13876 | The syntactic category is primary, and the ontological category is derivative |

15948 | Frege developed formal systems to avoid unnoticed assumptions |

9840 | Frege initiated linguistic philosophy, studying number through the sense of sentences |

8415 | Never lose sight of the distinction between concept and object |

9832 | Frege sees no 'intersubjective' category, between objective and subjective |

8414 | Keep the psychological and subjective separate from the logical and objective |

7740 | There exists a realm, beyond objects and ideas, of non-spatio-temporal thoughts |

8939 | We should not describe human laws of thought, but how to correctly track truth |

9821 | A definition need not capture the sense of an expression - just get the reference right |

13886 | Later Frege held that definitions must fix a function's value for every possible argument |

16877 | A 'constructive' (as opposed to 'analytic') definition creates a new sign |

9822 | Nothing should be defined in terms of that to which it is conceptually prior |

9845 | We can't define a word by defining an expression containing it, as the remaining parts are a problem |

9844 | Originally Frege liked contextual definitions, but later preferred them fully explicit |

11219 | Frege suggested that mathematics should only accept stipulative definitions |

10019 | Only what is logically complex can be defined; what is simple must be pointed to |

8623 | Proof reveals the interdependence of truths, as well as showing their certainty |

17495 | Proof aims to remove doubts, but also to show the interdependence of truths |

16878 | We must be clear about every premise and every law used in a proof |

8632 | You can't transfer external properties unchanged to apply to ideas |

19466 | The word 'true' seems to be unique and indefinable |

8187 | Frege was strongly in favour of taking truth to attach to propositions |

13881 | We need to grasp not number-objects, but the states of affairs which make number statements true |

19465 | There cannot be complete correspondence, because ideas and reality are quite different |

19468 | The property of truth in 'It is true that I smell violets' adds nothing to 'I smell violets' |

9154 | Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence |

9585 | Since every definition is an equation, one cannot define equality itself |

17745 | For Frege, 'All A's are B's' means that the concept A implies the concept B |

13455 | Frege did not think of himself as working with sets |

9157 | The null set is only defensible if it is the extension of an empty concept |

16895 | The null set is indefensible, because it collects nothing |

9835 | It is because a concept can be empty that there is such a thing as the empty class |

14238 | A class is an aggregate of objects; if you destroy them, you destroy the class; there is no empty class |

9854 | We can introduce new objects, as equivalence classes of objects already known |

9883 | Frege introduced the standard device, of defining logical objects with equivalence classes |

9834 | A class is, for Frege, the extension of a concept |

3328 | Frege proposed a realist concept of a set, as the extension of a predicate or concept or function |

7728 | Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) |

16881 | The laws of logic are boundless, so we want the few whose power contains the others |

7622 | In 1879 Frege developed second order logic |

9179 | Frege frequently expressed a contempt for language |

16867 | Logic not only proves things, but also reveals logical relations between them |

16863 | Does some mathematical reasoning (such as mathematical induction) not belong to logic? |

16862 | The closest subject to logic is mathematics, which does little apart from drawing inferences |

13473 | Frege thinks there is an independent logical order of the truths, which we must try to discover |

7729 | Frege replaced Aristotle's subject/predicate form with function/argument form |

4971 | I don't use 'subject' and 'predicate' in my way of representing a judgement |

4975 | A thought can be split in many ways, so that different parts appear as subject or predicate |

8645 | Convert "Jupiter has four moons" into "the number of Jupiter's moons is four" |

8490 | First-level functions have objects as arguments; second-level functions take functions as arguments |

8492 | Relations are functions with two arguments |

3319 | Frege gives a functional account of predication so that we can dispense with predicates |

6076 | For Frege, predicates are names of functions that map objects onto the True and False |

16865 | 'Theorems' are both proved, and used in proofs |

16891 | Despite Gödel, Frege's epistemic ordering of all the truths is still plausible |

16906 | The primitive simples of arithmetic are the essence, determining the subject, and its boundaries |

18772 | We can treat designation by a few words as a proper name |

8447 | In 'Etna is higher than Vesuvius' the whole of Etna, including all the lava, can't be the reference |

14075 | Proper name in modal contexts refer obliquely, to their usual sense |

10424 | A Fregean proper name has a sense determining an object, instead of a concept |

18773 | People may have different senses for 'Aristotle', like 'pupil of Plato' or 'teacher of Alexander' |

8448 | Any object can have many different names, each with a distinct sense |

4978 | The meaning of a proper name is the designated object |

10510 | Frege ascribes reference to incomplete expressions, as well as to singular terms |

18940 | It is a weakness of natural languages to contain non-denoting names |

18939 | In a logically perfect language every well-formed proper name designates an object |

13733 | Frege considered definite descriptions to be genuine singular terms |

9950 | A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) |

9991 | For Frege the variable ranges over all objects |

9871 | Frege always, and fatally, neglected the domain of quantification |

7730 | Frege introduced quantifiers for generality |

7742 | Frege reduced most quantifiers to 'everything' combined with 'not' |

9874 | Contradiction arises from Frege's substitutional account of second-order quantification |

14236 | Each horse doesn't fall under the concept 'horse that draws the carriage', because all four are needed |

13824 | Proof theory began with Frege's definition of derivability |

13609 | Frege produced axioms for logic, though that does not now seem the natural basis for logic |

16884 | Basic truths of logic are not proved, but seen as true when they are understood |

9462 | Frege is intensionalist about reference, as it is determined by sense; identity of objects comes first |

18936 | Frege moved from extensional to intensional semantics when he added the idea of 'sense' |

17624 | To understand axioms you must grasp their logical power and priority |

16886 | The truth of an axiom must be independently recognisable |

16866 | Tracing inference backwards closes in on a small set of axioms and postulates |

16868 | The essence of mathematics is the kernel of primitive truths on which it rests |

16870 | Axioms are truths which cannot be doubted, and for which no proof is needed |

16871 | A truth can be an axiom in one system and not in another |

16869 | To create order in mathematics we need a full system, guided by patterns of inference |

9886 | Cardinals say how many, and reals give measurements compared to a unit quantity |

18256 | Quantity is inconceivable without the idea of addition |

8640 | We cannot define numbers from the idea of a series, because numbers must precede that |

18252 | Real numbers are ratios of quantities, such as lengths or masses |

18253 | I wish to go straight from cardinals to reals (as ratios), leaving out the rationals |

9889 | Real numbers are ratios of quantities |

9838 | Treating 0 as a number avoids antinomies involving treating 'nobody' as a person |

9564 | For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined |

10551 | If objects exist because they fall under a concept, 0 is the object under which no objects fall |

8653 | Nought is the number belonging to the concept 'not identical with itself' |

8654 | One is the Number which belongs to the concept "identical with 0" |

8636 | We can say 'a and b are F' if F is 'wise', but not if it is 'one' |

8641 | You can abstract concepts from the moon, but the number one is not among them |

9989 | Units can be equal without being identical |

17426 | A concept creating a unit must isolate and unify what falls under it |

17429 | Frege says only concepts which isolate and avoid arbitrary division can give units |

15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted |

17437 | Non-arbitrary division means that what falls under the concept cannot be divided into more of the same |

17438 | Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage |

17428 | Frege says counting is determining what number belongs to a given concept |

17446 | Counting rests on one-one correspondence, of numerals to objects |

9582 | Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves |

10034 | The number of natural numbers is not a natural number |

18271 | We can't prove everything, but we can spell out the unproved, so that foundations are clear |

16883 | Arithmetical statements can't be axioms, because they are provable |

16864 | If principles are provable, they are theorems; if not, they are axioms |

10013 | Numerical statements have first-order logical form, so must refer to objects |

18181 | The Number for F is the extension of 'equal to F' (or maybe just F itself) |

18103 | Numbers are objects because they partake in identity statements |

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 |

9949 | There is the concept, the object falling under it, and the extension (a set, which is also an object) |

10623 | Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions |

9975 | Frege ignored Cantor's warning that a cardinal set is not just a concept-extension |

9586 | In a number-statement, something is predicated of a concept |

10553 | A number is a class of classes of the same cardinality |

10020 | Frege's biggest error is in not accounting for the senses of number terms |

10625 | Frege had a motive to treat numbers as objects, but not a justification |

13871 | Frege claims that numbers are objects, as opposed to them being Fregean concepts |

13872 | Numbers are second-level, ascribing properties to concepts rather than to objects |

9816 | For Frege, successor was a relation, not a function |

9953 | Numbers are more than just 'second-level concepts', since existence is also one |

9954 | "Number of x's such that ..x.." is a functional expression, yielding a name when completed |

17636 | A cardinal number may be defined as a class of similar classes |

10139 | Frege gives an incoherent account of extensions resulting from abstraction |

10028 | For Frege the number of F's is a collection of first-level concepts |

10029 | Numbers need to be objects, to define the extension of the concept of each successor to n |

9973 | The number of F's is the extension of the second level concept 'is equipollent with F' |

16500 | Frege showed that numbers attach to concepts, not to objects |

9990 | Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts |

7738 | Zero is defined using 'is not self-identical', and one by using the concept of zero |

13887 | Frege started with contextual definition, but then switched to explicit extensional definition |

13897 | Each number, except 0, is the number of the concept of all of its predecessors |

9856 | Frege's account of cardinals fails in modern set theory, so they are now defined differently |

9902 | Frege's incorrect view is that a number is an equivalence class |

17820 | If you can subdivide objects many ways for counting, you can do that to set-elements too |

17814 | The natural number n is the set of n-membered sets |

17819 | A set doesn't have a fixed number, because the elements can be seen in different ways |

17460 | A statement of number contains a predication about a concept |

16890 | Frege's problem is explaining the particularity of numbers by general laws |

8630 | Individual numbers are best derived from the number one, and increase by one |

11029 | 'Exactly ten gallons' may not mean ten things instantiate 'gallon' |

9956 | 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' |

17442 | Frege thinks number is fundamentally bound up with one-one correspondence |

11030 | The words 'There are exactly Julius Caesar moons of Mars' are gibberish |

10030 | 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor |

8690 | From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? |

10219 | Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) |

13889 | Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 |

9046 | Our definition will not tell us whether or not Julius Caesar is a number |

16896 | If numbers can be derived from logic, then set theory is superfluous |

8639 | If numbers are supposed to be patterns, each number can have many patterns |

13874 | Numbers seem to be objects because they exactly fit the inference patterns for identities |

13875 | Frege's platonism proposes that objects are what singular terms refer to |

7731 | How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? |

7737 | Identities refer to objects, so numbers must be objects |

8635 | Numbers are not physical, and not ideas - they are objective and non-sensible |

8652 | Numbers are objects, because they can take the definite article, and can't be plurals |

9580 | Our concepts recognise existing relations, they don't change them |

9589 | Numbers are not real like the sea, but (crucially) they are still objective |

17816 | Frege's logicism aimed at removing the reliance of arithmetic on intuition |

9831 | Geometry appeals to intuition as the source of its axioms |

8633 | There is no physical difference between two boots and one pair of boots |

9577 | The naďve view of number is that it is like a heap of things, or maybe a property of a heap |

9951 | It appears that numbers are adjectives, but they don't apply to a single object |

8637 | The number 'one' can't be a property, if any object can be viewed as one or not one |

9999 | For science, we can translate adjectival numbers into noun form |

9952 | Numerical adjectives are of the same second-level type as the existential quantifier |

11031 | 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' |

18166 | The loss of my Rule V seems to make foundations for arithmetic impossible |

8689 | Eventually Frege tried to found arithmetic in geometry instead of in logic |

8487 | Arithmetic is a development of logic, so arithmetical symbolism must expand into logical symbolism |

7739 | Arithmetic is analytic |

9945 | Logicism shows that no empirical truths are needed to justify arithmetic |

8782 | Frege offered a Platonist version of logicism, committed to cardinal and real numbers |

5658 | Numbers are definable in terms of mapping items which fall under concepts |

16905 | Arithmetic must be based on logic, because of its total generality |

8655 | Arithmetic is analytic and a priori, and thus it is part of logic |

16880 | Frege aimed to discover the logical foundations which justify arithmetical judgements |

18165 | My Basic Law V is a law of pure logic |

10607 | Frege's logic has a hierarchy of object, property, property-of-property etc. |

10831 | Frege only managed to prove that arithmetic was analytic with a logic that included set-theory |

10010 | Frege's belief in logicism and in numerical objects seem uncomfortable together |

9545 | Late in life Frege abandoned logicism, and saw the source of arithmetic as geometrical |

10033 | Why should the existence of pure logic entail the existence of objects? |

9631 | Formalism fails to recognise types of symbols, and also meta-games |

9887 | Formalism misunderstands applications, metatheory, and infinity |

8751 | Only applicability raises arithmetic from a game to a science |

8642 | Abstraction from things produces concepts, and numbers are in the concepts |

8621 | Mental states are irrelevant to mathematics, because they are vague and fluctuating |

11008 | Existence is not a first-order property, but the instantiation of a property |

8643 | Affirmation of existence is just denial of zero |

19470 | Thoughts in the 'third realm' cannot be sensed, and do not need an owner to exist |

5657 | Frege's logic showed that there is no concept of being |

8634 | The equator is imaginary, but not fictitious; thought is needed to recognise it |

18995 | Frege mistakenly takes existence to be a property of concepts, instead of being about things |

17443 | Many of us find Frege's claim that truths depend on one another an obscure idea |

17445 | Parallelism is intuitive, so it is more fundamental than sameness of direction |

10539 | Frege refers to 'concrete' objects, but they are no different in principle from abstract ones |

9578 | If objects are just presentation, we get increasing abstraction by ignoring their properties |

19471 | A fact is a thought that is true |

17431 | Vagueness is incomplete definition |

13879 | For Frege, ontological questions are to be settled by reference to syntactic structures |

10642 | Second-order quantifiers are committed to concepts, as first-order commits to objects |

10032 | 'Ancestral' relations are derived by iterating back from a given relation |

10606 | Frege treats properties as a kind of function, and maybe a property is its characteristic function |

4028 | Frege allows either too few properties (as extensions) or too many (as predicates) |

10317 | It is unclear whether Frege included qualities among his abstract objects |

10533 | We can't get a semantics from nouns and predicates referring to the same thing |

8647 | Not all objects are spatial; 4 can still be an object, despite lacking spatial co-ordinates |

10309 | Frege says singular terms denote objects, numerals are singular terms, so numbers exist |

10550 | Frege establishes abstract objects independently from concrete ones, by falling under a concept |

18269 | Logical objects are extensions of concepts, or ranges of values of functions |

8785 | For Frege, objects just are what singular terms refer to |

10278 | Without concepts we would not have any objects |

8489 | The concept 'object' is too simple for analysis; unlike a function, it is an expression with no empty place |

10535 | Frege's 'objects' are both the referents of proper names, and what predicates are true or false of |

9877 | Late Frege saw his non-actual objective objects as exclusively thoughts and senses |

17432 | Frege's universe comes already divided into objects |

9891 | The first demand of logic is of a sharp boundary |

9388 | Every concept must have a sharp boundary; we cannot allow an indeterminate third case |

16022 | The idea of a criterion of identity was introduced by Frege |

11100 | Frege's algorithm of identity is the law of putting equals for equals |

4893 | Frege was asking how identities could be informative |

3318 | Frege made identity a logical notion, enshrined above all in the formula 'for all x, x=x' |

9853 | Identity between objects is not a consequence of identity, but part of what 'identity' means |

17623 | To understand a thought you must understand its logical structure |

16885 | To understand a thought, understand its inferential connections to other thoughts |

9158 | For Frege a priori knowledge derives from general principles, so numbers can't be primitive |

8657 | Mathematicians just accept self-evidence, whether it is logical or intuitive |

16887 | Frege's concept of 'self-evident' makes no reference to minds |

9352 | An a priori truth is one derived from general laws which do not require proof |

16889 | A truth is a priori if it can be proved entirely from general unproven laws |

16894 | An apriori truth is grounded in generality, which is universal quantification |

16900 | Intuitions cannot be communicated |

16903 | Justifications show the ordering of truths, and the foundation is what is self-evident |

11052 | Psychological logic can't distinguish justification from causes of a belief |

16882 | The building blocks contain the whole contents of a discipline |

8624 | Induction is merely psychological, with a principle that it can actually establish laws |

8626 | In science one observation can create high probability, while a thousand might prove nothing |

8648 | Ideas are not spatial, and don't have distances between them |

8620 | Thought is the same everywhere, and the laws of thought do not vary |

9581 | Many people have the same thought, which is the component, not the private presentation |

8162 | Thoughts have their own realm of reality - 'sense' (as opposed to the realm of 'reference') |

19469 | We grasp thoughts (thinking), decide they are true (judgement), and manifest the judgement (assertion) |

9818 | A thought is distinguished from other things by a capacity to be true or false |

16379 | Thoughts about myself are understood one way to me, and another when communicated |

16876 | We need definitions to cram retrievable sense into a signed receptacle |

16875 | We use signs to mark receptacles for complex senses |

9870 | Early Frege takes the extensions of concepts for granted |

8622 | Psychological accounts of concepts are subjective, and ultimately destroy truth |

9947 | Concepts are the ontological counterparts of predicative expressions |

10319 | An assertion about the concept 'horse' must indirectly speak of an object |

8488 | A concept is a function whose value is always a truth-value |

18752 | 'The concept "horse"' denotes a concept, yet seems also to denote an object |

9839 | Frege equated the concepts under which an object falls with its properties |

9190 | A concept is a function mapping objects onto truth-values, if they fall under the concept |

13665 | Frege took the study of concepts to be part of logic |

13878 | Concepts are, precisely, the references of predicates |

18265 | We don't judge by combining subject and concept; we get a concept by splitting up a judgement |

7736 | A concept is a non-psychological one-place function asserting something of an object |

17430 | Fregean concepts have precise boundaries and universal applicability |

9948 | Unlike objects, concepts are inherently incomplete |

8651 | A concept is a possible predicate of a singular judgement |

4973 | As I understand it, a concept is the meaning of a grammatical predicate |

9846 | Defining 'direction' by parallelism doesn't tell you whether direction is a line |

9976 | Frege accepts abstraction to the concept of all sets equipollent to a given one |

10803 | Frege himself abstracts away from tone and color |

9988 | If we abstract 'from' two cats, the units are not black or white, or cats |

9579 | Disregarding properties of two cats still leaves different objects, but what is now the difference? |

9587 | How do you find the right level of inattention; you eliminate too many or too few characteristics |

9890 | The modern account of real numbers detaches a ratio from its geometrical origins |

9855 | Frege's logical abstaction identifies a common feature as the maximal set of equivalent objects |

10526 | Fregean abstraction creates concepts which are equivalences between initial items |

10525 | Frege put the idea of abstraction on a rigorous footing |

10556 | We create new abstract concepts by carving up the content in a different way |

9882 | You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables |

9881 | From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements |

5816 | Frege said concepts were abstract entities, not mental entities |

9588 | Number-abstraction somehow makes things identical without changing them! |

11846 | If we abstract the difference between two houses, they don't become the same house |

9167 | Frege felt that meanings must be public, so they are abstractions rather than mental entities |

9583 | Psychological logicians are concerned with sense of words, but mathematicians study the reference |

9584 | Identity baffles psychologists, since A and B must be presented differently to identify them |

7307 | A thought is not psychological, but a condition of the world that makes a sentence true |

4980 | The meaning (reference) of a sentence is its truth value - the circumstance of it being true or false |

16879 | A sign won't gain sense just from being used in sentences with familiar components |

8646 | Words in isolation seem to have ideas as meanings, but words have meaning in propositions |

7732 | Never ask for the meaning of a word in isolation, but only in the context of a proposition |

8446 | We understand new propositions by constructing their sense from the words |

4981 | The reference of a word should be understood as part of the reference of the sentence |

18778 | Every descriptive name has a sense, but may not have a reference |

7805 | Frege started as anti-realist, but the sense/reference distinction led him to realism |

4976 | The meaning (reference) of 'evening star' is the same as that of 'morning star', but not the sense |

4977 | In maths, there are phrases with a clear sense, but no actual reference |

4979 | We are driven from sense to reference by our desire for truth |

8449 | Senses can't be subjective, because propositions would be private, and disagreement impossible |

15155 | Expressions always give ways of thinking of referents, rather than the referents themselves |

4972 | I may regard a thought about Phosphorus as true, and the same thought about Hesperus as false |

9817 | Earlier Frege focuses on content itself; later he became interested in understanding content |

8171 | Frege divided the meaning of a sentence into sense, force and tone |

4954 | Frege uses 'sense' to mean both a designator's meaning, and the way its reference is determined |

7304 | Frege explained meaning as sense, semantic value, reference, force and tone |

7309 | Frege's 'sense' is the strict and literal meaning, stripped of tone |

7312 | 'Sense' solves the problems of bearerless names, substitution in beliefs, and informativeness |

11126 | 'Sense' gives meaning to non-referring names, and to two expressions for one referent |

8164 | Frege was the first to construct a plausible theory of meaning |

19467 | A 'thought' is something for which the question of truth can arise; thoughts are senses of sentences |

4974 | For all the multiplicity of languages, mankind has a common stock of thoughts |

16873 | Thoughts are not subjective or psychological, because some thoughts are the same for us all |

16872 | A thought is the sense expressed by a sentence, and is what we prove |

16874 | The parts of a thought map onto the parts of a sentence |

19472 | A sentence is only a thought if it is complete, and has a time-specification |

9370 | A statement is analytic if substitution of synonyms can make it a logical truth |

8743 | Frege considered analyticity to be an epistemic concept |

7725 | 'P or not-p' seems to be analytic, but does not fit Kant's account, lacking clear subject or predicate |

20295 | All analytic truths can become logical truths, by substituting definitions or synonyms |

7316 | Analytic truths are those that can be demonstrated using only logic and definitions |

8619 | To learn something, you must know that you don't know |

8656 | The laws of number are not laws of nature, but are laws of the laws of nature |

3307 | Frege put forward an ontological argument for the existence of numbers |

7741 | The predicate 'exists' is actually a natural language expression for a quantifier |

8644 | Because existence is a property of concepts the ontological argument for God fails |

8491 | The Ontological Argument fallaciously treats existence as a first-level concept |