modes of Thought / 9. Indexical Thought
Thoughts about myself are understood one way to me, and another when communicated
18. Thought / B. Mechanics of Thought / 5. Mental Files
We use signs to mark receptacles for complex senses
We need definitions to cram retrievable sense into a signed receptacle
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Early Frege takes the extensions of concepts for granted
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
A concept is a function mapping objects onto truth-values, if they fall under the concept
Frege took the study of concepts to be part of logic
We don't judge by combining subject and concept; we get a concept by splitting up a judgement
Concepts are, precisely, the references of predicates
A concept is a non-psychological one-place function asserting something of an object
Fregean concepts have precise boundaries and universal applicability
Psychological accounts of concepts are subjective, and ultimately destroy truth
Concepts are the ontological counterparts of predicative expressions
An assertion about the concept 'horse' must indirectly speak of an object
A concept is a function whose value is always a truth-value
'The concept "horse"' denotes a concept, yet seems also to denote an object
Frege equated the concepts under which an object falls with its properties
18. Thought / D. Concepts / 4. Structure of Concepts / a. Conceptual structure
Unlike objects, concepts are inherently incomplete
18. Thought / D. Concepts / 5. Concepts and Language / b. Concepts are linguistic
A concept is a possible predicate of a singular judgement
As I understand it, a concept is the meaning of a grammatical predicate
18. Thought / E. Abstraction / 1. Abstract Thought
Defining 'direction' by parallelism doesn't tell you whether direction is a line
18. Thought / E. Abstraction / 2. Abstracta by Selection
Frege accepts abstraction to the concept of all sets equipollent to a given one
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
Frege himself abstracts away from tone and color
If we abstract 'from' two cats, the units are not black or white, or cats
Disregarding properties of two cats still leaves different objects, but what is now the difference?
How do you find the right level of inattention; you eliminate too many or too few characteristics
The modern account of real numbers detaches a ratio from its geometrical origins
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects
Fregean abstraction creates concepts which are equivalences between initial items
Frege put the idea of abstraction on a rigorous footing
We create new abstract concepts by carving up the content in a different way
You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements
18. Thought / E. Abstraction / 8. Abstractionism Critique
Frege said concepts were abstract entities, not mental entities
Number-abstraction somehow makes things identical without changing them!
If we abstract the difference between two houses, they don't become the same house
19. Language / A. Nature of Meaning / 2. Meaning as Mental
Frege felt that meanings must be public, so they are abstractions rather than mental entities
Psychological logicians are concerned with sense of words, but mathematicians study the reference
Identity baffles psychologists, since A and B must be presented differently to identify them
19. Language / A. Nature of Meaning / 4. Meaning as Truth-Conditions
A thought is not psychological, but a condition of the world that makes a sentence true
The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false
19. Language / A. Nature of Meaning / 6. Meaning as Use
A sign won't gain sense just from being used in sentences with familiar components
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Words in isolation seem to have ideas as meanings, but words have meaning in propositions
Never ask for the meaning of a word in isolation, but only in the context of a proposition
We understand new propositions by constructing their sense from the words
19. Language / B. Assigning Meanings / 5. Fregean Semantics
Earlier Frege focuses on content itself; later he became interested in understanding content
Frege divided the meaning of a sentence into sense, force and tone
Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined
Frege explained meaning as sense, semantic value, reference, force and tone
Frege's 'sense' is the strict and literal meaning, stripped of tone
'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness
'Sense' gives meaning to non-referring names, and to two expressions for one referent
Frege was the first to construct a plausible theory of meaning
19. Language / C. Reference / 1. Reference theories
The reference of a word should be understood as part of the reference of the sentence
19. Language / C. Reference / 4. Descriptive Reference / a. Sense and reference
Frege started as anti-realist, but the sense/reference distinction led him to realism
The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense
In maths, there are phrases with a clear sense, but no actual reference
We are driven from sense to reference by our desire for truth
Senses can't be subjective, because propositions would be private, and disagreement impossible
Every descriptive name has a sense, but may not have a reference
19. Language / C. Reference / 4. Descriptive Reference / b. Reference by description
Expressions always give ways of thinking of referents, rather than the referents themselves
19. Language / C. Reference / 5. Speaker's Reference
I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false
19. Language / D. Propositions / 2. Abstract Propositions / a. Propositions as sense
For all the multiplicity of languages, mankind has a common stock of thoughts
Thoughts are not subjective or psychological, because some thoughts are the same for us all
A thought is the sense expressed by a sentence, and is what we prove
A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences
19. Language / D. Propositions / 5. Unity of Propositions
A sentence is only a thought if it is complete, and has a time-specification
The parts of a thought map onto the parts of a sentence
19. Language / E. Analyticity / 1. Analytic Propositions
A statement is analytic if substitution of synonyms can make it a logical truth
Frege considered analyticity to be an epistemic concept
'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate
19. Language / E. Analyticity / 2. Analytic Truths
All analytic truths can become logical truths, by substituting definitions or synonyms
Analytic truths are those that can be demonstrated using only logic and definitions
25. Society / E. State Functions / 5. Education / a. Education principles
To learn something, you must know that you don't know
26. Natural Theory / D. Laws of Nature / 6. Laws as Numerical
The laws of number are not laws of nature, but are laws of the laws of nature
28. God / C. Proofs of Reason / 1. Ontological Proof
Frege put forward an ontological argument for the existence of numbers
28. God / C. Proofs of Reason / 2. Ontological Proof critique
The predicate 'exists' is actually a natural language expression for a quantifier
The Ontological Argument fallaciously treats existence as a first-level concept
Because existence is a property of concepts the ontological argument for God fails