Conditionals, necessity, and contingence in mathematics class

Viviane Durand-Guerrier

CNRS-University Lyon 2-Equip COAST- UMR-GRIC
ENSL, 46, alle d'Italie, 69364 LYON cdex 07
CSJF Valence, Universite GRENOBLE 1
rue Barthemy de Laffemas 26000 Valence
vdurand@csjf.ujf-grenoble.fr

Introduction

The following analysis is derived from a didactic research about logic difficulties in mathematics for scientific students in high school and at University (first year). The focus point is on conditional statements which are so critical in the mathematical practice. A lot of studies, specially in psychology, show that conditional reasoning is difficult for the most part of adults, even for students. According with failure of most subjects in experiments such as Wason selection task, several authors argue that propositional logic is not a relevant model for human reasoning (Wason, 1977). Others suppose even that human subjects reason without logic (Johnson-Laird 1986), and emphasize the content-dependence effect on reasoning (Girotto 1991, Dumont 1982). Some others as Noveck (1991) argue that human reasoning is both logical and pragmatic. Although these analysis are relevant to explain these difficulties, we thought that we may also interpret them with logic itself. Indeed, all the previous authors consider that standard logic is propositional logic. Nevertheless, as referring to Aristotle, as they generally do, it is obvious, opposite with Piaget assertion (1949), that it is not enough; we need quantity to express Aristotelian logic, and so predicate logic is necessary (Hottois, 1989). We have shown (Durand-Guerrier 1994 & Appendix I) that if we translate Wason selection task in predicate logic language, it appears that it is not isomorphic to other tasks with realistic content generally supposed to be isomorphic (needing two conditionals sentences against only one for realistic ones); this complexity in formal language being related to a treatment's complexity as says Noveck (1991, p93) and as we show it in our own experiments. In mathematics too, it's clear that we need to express quantity, and most of time we use universal theorems. This allows in mathematics a kind of linguistic necessity: we will tell that a j is necessary a y if, in a certain theory, we can prove the generalized conditional "forall x (j(x) =>y(x)" (Dubucs, 1990). Consequently, we may meet contingent sentences in a meaning we will precise further.

Different notions of conditional and classical inference rules

Most often, in France, teachers don't distinguish clearly conditional as a propositional connective, generalized conditionals, and logically valid conditionals. However, the three acceptation are useful in mathematics, the third one corresponding to classical inference rules. And more over, sometimes teachers try to explain propositional conditionals with everyday life's sentences. According with Quine (1950), we claim that it's necessary to consider at least four species of conditional:

Necessity and contingence in mathematics class

In predicate logic, an open sentence as "P(x)", where P is a predicate, or "if p(x), then q(x)" , where p and q are predicates, has no truth value. Only non-open (closed) sentences have truth value ; there are two kinds of closed sentences: sentences without variable, for example "P(a)" where P is a predicate and "a" an element of a set for which P can be considered; or sentences in which every variable is quantified.

1. Contingent statement for a subject at a certain moment

There is, in predicate logic, a rule named " universal instantiation ". When "forall x F(x)", where F is a sentence with exactly one variable non-quantified, is true in a certain set, then for every element a of this set, we may infer "F(a)". According to this rule, we get an action rule for a subject solving a problem: as soon as a subject knows "forall x P(x)" is true in a certain set, he may infer P(a) for every element a of the set. More precisely, he can tell that, necessary "P(a)" is true. On the contrary, when the subject knows that "exists x P(x)" is false, he can infer for every element a of the set, that "P(a)" is false . On the other hand, when "forall x P(x)" is a false sentence and "exists x P(x)" a true sentence, it is possible that "P(a)" is true, and it is possible that "P(a)" is false. In that case, "P(a)", which has a truth value, is contingent for the subject as far as he is able to know the truth value of the sentence. So, for a subject solving a problem, at certain steps of his search, some sentences may be necessarily true, impossibly true (necessarily false) or contingent (possibly true, possibly false) according as he knows, or not , a convenient general theorem. We can illustrate this with an example (Durand-Guerrier 1995). It is an item abstracted from an evaluation concerning 15-16 years old pupils.

Description and analysis of labyrinth task:

This task is submitted to pupils 15-16, in mathematics class; it's an evaluation elaborated by teachers involved in didactic search and proposed by voluntary teachers to their own pupils. Subjects are told that a person named X managed to cross a labyrinth and never use twice the same door. The labyrinth is drawn. There are twenty rooms on four levels pointed by letters A, B, C, ... to T. Three ones have no door: A, B & P. Two have exactly one door: H & T. Three ones have three doors: L, N & R; one has four doors: I. The other ones have exactly two doors.

According with the configuration, you necessarily enter the labyrinth in room C and leave it crossing successively N, Q, R. The authors write:

We may state sentences relevant to the situation. For some of these sentences, we can state a truth value (TRUE or FALSE); for others, we don't have enough information to decide if they are true or not; (in that case, answer CAN'T TELL). For example, the sentence " X crossed C " is a true sentence. Indeed, we affirm that X crossed the labyrinth and C is the only entrance room.
Then they propose the six following sentences:

Sentence one is necessarily false; indeed, P has no door.

Sentence two is necessarily true as we said before.

Sentence three has a truth value; but we can't know it without further information; the right answer is "CAN'T TELL".

Sentence four is necessarily true; indeed O is a room with exactly two doors and one is common with F;

Sentence five is necessarily true for a similar reason.

For sentence six, we can't know the truth value; indeed, you can cross the labyrinth, crossing successively C,D,I,L,M,N,Q,R; in that case the sentence 6 is false; or you can cross it, crossing successively C,D,I,J,L,M,N,Q,R, and in that case the sentence is false; so, the right answer is "CAN'T TELL".

According to the authors, most of pupils (71%) answered " CAN'T TELL "; the surprise comes from the teachers themselves who consider that this answer is wrong! They give as an example of false reasoning the following argues:

The sentence number six is neither true nor false. We can't tell. For X might crossed through K, but might also cross through I, a room which has a common door with L, avoiding so K.

Except for the fact that the sentence number six has a truth value, we agree with this answer. However, we can understand the teacher's point of view through this notice concerning the conditionals sentences number 4 to 6: "Are they mathematical statements, which we must understand in their whole? In that case, the important matter is the bound between the two sentences and not the particular truth value of each one."

So, for the teacher, the conditional statement is clearly the Russell's generalized conditional, and in the sentences number four to six, X is a quantified variable, which is not the case in the sentence three for which they expect the answer "CAN'T TELL". In fact, although the person is named X, X is not here a variable; we might have call her Paul or John or every else. More over, there is no referee population; endless, to describe the situation in logical language, the relevant variable is the "crossing", as it appears in spontaneous treatment. Doing this (a crossing is a succession of letters among the letter from A to T, with some rules), we can see that for sentences three and six, the formal open sentences corresponding lead to a false universal sentence and a true existential sentence; so, the formalization of the task allows us to make clear that point: the truth values of sentences three and six are not blocked by the situation.

The teacher's point of view corresponds to a very common practice in french mathematics classes, and also in mathematical article. Indeed, it is nearly never assume that some sentences may be contingent for the subject. However, this experiment and others (see Noveck 1991, p 95) shows that when "can't tell"'s choiceis given, pupils use it. So, in a certain way, implicit quantification in mathematics class prevents the emergency of contingent statements which are rather "natural" for pupils and students. Even for students arriving in scientific university, this practice is not shared by many of them: we asked about two hundred fifty arriving students at the beginning of the university year, in October 1992, whether a quadrilateral with perpendicular diagonals was a rhomb or not.

According to scholar habits, we could thought that most of students will answer as if the question was "Is it true that every quadrilateral with perpendicular diagonals is a rhomb?" which leads to answer "no". But only 38% answer "no"; 24% answer "yes", 4% don't answer, and 34% give an answer expressing contingence . This experiment show that even when "can't tell"'s choice is not given, some students use it, neglecting the invited inference "false" for "p(a)" when knowing that "q(a)" and "forall x (p(x) =>q(x) )" are both true whereas "forall x ( q(x) => p(x) )" is false. In th= is experiment, subjects answering "can't tell" and subjects answering "false" give exactly the same reasons. In the labyrinth task, the authors noticed that failing in the task (especially for sentence number six), was not related to failure in mathematics reasoning in classical mathematics tasks, and couldn't understand why it was so. As we will show now, to recognize cases where "can't tell " is the right answer is necessary to understand logical conditional, and then to lead correct reasoning.

2. Lack of inference rule with conditional sentences

Let us consider a subject solving a problem; he knows some theorems he will use to prove some statement. As we already told, we have two main inference rules: Modus Ponens and Modus Tollens. Actually, with a general conditional, we may introduce necessity as following. When "forall x (P(x) => "Q(x))" and "P(a)" are both true in a certain set, necessarily "Q(a)" is true; when "forall x (P(x) => Q(x))" and "non-Q(a)" are both true in a certain set, necessarily "non-P(a)" is true. Now, when a theorem is not an equivalence, in the two others cases, we have a lack of inference rule. Polya (1965) introduces for this two cases the two heuristic syllogisms:

  1. When "forall x (P(x) => Q(x))" and "Q(a)" are both true in a certain set, "P(a)" is more credible.
  2. When "forall x (P(x) => Q(x))" and "non-P(a)" are both true in a certain set, "Q(a)" is less credible.
These reasoning forms appear when you have to treat conjectures (a conjecture is a contingent statement with an epistemic value: the subject thinks that the statement is true). Knowing a theorem for which the conjecture you are studying is a case of the premise, you can look for the consequence's truth value; if it's false, you infer your statement is false and give up your conjecture; then you can stop your search. If it is true, your conjecture gains credibility; it succeeds in the first test. You ought to go on searching.

We have a lot of examples of such reasoning at University, first year. For example, if you look for an eventual limit for a two variables function in (0,0), you can first look for partial functions obtained when you consider one of the two variables as a parameter. If for all of these functions, you find a limit, the same for all of them, you can go on searching; in the other case, you can affirm that there is no limit, using a conditional theorem asserting that "if such a function admit a limit every partial function admit the same limit.", the generalized converse conditional sentence is not a theorem. In this case, it may be relevant to do so, for it is sometimes rather difficult to prove the existence of a limit; this test allow us to engage calculation only when the existence of a limit is rather credible; and more other, it indicates which value is a "candidate" for limit. This kind of reasoning appears also very early, when children do some verifications. For example, in elementary school, pupils learn how to control their multiplication with the "nine-proof". When the "nine proof" is false, they know that there is a mistake and do again their calculation. When it is right, they are not sure that there is no mistake. Of course, they generally trust their calculations; however, it is rather common that "nine-proof" is right and multiplication wrong.

3. Which inference rules for mathematics class?

We may notice that there is no lack of inference with a characteristic property ; so subjects do not meet contingent statements with biconditionals theorems. Yet, in France, teachers are invited to separate every biconditionals in two conditionals. Then, for deduction, they give only one rule: the Modus Ponens. And children are expected to understand the difference in using one conditional or the other one for concluding. For example with Pythagorean theorem, let us imagine a subject who has to decide if a triangle is squared or not. He compares hypotenuse's square with the sum of the two others side's squares. According with non-equality or equality, he will use either the statement "if non-q, then non-p", which is logically equivalent to "if p, then q", or the statement "if q, then p". Both lead to conclusion. Generally, it is expected that children distinguish the four statements; but most of them do not. As for us, we think that one biconditional statement is enough, but with two inference rules: Modus Ponens and Modus Tollens. Noveck (1991) says that, usually, children give the correct answer when knowing "not-q" and "if p, then q". We thought that this inference rule might be early given and use in mathematics class, by developing verification's practice, which is generally neglected. Doing that allows to distinguish between statements with no lack of inference, conjunctions and biconditionals, and statements with lack of inference, disjunction and strict conditional; which is clearly necessary to understand logical conditional.

4. About tertium non datur

Most often, teachers assume, as a law, that in Mathematics, every sentence is either true, or false. This rule is generally identified with tertium non datur principle; yet this is not exactly tertium non datur. In predicate logic, "p(x) or non-p(x)", where p is a predicate, is a tautology corresponding to tertium non datur principle, although neither "p(x)" nor "non-p(x)" can receive a truth value. Aristotle, already, distinguished between the two principles: the first one characterizes propositions, the second one can be applied to statements without truth value, and more over, you can assume tertium non datur even when you don't know which sentence, among p and non-p, is true (except, in certain cases, if you are intuitionist). As we saw before, open statements do not have truth value. An important activity for mathematicians is to determinate for an open statement which objects satisfy it, and which do not. According to Lakatos ( 1976), looking for conjecture's counter- examples is very important for mathematics discovery.

As we can see in some experiments (Arsac, 1989), some 13-14 years old children don't accept to declare false a statement with only one counter-example, or a class of counter-examples (every multiple of a certain number, for example); for these children, a statement may be neither true, nor false. So they implicitly assume contingent statement, consequently they generally do not understand why teachers give up a statement with only one counter-example. This difference between children's point of view and teacher's one emphasizes the difficulties with conditionals theorems which are not biconditionals. In that case, teachers say that the converse theorem is false; yet usually, the converse open statement has many examples, and even advanced students do not agree with saying it is false. Then they do not recognize the lack of inference and may assume non valid deductions. So, it is necessary to oppose generalized conditionals statements which lead to necessary conclusion; and open conditionals statements, without truth value, which lead to sentences "contingent for the subject".

Conclusion

Propositional logic and predicate logic, although they are often criticized for their inadequacy for human reasoning, give us possibilities to analyze reasoning processes in mathematics activity at the considered levels. The previous analysis shows that it is necessary to introduce largely explicit quantity in mathematical sentences, and to let emerge statements whose truth value is not blocked by situation. This appears as a condition "a minima", so that the logical conditional between propositions as used in mathematics can be understood by pupils and students. According to the importance of conditionals and inference rules in solving problems, it is obvious that understanding this notion is absolutely necessary for a scientific education. We let others say if it is the same in some others fields of human activity.

However, as told Wittgenstein (1921), proof in logic and a logic proof for proposition of a certain field, are two different things. So, learning how to recognize tautology will not help children for mathematical reasoning. The meaning of logical conditional will appear progressively while practicing mathematics, in relevant situations, even in elementary school (0rus-Baguena, 1992), and all over scholarship time. According to the mentioned studies, it is obvious that rules for mathematical reasoning have to be taught; specially, we can claim that without a specific attention paid to conditionals, the common notion (to assume a conditional, is to assume his premise) will persist till University, constituting a pregnant obstacle to go on studying mathematics.

At every level considered, logic appears as a method and not as theory. So it is not necessary to teach logic to young children; however it is necessary that teachers know enough logic to be able to analyze the reasoning processes playing in the classroom, and to build relevant situations for their pupils. So, it is necessary to include a logic course, about connectives, specially conditionals, inference rules, quantity, syntax and semantic in predicate logic, insisting on the difference between truth in a theory and logical validity. Even if someone may object that this distinction is not relevant in logic, it is clear that using logic as a method supposes that the difference between a logical theorem (a tautology) and a theorem in algebra, geometry or analysis is clearly exposed.

As for us, according with these analysis, we give, since the present year, a course for voluntary post-certifcate scientifically students entitled "Logic and mathematical reasoning". In this course, we approach all the above mentioned points through some philosophical short texts (Aristotle, Russell, Quine ..), and through mathematical texts derived from their own mathematical course. We pay a special attention to the relevance of predicate logic for mathematics.Though it is too early to give any conclusion, we consider that it is rather cheering: indeed, most of the thirty concerned students are very interested and some of them claim that this course help them to understand mathematics and specially symbolic language employment.(See Appendix II).

Appendix I

About Wason selection task

Presentation of the task from Noveck et al (1991, pp. 90-91))
The Wason selection task has been, over the last two decades, the most investigated deductive reasoning problem. In the standard version of the task subjects are presented four cards showing, e.g., A, D, 4, and 7 respectively, are informed that each card has a letter on one side and a number of the other side, and are given the rule If a card has a vowel on one side, then it has an even number on the other side. Subjects are then told that the rule may be or not be true and are required to select for inspection those cards, and only those cards, that can provide a test of whether the rule is true. Because only the cards showing A (a vowel) and 7 (an odd number) can lead to potentially falsifying evidence, these are the only cards that should be selected. Subjects rarely make the correct selection, instead tending to select the cards showing A and 4, or sometimes just the card showing A.
Which logic to analyze this task?

Most of authors say that this task requires an elementary propositional reasoning, using only Modus Ponens; because most of subjects fail in solving this task, they argue that this is not an universally available inference schema. Noveck and al (1990), as for them, say that

The selection requires a complex indirect reasoning strategy, and a logical competence alone does not provide such a strategy."
As for us we agree with Noveck, and also say that the propositional calculus is not relevant to formalize this task. Formalizing it in predicate calculus shows clearly that some non logical process is required. We can't define predicates for the cards; indeed, the two properties are not independent. Clearly, we must apply predicates to sides, and not to cards themselves. Then, we need two predicates: "to have a vowel", and "to have an even number", noted p & q, applied to variable x for "sides". We need also an involution f defined by "f(x) is the opposite side of x". The rule becomes: "If p(x), then q(f(x), for every x.".

Let us now consider a subject with cards before him. He sees four sides. Considering one side a, he has to control the rule: "If p(a), then q(f(a))". This rule may be falsified in only one case: the side a has a vowel. So, he has to control the card with A. However, this is not enough to be sure that the rule is satisfied. There are eight sides, and the subject must consider too the hidden sides. For the considered side a, he must control the rule for the opposite side; so he has to control the rule: "If p(f(a), then q(a)". This rule may be falsified in only one case: the side a has an odd number. So, you have to control the card with 7. Finally, for each side he sees, the subject has to control two rules, and not only one.

This is different for others classical versions of the selection task as "postal version" of Johnson-Laird: "For example, Johnson-Laird, Legrenzi, and Legrenzi (1972) reported that over 80% of subjects make correct selections when presented with a postal rule, If an envelope is sealed, then it must have at least a 50 lira stamp, and four envelopes, one sealed, one unsealed, one with a 50 lira stamp and one with a 30 lira stamp." Content-dependent theories are proposed to account this facilitative content effect. But according with Noveck and al, in an over selection task with arbitrary content, the Grigg's one, 80% of subjects make the correct selection. The analyze we give for Wason selection task allows us to point a logical difference between the two tasks: indeed, for postal rule, the predicates apply to envelopes themselves: "to be sealed" and "to have at least a 50 lira stamp"; so there is only one rule to control for each envelope. This shows that these two selection tasks are not logically isomorphic, as it is generally considered; the propositional calculus is not the relevant system to interpret these tasks. More over, as it appears in an other task, for some subjects, the necessity of applying the rule to the opposite face is not recognized.

A rather near task: the cube. Argaud (1987; Durand-Guerrier (1991)

Subjects are presented the following rule: If on the side of the cube you can see is a triangle, then their is a circle on the opposite side. The subjects know that on each side of the cube, there is either a square, or a triangle, or a circle and that the mentionned cube follows the rule. Then, they are asked to imagine the cube is lying on a table in front of their eyes and to answer the following questions (among others): If on the side I can see there is a square, then it's possible that on the opposite face there is a square? a circle? a triangle?

According to us, the task is rather near of selection one because to formalize it in predicate logic we need also the variable "sides", the involution f for opposite side, and two sentences for each side considered. If p is "To have a triangle" and q "to have a circle", the rule becomes : "If p(x), then q(f(x)" and for a considered side a we have the two rules: "If p(a), then q(f(a))" and "If p(f(a), then q(a)". For a side a with a square, the first sentence is necessarily true; so, considering only this part of the rule, you can have any figure on the opposite side. But the second sentence may be wrong: if on the opposite side there is a triangle. So, you can't have a triangle on the opposite side.

In the first experiment, 10 subjects among 17 answered that it's possible to have a triangle. In the second experiment, 134 scientific students were presented this task; about 30% answered that it's possible to have a triangle.

In the second experiment, we recorded a dialog between to students, one (E1) answering no triangle on the opposite face, the other (E2) answering it's possible to have a triangle:

E2: it depends how you take your cube; it depends if you agree to can look at it only on one side, or if you say you can look at it on every side.
E1: ben, the opposite side (...) if you go behind on the other side, you see a triangle.
E2: yes but it' s not
E1: So, according with the rule on the other side, you have a circle
E2: yes but it's not said in the text that you can go behind on the other side. (...) Therefore you see only one face, so you ought to reason only from the side you see.

We can see that the critical point is the possibility, or not, to consider the opposite side.The student E2 does not want to do it. So, she can't consider that cases necessitating to apply the rule to the opposite face are counter-examples for the rule. The answer she gives is coherent, considering the way she reads the rule. We can thought that for Wason selection task, some subjects don't consider that the rule must be applied to the opposite sides of the cards. In that case, controlling only the card with A is coherent.

Appendix II

Logic and mathematics reasoning A course for scientific students

This course was given for the first time this year, it concerns twenty-nine voluntary [They had to chose a course among several (demography, economy, astronomy, aeronautics, ...)] students following scientific courses at University (first year). They worked in permanent groups of four students and in the second part of the year, they had to treat a special subject they chose.

A- The content of this course:

First part: Some epistemological contributions:

Students were presented some short texts that they had to read and discuss in their group. They were also asked, when possible, to give mathematical examples of the different points mentioned in these texts.

I-Logic in the ancient Greece: Aristotle and Stoics

Texts from Organon, Topics and Analytics, about propositions, negation, tertium non datur, syllogism. For Stoics, texts from Diogene Laerce and Sextus Empiricus about proposition, and tropes.

II- George Boole, The laws of thought (1854, 1992).

Analyze of some texts from chapters one and two and presentation of the system from Gochet-Gribomont (1990)

Second part: Propositional calculus

III-1 Bertrand Russell (1903, 1989): Principles of mathematics (Chapters 2 and 3)

Texts about the material and formal implications

III-2 Wittgenstein (1921, 1961): Tractacus logico-philosophicus

Aphorisms about proposition, truth value tables, tautology;

III-3 Quine (1952, 1972): Methods of logic

Chapters 3 (Conditional) and 7 (implication) in their whole. III-4 The system itself from Cori & Lascar (1993)

Third part: Predicate Calculus

I- Quine (1952, 1972): Methods of logic

Chapter 21 as an introduction to quantification.

II-Hottois (1989): Penser la logique

Description of Copi's rules- Using for analyzing mathematical demonstration issued from the mathematical course.

III- The system itself from Cori et Lascar (1989)

Syntax; semantic; elementary model-theoretic point of view. classical tautologies and universally valid formula.

Fourth part: About recurrence

IV-1 Short texts from Pascal and Poincarre
IV-2 Douglas Hofstadter, Goedel, Escher, Bach (1979, 1985)

Chapter 8 in the whole: "An explosive system: T.N.T.", in which the author describes a formal system for arithmetic: a typographic number theory. In this text, the author shows the necessity to introduce a special rule for "complete induction", and gives some explanations about sentences which truth value can't be decide in the theory itself.

B- Subjects chosen by students:

C- Evaluation of he course by students

The students were presented at the end of a year some questions about this course. Twelve of them answered.

The reasons of the choice: try to understand mathematics reasoning; interest for philosophical texts. All of them, except one, tell that the course generally satisfied them. The possibility to read and understand the texts surprised them, and they were very interested to discuss the different points of view. Most of them tell that predicate calculus is difficult and rather technical. Eight of them declare that there is an important effect on their mathematical ability. Most students should like that this course may go on next year. One declares that he wants to study logic later.

References