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Loughborough University
Wednesday, December 10th 2008
Semantic perspective in mathematics
education
A model-theoretic point of view
Viviane DURAND-GUERRIER
[email protected]
Université de Lyon, Université Lyon 1, IUFM de Lyon & LEPS EA 4148 LIRDHIST
http://lirdhist.univ-lyon1.fr
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Preliminary
We assume a logical perspective on semantics, that suits
with the following definitions referring to Morris (1938)
or Eco ( 1971)
Semantics concerns relation between signs and objects they
refer to.
Syntax concerns the rules of integration of the signs in a
given system
Pragmatics concerns the relationship between subjects and
signs : signs perceived according to their origin, the
effects they produce, and the way they are used.
According with Da Costa (1997), it is necessary to take in
account these three aspects for a right understanding of
logical mathematical field.
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An example : addition of integers
Semantics : the addition of two integers is defined as the cardinal of the
union of two relevant discrete collections ; the result is independent
of the nature of the involved object (with respect that mixing these
objects will preserve their integrity)
Syntax : Addition is defined as the iteration of the successor ; it does not
necessitates reference to quantities. This provides algorithmic rules
in a given system of numeration.
Pragmatics : The articulation between both aspects is build by a forth
and back between calculation (syntax) and effective counting
(semantics).
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Negation, between syntax and semantics
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Negation (1)
Semantics : the negation of a proposition exchanges the truth value; the
negation of a property exchange those objects which own the
property.
3 is an even number / 3 is not an even number
To be an even number / not to be an even number
Syntax : the negation of a given proposition follows precise rules with
respect of the concerned language. In French :
-for singular proposition, and universal propositions, we apply “ne
..pas” on the verb :
3 est un nombre pair/3 n’est pas un nombre pair
Tous les nombres sont pairs/Tous les nombres ne sont pas pairs
-for existential proposition, we use the quantifier “aucun” (no).
Certains nombres sont pairs / aucun nombre n’est pair
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Negation (2)
Pragmatics : Some linguistic forms may lead to referential ambiguities
“Tous les nombres ne sont pas pairs” (1) (not all numbers are even) is
sometimes interpreted as “Aucun nombre n’est pair” (2) (No number
is even), the contrary in Aristotle’ sense.
This interpretation is reinforced by the possibility of changing “ ne sont
pas pairs” (are not even) in “sont impairs” (are odd) in sentence (1),
that gives “Tous les nombres sont impairs”, synonym of (2).
Generally, but not always, the context permits to choose the right
interpretation.
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Concerning negation, it is likely that it is considered by most
mathematics teachers as a rather simple notion, met and used early
by young children, and build through practise of mathematics.
However, negation, as soon as it operates on quantified statements,
involved both syntactic and semantics criteria, that interoperate
according to the specificity of the concerned language.
This leads to actual difficulties for students in practising
mathematics, in particular, confusion between negation and contrary
that are largely underestimated in the teaching of mathematics,
whatever the level.
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Langage ordinaire
Predicate calculus
Ordinary language
Toutes les boules sont
rouges
x R(x)
All the balls are red
Toutes les boules ne sont
pas rouges
x R(x) ou
xR(x) ?
All the balls are not red
Il existe (il y a ) au
xR(x)
moins une boule qui n’est
pas rouge
Exists at least a ball
which is not red
Certaine boule n’est pas
rouge / Certaines boules
ne sont pas rouges
xR(x)
Some ball is not red
Some balls are not red
Il n’y a pas de boule
rouge /Aucune boule
n’est rouge
x R(x)
x R(x)
There is no red ball/
No ball is red
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In French a sentence such as
“Tous les A ne sont pas B” (“All A are not B”) is
ambiguous.
According with the French linguistic norm, its meaning
is:
It is false that “all A are B”,
eg. “not all A are B”,
that means “some A is (are) not B”.
However, it is often used to express that
« No A is B » (the contrary, following Aristotle)
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The French linguistic norm does not suit with logical syntax and does
not respect the substitution principle
Exemple :
Tous les diviseurs de 12 sont pairs (1)
Tous les diviseurs de 12 ne sont pas
pairs (2)
Tous les diviseurs de 12 sont impairs (3)
Selon la norme linguistique française
comme (1) est Faux, (2) est Vrai
Selon le principe de substitution
(2) et (3) ont la même valeur de vérité
Example
All divisors of 12 are even (1)
All divisors of 12 are not even (2)
Or (3) est Faux
Ceci renforce l’interprétation « Aucun »
But (3) is False
This reinforce the interpretation by “None”
All divisors of 12 are odd (3)
According with the French norm, as (1) is
False, thus (2) is True
According with the substitution principle, (2)
and (3) have the same truth-value
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Effective difficulties for pupils and students
An inquiry of the CI2U, a national commission of the IREM (Institute
for Research on Mathematics Education)
Questionnaire about absolute value-limits-logic
Students at the very beginning of their first university year (340 answers
analysed)
Exercise : Give the mathematical negation of the following sentences
• 1 - Toutes les boules contenues dans l’urne sont rouges.
All the balls in the urn are red
• 2 - Certains nombres entiers sont pairs.
Some integers are even
• 3 - Si un nombre entier est divisible par 4, alors il se termine par 4.
If an integer is divisible by 4, then the last digit is 4
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Toutes les boules contenues dans l’urne sont rouges
All the balls in the urn are red
• C0 : A sentence synonym of the affirmative answer 13%
• C1 : A sentence synonym of “Il y a au moins un balle qui
n’est pas rouge” (“There exits at least a ball in the urn
that is not red”) 38%
• C2 : The ambiguous sentence: “Toutes les boules ne sont
pas rouges” 6%
• C3 : A sentence synonym of “Aucune boule n’est rouge”
(“No ball is red”) 21%
• Other answers : 10%
• No answer :12%
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Certains nombres entiers sont pairs.
Some integers are even
• C0 : The affirmative sentence “Tous les entiers sont pairs” (“All
integer are even”) 5,5
• C2: The ambiguous sentence “Tous les entiers ne sont pas pairs” (All
integers are not even) 5,5%
• C4: The sentence “Tous les entiers sont impairs”(“All integers are
odd”) 15,5%
• C5: The sentence “Aucun entier n’est pair”(“No integer is even”)
13%
• C6: The sentence “Certains entiers ne sont pas pairs (sont impairs)”
“Some integers are not even (are odd)” 34%
• Other answers 11,5%
• No answer : 15%
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Si un nombre entier est divisible par 4, alors il se termine par 4
If an integer is divisible by 4, then the last digit is 4
For this item, 98 students (29%) did not gave an answer;
only 34 students (10%) gave a correct answer, synonym
of “There exists an integer dividable by 4 for which the
last digit is not 4”.
In 155 answers, there is an implication with various
position for the negation;
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• In our population, a large number of students face difficulties to
provide negation of quantified sentences, especially in the case of the
sentence involving an implication.
• As the population comes from various universities, we can suspect
that many fresh students in France face these difficulties.
• Due to the importance of negation, implication and quantification in
the process of conceptualisation in mathematics, our results indicate
that it is necessary to take in account these logical questions in the
teaching of mathematics.
• An international perspective could be to explore theses questions in
various languages, and particularly in context where students learn
mathematics in a foreign language, either in their own countries, or
in a country where they are studying.
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A model-theoretic point of view
Some insights
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• Model theory is developed by Tarski in the
continuity of his work on the semantic definition
of truth for quantified logic.
• A semantic definition of truth put in relation a
formalized language with interpretative
structures of this language.
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A semantic version of propositional
calculus (Wittgenstein, 1921)
In Tractatus logico-philosophicus, Wittgenstein proposes a
formalization of the notion of proposition.
A proposition is a linguistic entity that is either true or false.
The components of the system are “propositional variables”, that could
be interpreted as propositions in some particular piece of discourse.
There are two principles
- First, the principle of bivalence, proposing that there are exactly two
truth values in the system
- Second, the principle of extension, which asserts that the truth-value
of a complex sentence is entirely determined by the truth values of
its elementary components.
The truth value of a proposition expresses its agreement or its
disagreement with the facts or the state of things that it pretends
describing.
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A tool and two fundamental notions
Wittgenstein introduced the truth-tables, and consider all the possible
distributions of truth values, in particular with two propositions (16
distributions among them those of the classical connectors).
With the principle of extension, it is possible to determine the truthtable of any complex sentence. Among them, two play a particular
role:
• Tautology : a statement of the system true for any distribution of
truth-value; so true for any interpretation in any piece of discourse.
• Contradiction : a statement of the system false for any distribution of
truth-value; so false for any interpretation in any piece of discourse.
Tautologies such as « S S’ » support the classical inference rules.
Example : S : « p  (pq) » et S’: « q » ; « S  S’ » is a tautology
associated to Modus Ponens (P; and If P, then Q; hence Q)
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The semantic conception of truth
(Tarski, 1933, 1944)
• The main problem is to give a definition of truth
materially adequate and formally correct (Tarski, 1944,
1974, p.269)*.
• In this study, I only look for grasping the intuitions
expressed by the so named « classical » theory of truth,
i.e. this conception that “truly”as the same signification
as “in agreement with reality” (contrary with the
“utilitariste” conception that “true” means useful under
such or such aspect (Tarski, 1933, 1972, p. 160)*.
* Our translation
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A recursive definition of truth
• Interpretation of a propositional function (a predicate) of
a given formal language in a « domain of reality » by an
open sentence.
• Satisfaction of an open sentence by an object (an
individual) of the discourse’s universe.
– For all a, a satisfies the propositional function « x is white » if
and only if (it is the case that) a is white
• Definition of the truth of a complex sentence
– Propositional connectors (Wittgenstein)
– Quantifiers (in agreement with common sense)
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Théorie des modèles
(Tarski 1955)
• A formalized language L, a syntax, well-formed
statements (formulae) : F, G, H …..
• An interpretative structure  (a domain of reality, a
mathematical theory).
•  is a model of a formula F of L if and only if the
interpretation of F in  is a true statement.
• A formula H is a logical consequence of a formula G
if and only if any model of G is a model of H.
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An example
F : xy (S(x,y) S(y,x)  x=y)
 : set of ordered real numbers in which S is interpreted
as the relationship ‘to be inferior or equal’
The interpretation of F in  expresses that « the
relationship ‘to be inferior or equal’ is anti
symmetric ». This is true.
Hence,  is a model of F
’ : set of ordered real numbers in which S is interpreted
as the relationship ‘to be equal’.
’ is not a model of F
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About logical consequences
Logical consequences support validity, and hence classical mode of
reasoning in mathematics (Quine, 1950)
1. G : p(x)  (p(x)  q(x))
H : q(x)
H is a logical consequence of G (Modus Ponens)
2. F : p(x)
G :  x p(x)
H : x p(x)
F is a logical consequence of G / G is not a logical consequence of F
H is a logical consequence of F / F is not a logical consequence of H
3. F: xy p(x, y)xy q(x, y)
G: xy (p(x,y)q(x,y))
F is a logical consequence of G / G is a not a logical consequence of F
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Deduction theorem
• Every theorem of a given deductive theory is satisfied by
any model of the axiomatic system of this theory;
moreover at every theorem one can associate a general
logical statement logically provable that establishes that
the considered theorem is satisfied in any model of this
type. (Deduction theorem)
• All the theorems proved from a given axiomatic system
remain valid for any interpretation of the system.
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Methodology of deductive sciences
(Tarski, 1960, Introduction ˆla logique)
Axiomatic formal system
(without reference to objects)
Deductuve Mini-thory
Models
Primitive terms
DiscourseΥ
s universe
Defined terms
Interpretation
Axioms
of letters
Theorems
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Proof by interprétation
• A proof that a given statement is not a logical
consequence of the axioms of a theory consists in
providing a model of the theory that is not a model of the
formula associated with the statement in question.
• Example : le fifth Euclid’s postulate and the nonEuclidean geometries.
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Didactic perspectives
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Objects and properties versus statements
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Objects and properties versus statements (1)
An example from Arsac & al. (1992)
A general statement :
n2-n+11 is a primary number for every n (false)
A property of some objects (an open sentence) :
n2-n+11 is a primary number
True for integers from 1 to 10
False for every multiple of 11 ; False for 25
Exploring the statement in grade 7 (12-13), students work
with objects (integers).
Some of them declare the statement false as soon as they
find 11 is a counter example
Others state that the statement is neither true, nor false, or
both true and false, or look for a domain where it is
always true.
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Objects and properties versus statements (2)
• (40) Student : there is an exception, hence it is not
always
• (41) Marie : That has been established. Except for this, it
is always a prime number. What if we eliminated 11…?
(…)
• (63) Marie : Yes but 22 is twice 11; we can maybe try 33;
I think that this will also be an exception.
• (64) Marie : I think they have won, because 25 is also an
exception.
• (76) Marie: They are no longer exceptions because 22,
33, are all multiples.
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Objects and properties versus statements (3)
For Marie, the argument is not the number of
counterexamples, but their relationship to the
situation. She adds that to be sure of getting a true
sentence, it is necessary to be under hundred.
The authors report that asking the question anew
some days later, several pupils declare the statement
is false, citing the two counterexamples 11 and 25.
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Objects and properties versus statements (4)
• Our interpretation is that those students who do not want
to declare that the statement is false as soon as a
counterexample is found are not considering the closed
statement. They are working with the open statement
“n2-n+11 is a prime number”, in which they substitute
numerical values for n.
• The group of discussions reveals a disagreement between
those students who consider the general statement and
insist on the fact that “it is not always true, so it is false”,
and those students who remain focussed on the particular
cases they have used to make up their minds.
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Objects and properties versus statements (5)
This activity may lead students to make various true
assertions:
• the sentence is false; the sentence is not always true;
• the sentence is true for all integers from one to ten;
• the sentence is sometimes true, sometimes false;
• the sentence might be true and might be false;
• the sentence is true except for 11; the sentence is true
except for the multiples of 11;
• the sentence is false for every multiple of 11;
• it is impossible to determine all the numbers for which
the sentence is true (or false)
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An invalid inference rule
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An invalid inference rule (1)
To prove
• “Given two functions f and g defined in a subset A
of the set of real numbers, and a an adherent
element of A, if f(t) and g(t) have h and k
respectively for limits as t tends to a remaining in
A, then f+g has h+k for a limit in a”.
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An invalid inference rule (2)
A proof (Houzel, 1996)
• “ By hypothesis, for all >0, there exists >0 such that t
A and ta  imply
• f(t) - h and g(t) - k ; thus we have
f(t) + g(t) – (h + k) 
 f(t) – h + g(t) - k
f(t) - hg(t) – k  ”
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An invalid inference rule (3)
The first assertion could be interpreted as the application of
the invalid inference rule* :
• “for all x, there exists y, such that F(x, y)”,
• and
• “for all x, there exists y, such that G(x, y)”,
• hence
• “for all x, there exists y such that F(x, y) and G(x, y)”
* In some interpretations, it is possible that the two premises
are true and the consequent is false
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An invalid inference rule (4)
The use of this invalid rule can be found in many situations,
providing
• a proof with a gap that can be easily completed,
• an incorrect proof for a true statement,
• An incorrect proof of a false statement.
This may be encountered either in history (Abel, Cauchy,
Liouville, Seidel) or in undergraduates or graduates
students’ proofs (Durand-Guerrier &Arsac, 2005)
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An invalid inference rule (5)
This enlightens a very important difference between an
expert and a novice in mathematics:
an expert in a mathematical field knows when it is
dangerous to slack off the rigorous application of rules
of inference, while novices have to learn this at same
time as they acquire the relevant mathematical
knowledge.
These two aspects of mathematics cannot be learned
separately.
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Conclusion
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Adopting a semantic point of view, and being
situated at a meta mathematic level, a modeltheoretic point of view provides on the one hand
a frame to analyse a priori the situations under
both mathematical and didactical aspects, and on
the other hand to analyse students’ activity, in
particular by providing to the researcher a
methodology to identify and study the place and
the role of objects, besides statements, in the
process of conceptualisation in mathematics, and
to take in account the articulation between syntax
and semantics, and truth and validity.
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« La logique semble bien,
contrairement à ce que pensait
Wittgenstein, un indispensable
moyen, non de « fonder » mais de
comprendre l’activité
mathématique. C’est-à-dire pour
une part, explorer la relation de
l’implicite à l’explicite d’une
théorie.(…) Une part essentielle de
l’analyse épistémologique est
ainsi ouvertement prise en charge
par l’analyse logique. (…) En
même temps elle apparaît comme
une épistémologie effective dans la
mesure où la réflexion est orientée
vers et investie dans l’agir.»
(Sinaceur, H. 1991, Logique : mathématique
ordinaire ou épistémologie effective ?, in
Hommage à Jean Toussaint Desanti, TER)
“Logic seems, opposite with
what Wittgenstein thought, an
indispensable mean, not of
‘founding’ but of understanding
mathematical activity. That
means for a part to explore the
relation from implicit to explicit
in a theory (…). An essential
part of the epistemological
analysis is so openly taken in
account by logical analysis. (…).
So it appears as an effective
epistemology in the measure that
the reflection is oriented and
invested in action.*
* Our translation
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Some references
• Durand-Guerrier, V. : 2003, Which notion of implication is the right one ?
From logical considerations to a didactic perspective, Educational Studies in
Mathematics 53, 5-34.
• Durand-Guerrier, V. Logic and mathematical reasoning from a didactical
point of view. A model-theoretic approach. in electronic proceedings
CERME 3 (Conference on European Research in Mathematic Education,,
Bellaria, Italy, Februar 2003.
http://www.lettredelapreuve.it/CERME3Papers/TG-Guerrier.pdf
• Durand-Guerrier, V. & Arsac, G. : 2005, An epistemological and didactic
study of a specific calculus reasoning rule, Educational Studies in
Mathematics, 60/2, 149-172
• Durand-Guerrier, V. : 2008, Truth versus validity in proving in
mathematics, Zentralblatt für Didaktik der Mathematik, 40/3, 373-384
• About logic, language and reasoning at the transition between French upper
Secondary school and University.Negation, implication and quantification,
ICME 11, Monterrey, 13-07-08, on line.
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