Title: Freges logic: An introduction
1Freges logic An introduction
- Dr. Chin-mu Yang
- Department of philosophy
- National Taiwan University
- Email cmyang_at_ntu.edu.tw
2Contents
- 1. Historical background
- Why Frege had to construct a new logic?
- 2. Freges primary interests and his logicism
- 3. Two main trends in logic in the19th century
- English vs German
- 4. Main contributions of Begriffsschrift and
Freges main works - 5. What, and why, is linguistic turn?
- 6. Some other issues
3Two main bedrocks of Western culture/civilization
- (i) Aristotles logic (syllogism)
- (ii) The Euclidean axiomatic approach to
mathematical theories
4Kants appraisal of Aristotles logic
- That from the earliest times logic has traveled
this secure course can be seen from the fact that
since the time of Aristotle it has not had to go
a single step backwards, unless we count the
abolition of a few dispensable subtleties or the
more distinct determination of its presentation,
which improvements belong more to the elegance
than to the security of that science.
5Kants appraisal of Aristotles logic
- What is further remarkable about logic is that
until now it has also been unable to take a
single step forward, and therefore seems to all
appearance to be finished and complete. - Immanuel Kant, Preface to the second edition of
Critique of Pure Reason (Bviii)
6A historical problem!!!
- Why a new logic is required?
- Something wrong with Euclidean geometry!?
7Method of axiomatizationThe construction of an
axiom system
- Fix a given subject matter
- Get some naïve facts by intuition or observation
- Describe those facts by sentences of an
appropriate language, taken as true statements,
expressing some truths - Pick out some of them as axioms
- Put forward some rules of inference
- Specify the construction of proofs (derivations),
especially for theorems.
8Axioms and postulates of Euclidean geometry
- Axioms
- 1. Things which are equal to the same thing are
equal to one another - 2. If equals be added to equals, the wholes are
equal - 3. If equals be subtracted from equals, the
remainders are equal. - 4. Things which coincide with one another are
equal to one another - The whole is greater than the part.
9Axioms and postulates of Euclidean geometry
- Postulates
- 1. To draw a straight line from any point to any
point. - 2. To produce a finite straight line continuously
in a straight line - 3. To describe a circle with any center and (any)
distance - 4. That all right angles are equals to one
another
10The dispute over the independence of Euclids
fifth postulate (the postulate of Parallels)
- 5. The fifth postulate (The postulate of
Parallels) - Through any point not falling on a straight
line, one straight line can be draw that does not
intersect the first.
11The issue
-
- Is the postulate of Parallels independent of the
others? Or, can the postulate of Parallels be
derived from the other postulates?
12Towards non-Euclidean geometry
- After mathematicians had tried but in vain to
prove the independence of the postulate in
question for several centuries, some brilliant
mathematicians started to take a different
approach toward the very issue by asking - if the fifth postulate is rejected, would the
system still remain consistent?
13Non-Euclidean geometries
- Surprisingly, this line of thought gives rise to
the establishment of so-called non-Euclidean
geometries. Two typical versions - N.I.Lobachevsky (1820s) there are several such
lines - G.F.B. Riemann (1822-66) no such lines exist.
14The establishment of non-Euclidean geometries
- By the middle of the 19th century, it was proved
that Euclidean geometry and non-Euclidean
geometries are co-consistency in the sense that
if Euclidean geometry is consistent, so are
non-Euclidean ones.
15The impact of non-Euclidean geometry
- The establishment of non-Euclidean geometries
suggests that the axiomatic method would not
suffice to secure the alleged rigour and
preciseness of mathematical theorems, hence that
of mathematical theories. - Mathematicians realized that
- Something has gone wrong!?
16Way out
- It is therefore a natural inclination to stress
that - Something should be done!
- What should be done? The intuition is to search
for a solid foundation of Mathematics.
17Plausible approaches to a solution
- It was soon realized that there are two main
aspects of axiomatic theory - (i) the truth of axioms
- (ii) the deduction of theorems
18Reconstruction of the foundation of mathematics
- Now, if something has gone wrong with an
axiomatic mathematical theory, then - (i) either the former is questionable or
- (ii) the latter needs a refinement.
- Interesting enough, this observation leads to
different approaches toward the reconstruction of
the foundation of mathematics.
19Frege vs. Boole
- Boole focused on (i) the truth of axioms
- Frege paied more attention to (ii) the deduction
of theorems
20Logic in the 19th century
- Boole is correctly regarded as the father of
modern symbolic logic. Frege shared two concerns
of many nineteenth-century mathematicians
avoiding incorrect derivations and providing
rigorous and clear foundations for the
infinitesimal and derivative calculus and
therefore sought to develop a very clear notion
of mathematical proof.
21Major topics in logic in the 19th century (I)
- 1. the purpose or scope of logic,
- 2. the extent to which logic can usefully be
symbolized or diagrammed, - 3. the organization or codification of basic and
derivative logical principles, such as in an
axiomatization, - 4. the relationship of logic to mathematics,
22Major topics in logic in the 19th century (II)
- 5. the relationship of types of logic, such as
deductive and inductive, to each other, and to
other ideal forms of reasoning, such as the
scientific method, - 6. how far logic should depart from the account
of quantification in Aristotles theory of the
categorical syllogism,
23Major topics in logic in the 19th century (III)
- 7. the relationship of types of deductive logic,
especially of categorical to propositional logic
(usually called hypothetical logic in the
nineteenth century), - 8. the extent to which logic needs to treat
relations in a special way.
24Some hot issues in logic in the 19th century
(i)--Extensional vs. Intensional
- Whether logic should be extensional or
intensional, opinion eventually settling on a
purely extensional conception. Extensional logic
treats terms only according to the concrete
entities to which they refer intensional logic
treats terms according to their meaning,
variously called intension, comprehension or
connotation. C.I. Lewis (1918) claimed that the
extensive symbolic development of logic required
this extensional conception.
25Logic of individuals vs. logic of concepts
English tradition vs Germany
- English (nominalistic) tradition, which paid much
more attention to a logic of individual, concrete
things was further solidified in works by
Whately, Boole, Mill, De Morgan, Venn and Peirce
except for W.S. Jevons (1864) intensional logic.
- German logic (idealist tendencies) had
traditionally gravitated towards the approach
that logic deals with concepts, not with
things. Works by Grassmann (1844), Schröder
(1877), and the monumental Vorlesungen of
18901905) had reversed this course.
26Logic of individuals vs. logic of concepts
Frege in English tradition vs Germany
- The works of Frege fall squarely within the older
German intensional tradition, but this feature
was virtually ignored by his English
popularizers, such as Russell, and was later
conflated with the firmly extensionalist,
set-theoretic views of Dedekind and Cantor that
arose contemporaneously in mathematics.
27Some hot issues in logic in the 19th century
(ii) --The scope of logic
- Was logic to cover all forms of reasoning, or
only deductive reasoning? - Was logic to be a theoretical account of the
patterns of valid inference, or a practical,
how-to manual to help readers reason better or
identify the faulty reasoning of others?
28Deduction vs. induction English tradition vs
Germany
- The scope of logic in English tradition vs. in
German tradition - The broad scope The Reform and English-textbook
traditions included treatments of deductive and
inductive logic, together with other types of
reasoning (analogy, causation, or scientific
reasoning). - German logic was always focused exclusively on
deductive logic. This included authors as diverse
as Kant, Moritz Drobisch, Ernst Schröder and
Frege.
29Kants conception of logic (i)
- Kants views had considerable impact on German
logic in the early nineteenth century and
sharpened the tension between traditional,
narrower conceptions of logic and a
psychological logic about the way humans must
reason. Kants conception of logic as the science
of a priori judgments of all sorts, popularized
by William Hamilton, led many English writers
(Boole, Alexander Bain, Mill) to speak of the
laws of thought.
30Kants conception of logic (ii)
- a transcendental or normative theory of judgment
and reasoning, rather than being simply empirical
psychology.
31Freges goal in his new logic
- Freges goal is to display in a perspicuous way
the relationships between concepts and
propositions The goal of the whole of logic is to
demonstrate the correctness of deductions without
gaps between premises and conclusions, using
acknowledged formal and precise rules of
inference. Freges focus seems initially to have
been on determining the correctness (and hence
non-synthetic, a priori nature) of mathematical
proofs, rather than examining reasoning of all
sorts, as had traditionally been the subject of
logic.
32Frege in math. Tradition in the 19th century
- Logic before Frege had not devoted itself only,
or even extensively, to reasoning in mathematics
mathematicians were thought to be those least in
need of help. Frege seems to have shared two
concerns of many nineteenth-century
mathematicians avoiding incorrect derivations,
like many faulty proofs of the parallel
postulate from other axioms and postulates in
Euclidean geometry and providing rigorous and
clear foundations for the infinitesimal and
derivative calculus.
33Freges solution
- His solution was, in essence, to develop a very
clear notion of what counts as a mathematical
proof.
34Freges contribution
- (i) He developed the first theory of
quantification. - The Boolean quantifiers of Peirce and Schröder
were typically only class abstraction operators,
with implicit rules given for their use. - (ii) Frege gave a propositional logic (using
notational devices for the material conditional
and for negation) and made it the core of his
theory whereas for Aristotelians and Booleans,
propositional logic was dependent on the logic of
categorical statements, understood as a theory of
classes.
35Freges contribution
- (iii) Frege used this theory to develop an
account of the nature of numbers (1893, 1903)
that was to have an enormous impact on Russell,
and on the philosophy of mathematics. - (iv) Frege contributed a great many concepts that
have become part of the philosophy of logic what
we now call predicates and propositional
functions the use of functions in logic the
distinction between sense and reference, and many
others.
36Freges primary interest
- To understand both the nature of mathematical
truths and the means whereby they are ultimately
to be justified. - The appeal to reason What justifies mathematical
statements is reason alone their justification
proceeds without the benefit or need of either
perceptual information or the deliverances of any
faculty of intuition. - The Task To articulate an experience- and
intuition-independent conception of reason.
37Freges goal- Logicism
- To show that most of mathematics could be reduced
to logic, in the sense - (i) that the full content of all mathematical
truths could be expressed using only logical
notions and - (ii) that the truths so expressed could be
deduced from logical first principles using only
logical means of inference.
38Freges main works
- (i) Begriffsschrift (Conceptual Notation) (1879)-
New logic is firstly presented - (ii) Die Grundlagen der Arithmetik (The
Foundations of Arithmetic) (1884)- to outline his
strategy to reduce arithmetic to logic and then
to provide the reduction with a philosophical
rationale and justification - (iii) Grundgesetze der Arithmetik (Basic Laws of
Arithmetic) ( volumes 1, 1893, and 2, 1903)- to
carry out the logicism-programme in detail.
39Freges main works
- (iv) A series of philosophical essays on
language, the most important of which are - -- Funktion und Begriff (Function and Concept)
(1891), - -- Über Sinn und Bedeutung (On Sense and
Reference) (1892a), - -- Über Begriff und Gegenstand (On Concept and
Object) (1892b) and - -- Der Gedanke eine logische Untersuchung
(Thoughts ) (1918/9).
40The contributions of Begriffsschrift (Conceptual
Notation) (1879)-
- In 1879, with extreme clarity, rigour and
technical brilliance, he first presented his
conception of rational justification. - (i) A deep analysis was possible of deductive
inferences involving sentences containing
multiply embedded expressions of generality (such
as Everyone loves someone). - (ii) he presented a logical system within which
such arguments could be perspicuously
represented this was the most significant
development in our understanding of axiomatic
systems since Euclid.
41Freges approach to philosophical problems Three
guiding ideas (i)
- Shaping the primary concerns and methods of
analytic philosophy. - (i) lingua-centrism The linguistic turn
- Frege translates central philosophical problems
into problems about language for example, the
epistemological question of how we are able to
have knowledge of objects which we can neither
observe nor intuit, such as numbers, will be
replaced with the question of how we are able to
talk about those objects using language.
42Freges approach to philosophical problems Three
guiding ideas (ii)
- (ii) the primacy of the sentence
- It is the operation of sentences that is
explanatorily primary the explanation of the
functioning of all parts of speech is to be in
terms of their contribution to the meanings of
full sentences in which they occur.
43Freges approach to philosophical problems Three
guiding ideas (iii)
- (iii) anti- psychologism
- We should not confuse such explanations with
psychological accounts of the mental states of
speakers inquiry into the nature of the link
between language and the world, on the one hand,
and language and thought, on the other, must not
concern itself with unshareable aspects of
individual experience.
44Three fundamental principles
- the Preface to The foundation of Arithmetic
- (i) always to separate sharply the psychological
from the logical, the subjective from the
objective - (ii) never to ask for the meaning of a word in
isolation, but only in the context of a
propositionThe context principle - (iii) never to lose sight of the distinction
between concept and object. (1884 x)
45What is, and why, linguistic turn?
- The setting
- Frege rejects the physicalist view of numbers,
and claims that numbers are not physical objects
nor does he (against Kant) accept the view that
they are objects of intuition. - The starting point The epistemological problem
of arithmetic objects/truths - (EPA) How we can have knowledge of the objects
(and a fortiori, the truths) of arithmetic?
46Freges solution to the problem
- Stick to the context principle never to ask for
the meaning of a word in isolation, but only in
the context of a proposition, frege reformuated
The epistemological problem of arithmetic
objects/truths as - How to characterize/specify the sense of a
proposition in which a number word occur?
47The main theme of linguistic turn
- Our ability to refer to numbers, both
metaphysically and epistemologically
(mistakenly?) taken as objects, should be
explained in terms of our understanding of
complete sentences in which names are used. - Epistemological/metaphysical problems are thus
reformulated in terms of semantic problems-
problems about language
48The contributions of Begriffsschrift
- A revolution of the study of deductive inference.
- (i) A satisfactory logical treatment of
generality and - (ii) The development of the first formal system.
49Quines comment on Begriffsschrift
- "Pinpointed, the logical renaissance might be
identified with the publication of Frege's ,
Begriffsschrift in 1979 - a book which is no
older today than was Copernicus's De
revolutionibus in the heyday of Galileo. " -
W.V.O.Quine, Preface to J.T. Clark, Conventional
Logic and Modern Logic (Woodstock, Maryland
Woodstock College Press, 1952, pp. v-vii).
50Singular terms/concept-words The underlying
structure of sentences
- Frege illuminates the important features of
languages underlying structure by indicating
that traditional grammatical categories have no
logical significance and urging instead the
consideration of the categories of singular
terms (which he calls logical subjects) and of
predicates (which he calls concept-words).
51Singular terms vs.perdicates
- A singular term is a complete expression, one
which contains no gaps into which another
expression may be placed a predicate such as (
) was written by Virginia Woolf is something
incomplete it is a linguistic expression which
contains a gap and which becomes a sentence once
this gap is filled by a singular term.
52Two characteristics of the singular-term/concept-w
ord structure
- (i) In the categories of reality, there are
counterparts to the linguistic categories of
singular term and predicate he calls these
ontological categories object and concept,
respectively. - A singular term refers to, or designates, an
object. A predicate refers to, or designates, a
concept.
53(ii) Concepts as functions
- (ii) The notion of a concept is to be construed
as functions in mathematics. A first-level
concept is said to be true or false of an object
or, as Frege puts it, an object falls under
or fails to fall under a concept. - Hence, Frege calls concepts unsaturated unlike
objects, they await completion, whereupon they
yield one of the two truth-values, which Frege
takes to be objects the True and the False.
54- In short, concepts are a kind of function, namely
those that take as their only values the True or
the False.
55The construction of a formal system
- A formal system, as Frege conceives it, has three
parts - (i) a highly structured language in which
thoughts may be expressed - (ii) certain specified axioms, or basic truths,
about the subject matter in question - (iii) rules of inference governing how one
sentence may be inferred from others already
established.
56The annoucment of logicism
- In the Preface to Begriffsschrift, Frege
announced his interest in determining whether the
basic truths of arithmetic could be proven by
means of pure logic. In short, the truths of
arithmetic are truths of logic - Kant the truths of arithmetic are synthetic a
priori for example, knowledge of 7 5 12
requires appeal to intuition
57The contributions of The foundation of
Arithmetic (i)
- (i) One of Freges main goals in The Foundations
of Arithmetic was to refute Kants view by giving
purely logical proofs of the basic laws of
arithmetic, thereby showing that arithmetical
truths can be known independently of any
intuition.
58The contributions of The foundation of
Arithmetic (ii)
- Grundlagen is arguably the first work of analytic
philosophy. At crucial points in the book, Frege
makes the linguistic turn that is, he recasts
an ontological or epistemological question as a
question about language. Unlike some linguistic
philosophers, his purpose is not to dissolve the
philosophical problem to unmask it as a
pseudo-problem but to reformulate it so that
it can be solved.
59Topics in the philosophy of language
- Sense and reference
- Thoughts
- Objectivity of thoughts
- Freges notion of Truth
- Freges notion of existence
- Freges treatment of indirect contexts
- Slingshort argument, etc.
60Further readings (I)
- Dummett, M. (1973), Frege Philosophy of
Language, London Duckworth. - -----(1991), Frege Philosophy of Mathematics,
London Duckworth. - ----- (1991), Frege and Other Philosophers,
Oxford Clarendon Press.
61Further readings (II)
- Burge, T. (2005), Truth, Thought, Reason Essays
on Frege, Oxford Clarendon Press. - Carl, W. (1994), Frege's Theory of Sense and
Reference - Its Origins and Scope, Cambridge
Cambridge University Press. - Kenny, A. (1995), Frege, London Penguin.
- Mendelsohn, R. L. (2005), The Philosophy of
Gottlob Frege, Cambridge Cambridge University
Press. - Noonan, H. W. (2001), Frege A Critical
Introduction, Oxford/Cambridge Polity