Freges logic: An introduction

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Freges logic An introduction

• Dr. Chin-mu Yang
• Department of philosophy
• National Taiwan University
• Email cmyang_at_ntu.edu.tw

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Contents

• 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

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Two main bedrocks of Western culture/civilization

• (i) Aristotles logic (syllogism)
• (ii) The Euclidean axiomatic approach to
  mathematical theories

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Kants 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.

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Kants 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)

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A historical problem!!!

• Why a new logic is required?
• Something wrong with Euclidean geometry!?

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Method 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.

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Axioms 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.

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Axioms 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

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The 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.

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The issue

• Is the postulate of Parallels independent of the
  others? Or, can the postulate of Parallels be
  derived from the other postulates?

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Towards 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?

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Non-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.

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The 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.

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The 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!?

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Way 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.

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Plausible 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

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Reconstruction 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.

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Frege vs. Boole

• Boole focused on (i) the truth of axioms
• Frege paied more attention to (ii) the deduction
  of theorems

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Logic 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.

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Major 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,

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Major 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,

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Major 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.

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Some 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.

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Logic 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.

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Logic 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.

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Some 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?

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Deduction 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.

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Kants 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.

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Kants conception of logic (ii)

• a transcendental or normative theory of judgment
  and reasoning, rather than being simply empirical
  psychology.

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Freges 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.

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Frege 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.

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Freges solution

• His solution was, in essence, to develop a very
  clear notion of what counts as a mathematical
  proof.

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Freges 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.

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Freges 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.

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Freges 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.

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Freges 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.

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Freges 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.

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Freges 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).

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The 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.

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Freges 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.

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Freges 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.

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Freges 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.

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Three 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)

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What 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?

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Freges 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?

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The 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

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The 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.

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Quines 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).

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Singular 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).

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Singular 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.

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Two 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.

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(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.

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• In short, concepts are a kind of function, namely
  those that take as their only values the True or
  the False.

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The 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.

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The 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

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The 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.

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The 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.

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Topics 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.

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Further 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.

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Further 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