Adapting Formal Logic for Everyday Mathematics
Antti Valmari
a
University of Jyv
¨
¨
a, Faculty of Information Technology, PO Box 35, FI-40014 University of Jyv
¨
¨
a, Finland
Keywords:
High School Mathematics, Elementary University Mathematics, Higher Order Thinking Skills.
Abstract:
Although logic is considered central to mathematics and computer science, there is evidence that teaching
logic has not been a great success. We identify three issues where what is typically taught conﬂicts with what
is needed by those who are supposed to apply logic. First, what is taught about the notion of implication often
disagrees with human intuition. We argue that in some cases human intuition is wrong, and in some others
teaching is to blame. Second, the formal concepts of logical consequence, logical equivalence and tautology
are not the similar concepts that everyday mathematicians and computer scientists need. The difference is
small enough to go unnoticed but big enough to cause confusion. Third, how to deal with undeﬁned opera-
tions such as division by zero is left informal and perhaps fuzzy. These problems also harm development of
computer tools for education. We present suggestions about how to address them in teaching.
1 INTRODUCTION
Logical skills are widely believed to be important
for mathematics and computer science. For instance,
please see (Bronkhorst et al., 2020), (Hammack,
2018) and (Association for Computing Machinery
(ACM) and IEEE Computer Society, 2013).
Despite this, logic has never been a great success
in computer science education. Among the 75 top-
ics whose importance were surveyed by (Lethbridge,
2000), it is perhaps not a surprise that Speciﬁc pro-
gramming languages, Data structures, Software de-
sign and patterns, and Software architecture were
considered the four most important. More surpris-
ingly, typical engineering mathematics topics such as
Differential / integral calculus were near the oppo-
site end. Logic was in the middle ground. More
see (Niemel
¨
a et al., 2018) for a discussion.
The situation of logic in computer science is pre-
sented vividly in the Preface to 2003 edition of a
textbook that was originally published commercially
as (Reeves and Clarke, 1990). The 2003 edition was
made, because of repeated demands from around the
world for more copies of the book. However, it was
able on the web, because “One is that no company,
today, thinks it worth publishing . . . The publishers
look around at all the courses which teach short-term
a
https://orcid.org/0000-0002-5022-1624
skills rather than lasting knowledge and see that logic
has little place, and see that a book on logic for com-
puter science does not represent an opportunity to
make monetary proﬁts.
(Mathieu-Soucy, 2016) wrote on the situation of
logic in mathematics: “This paper shows once again
that extensive knowledge of formal logic is not neces-
sary to do mathematics. However, what this research
brings is the whole idea of alertness to logical charac-
teristics, which is an interesting asset for mathematics
students. . . . This brings us to expand our reﬂection to
the teaching of logic: what kind of knowledge should
be taught and in what way to promote students’ un-
derstanding and diminish logical mistakes, in order to
make logic courses as efﬁcient as possible?”
Many concepts in formal logic have been devel-
oped for theoretical studies on what can be expressed
and proven, not for actually carrying out proofs. As a
result, some of them do not serve applications well.
For instance, while 1 and < have a ﬁxed meaning
in everyday mathematics, the formal notion of logi-
cal consequence assumes that their meanings can vary
within the limits set by the assumptions.
Therefore, “if x < 1 then x 1” is not a logical
consequence. It fails, for instance, if the meanings of
< and are swapped. Instead, x 1 is a logical con-
sequence of the two formulas x < y x y¬(x = y)
and x < 1. It is denoted by {x < y x y ¬(x = y),
x < 1} |= x 1. Indeed, formal logic has notation
for “I derive x 1 from the laws of real numbers and
Valmari, A.
Adapting Formal Logic for Everyday Mathematics.
DOI: 10.5220/0011063300003182
In Proceedings of the 14th International Conference on Computer Supported Education (CSEDU 2022) - Volume 2, pages 515-524
ISBN: 978-989-758-562-3; ISSN: 2184-5026
c
515
Figure 1: A proof from (Stefanowicz, 2014) page 14.
x < 1” but not for “I derive x 1 from x < 1, taking
the laws of real numbers for granted”.
So formal logic lacks practical notation for ex-
pressing reasoning chains. This has made many to use
for the purpose. Figure 1 shows an example. How-
ever, when is taught, this meaning is almost never
taught. Instead, is often taught to mean so-called
material implication and sometimes to mean logical
consequence. This causes confusion. The situation is
made worse by the fact that some correct aspects of
implication seem counter-intuitive to many at ﬁrst.
In Section 2 we introduce material implication,
implication as an operator for expressing a reasoning
step or reasoning rule, and logical consequence. It
is a main claim of this paper that one of the reasons
why students have difﬁculties with logic is failure to
clarify the differences between these three concepts.
Some difﬁculties with material implication are
discussed in Section 3. First, Wason’s famous selec-
tion task is used to argue that original human intuition
on implication is not necessarily correct. However,
after explanation, there is no disagreement about the
correct solution. This is different from the next topic:
principle of explosion. It is so central to everyday
mathematics that it cannot be avoided. It is also so
counter-intuitive that there has been a long debate re-
lated to it among researchers. Finally an example is
analysed where “if . . . then . . . cannot be intepreted
as material implication but can as a reasoning rule,
demonstrating that the difference between these con-
cepts is important to understand.
Section 4 presents as a reasoning operator. Its
meaning is explained in a way which we believe eas-
ier for students than contrasting to logical conse-
quence. Our has proven suitable for educational
software that checks reasoning (Valmari, 2021).
A different problem for both humans and educa-
tional software is that standard systems of logic lack
machinery for undeﬁned operations, such as division
by 0. As a consequence, teaching about how to deal
with them in reasoning is scarce and mostly consists
of informal rules. This is unsatisfactory, because un-
deﬁned operations are ubiquitous in mathematics and
computer science. Section 5 addresses this problem.
The paper ends with a brief conclusions sec-
tion. Throughout the paper we make recommenda-
in teaching.
2 IMPLICATION AND RELATED
CONCEPTS
Let us denote the truth values with F (false) and T
(true). Implication as a connective is a propositional
logic connective that inputs two truth values and out-
puts a truth value according to the following table:
F T
F T T
T F T
It is also called conditional, material implication and
material conditional. It is usually denoted with
(especially in theoretical sources) or . We use .
The only case when material implication yields F
is when its ﬁrst argument yields T and the second ar-
gument yields F. That is, P Q is logically equiva-
lent to ¬P Q. (That they are logically equivalent, is
sometimes called the rule of material implication.)
The connective or is called biconditional,
material equivalence or material biconditional. We
have P Q if and only if (P Q) (Q P).
The symbols and are also often used infor-
mally in a somewhat different meaning, to express a
reasoning step that may be chained to other reason-
ing steps, like in Figure 1. On page 1680 (Bronkhorst
et al., 2020) write “If people smoke or inhale partic-
ulate matter, then it will affect their health and thus
shorten their life”, and then represent it as “(smoking
inhaling particulate matter) unhealthy shorter
life”. When solving a pair of equations, it is handy to
write 2x + 2y = 8 3x 2y = 7 5x = 15 x = 3.
These symbols are also often used to express rules
that justify such steps, such as x < y x y x 6= y.
In mathematics, “if x < 1 then x 1” and “if x 1
then x < 1” may be interpreted as material implica-
tions x < 1 x 1 and x 1 x < 1. The former
yields T for all values of x, and the latter yields F
when x = 1 and T for all other values of x. However,
it is often more natural to interpret them as a correct
reasoning rule x < 1 x 1 and an incorrect rule
x 1 x < 1. The former is correct, because x 1
is a mathematical consequence of x < 1. The latter is
incorrect, because x < 1 is not a mathematical conse-
quence of x 1, since x = 1 is a counter-example.
In everyday mathematics, the difference between
material implication and implication as a reason-
ing operator is often ignored or confused. For in-
stance, (Hammack, 2018) writes on page 44: “In
mathematics, whenever we encounter the construc-
tion “If P, then Q, it means exactly what the truth
table for expresses. This clearly refers to material
implication. On the other hand, on page 57 we have
“If P, then Q,is a statement. This statement is true
if it’s impossible for P to be true while Q is false. It is
CSEDU 2022 - 14th International Conference on Computer Supported Education
516
false if there is at least one instance in which P is true
but Q is false. This is deﬁnitely not material impli-
cation. It resembles much more a reasoning rule.
Hammack’s idea becomes understandable on page
56: “In mathematics, whenever P(x) and Q(x) are
open sentences concerning elements x in some set X
(depending on context), an expression of form P(x)
Q(x) is understood to be the statement x X , P(x)
Q(x)”. That is, he always treats in a reasoning
operator -like fashion, but for closed formulas it is
equivalent to material implication, and he uses this
fact to explain his approach. Around page 44, he tried
to restrict P and Q to closed formulas, but failed. (A
formula is closed if and only if every occurrence of
every variable in it is quantiﬁed by or . Ham-
mack’s “statement” is a slightly different notion, but
this is not essential for the present discussion.)
We recommend using for material implication
and as a reasoning operator. In this notation (and :
instead of ,), Hammack’s idea is that P(x) Q(x) is
correct if and only if x X : P(x) Q(x) holds.
This idea works well in the basic case. Indeed, we
will show in Section 3.4 that it resolves a much de-
bated paradox. On the other hand, it runs into trouble
in some other cases. For instance, it is true that if x is
non-negative, then if x
2
1 then x 1. It might seem
natural to express this as x 0 (x
2
1 x 1).
However, transforming it as above yields x : x 0
(x : x
2
1 x 1). Because (1)
2
1 but not
1 1, the closed formula x : x
2
1 x 1 yields
F, making the translated formula false as a whole.
Also another difﬁculty remains. We will address
it in Section 5. It can be illustrated with the following
reasoning. Its conclusion is unacceptable, although
each step matches the interpretation of discussed
above: On real numbers x > y if and only if not x y.
Furthermore, if
1
x
> 0, then x > 0. By contraposition,
if x 0, then
1
x
0. We can similarly derive that if
x 0, then
1
x
0. Because 0 0, letting x = 0 we get
both
1
0
0 and
1
0
0. So
1
0
= 0.
Yet another potential problem is confusing
with the notion of logical consequence in formal
logic. Roughly speaking, the latter means a conse-
quence that holds for every possible interpretation of
all proposition, constant, function and relation sym-
bols other than =, for all value combinations of vari-
ables. The precise deﬁnition is too complicated to be
shown here, because it needs the notions of signature,
domain of discourse, structure, assignment, interpre-
tation and model. Two (sets of) formulas are logically
equivalent, if and only if in both directions, one is a
logical consequence of the other.
The essential difference is that in everyday math-
ematics, constant symbols (for instance, 35 and π),
function symbols (for instance, + and
) and rela-
tion symbols (for instance, ) are assumed to have
ﬁxed meanings, while in formal logic their meanings
may vary (with the exception that many authors ﬁx
the meaning of = to its familiar meaning in mathe-
matics). As was explained in Section 1, x 1 is not
a logical consequence of x < 1, because and < can
be given meanings and x a value such that x < 1 holds
but x 1 does not (for instance, swap their standard
meanings and choose x = 1).
It seems that “logical consequence”, “logically
equivalent” and “tautology” are typically taught via
truth tables. This matches their meaning in formal
propositional logic, but leaves it open how to use them
elsewhere. Because the idea of varying the meaning
of 0, + and is strange to students, and because no
other phrase has been taught, this runs the risk of stu-
dents thinking that, for instance, x 1 is a logical con-
sequence of x < 1. We recommend teaching that it is
a mathematical consequence. When the meaning of
0, +, and so on may vary, we have “logical conse-
quence”, “logically equivalent” and “tautology”; and
when their meaning is ﬁxed to the standard meaning,
we suggest using mathematical consequence, mathe-
matically equivalent and mathematical fact.
Logical consequence is denoted with Γ |= P,
where Γ is a set of formulas and P is a formula. For
the time being, we deﬁne P Q as Γ ϒ {P} |= Q,
where Γ speciﬁes the properties of the mathematical
system in question (such as real numbers) and ad-
ditional symbols (such as < in terms of ), and ϒ
lists the local assumptions (such as the speciﬁcation
of a case in a proof by cases). So and denote
mathematical consequence and mathematical equiva-
lence. “Reasoning step / rule” refer to applications of
and . Formally, step / rule are the same, but typi-
what justiﬁes it. We will ﬁne-tune the deﬁnition and
present it in a more student-friendly form in Section 4.
In formal logic, Γ `P denotes that P can be proven
from Γ. Modus ponens is the rule that {P, P Q} `
Q. Deduction theorem says that if Γ {P} ` Q, then
Γ ` P Q. Both hold in standard systems of logic.
Together they constitute a tight link between and
, partially explaining why and are often con-
fused with each other. Modus ponens will remain
appreciated throughout this paper, but we will ﬁnd a
problem with Deduction theorem in Section 5.
As evidence to the fact that may be confused
on Logical equivalence: “The logical equivalence of
p and q is sometimes expressed as p q, . . . “or
p q, depending on the notation being used.
However, these symbols are also used for material
Adapting Formal Logic for Everyday Mathematics
517
equivalence, so proper interpretation would depend
on the context. Logical equivalence is different from
material equivalence, although the two concepts are
intrinsically related.
3 PROBLEMS WITH
IMPLICATION
Wason’s selection task (Wason, 1968) is a fa-
mous psychological experiment on material implica-
tion. The literature discussing it is extensive. We
used (Ragni et al., 2017) as our source.
The original experiment was of the following
kind. There are four cards on the table. The partic-
ipants are told that each card has a letter on one side
and a number on the opposite side. The visible sides
of the cards on the table show D, K, 3 and 7. The par-
ticipants are asked to choose as few cards as possible
so that by looking at their hidden sides one can check
whether or not the four cards obey the following rule:
If a card has D on one side, then it has 3 on the
other side.
The only way to violate this rule is to have D on the
letter side and some other number than 3 on the num-
ber side. So the correct answer is D and 7.
in a bare-boned form. The task does not depend on
whether implication is thought of as a connective or as
a reasoning operator. Those issues do not arise where
material implication is an inapproriate formalization
of implication in natural language.
In the original experiment, less than 10 % of
the participants chose the right cards. In the meta-
analysis by (Ragni et al., 2017), 19 % solved a similar
task correctly; 36 % chose only the equivalent of D,
39 % chose the equivalent of D and 3, and 5 % chose
the equivalent of D, 3 and 7.
However, the results improve dramatically, if the
abstract letters and numbers are replaced by concrete
familiar items. For instance, one side of each card
could contain the name of a drink (beer or orange
juice), the opposite side the age of a person (16 or
25), and the rule could be
If a person is drinking beer, then the person
must be over 19 years of age.
Now in the meta-analysis, 64 % chose the right cards,
13 % chose only the equivalent of beer, 19 % chose
the equivalent of beer and 25, and 4 % chose the
equivalent of beer, 16 and 25.
(Ragni et al., 2017) have recognized 15 distinct
theories that aim at explaining this or related results.
We believe that the result can be explained by the idea
of two modes of human thinking: a fast instinctive
mode and a slower more rational mode (Kahneman,
2011). Apparently the fast mode of many people lacks
correct treatment of even the bare-boned material im-
plication in its abstract form, while has rules for con-
crete applications in familiar situations such as age
and alcohol.
On the other hand, the slower mode seems to mas-
ter the bare-boned version. In the words of (van
Benthem, 2008): A psychologist, not very well-
disposed toward logic, once confessed to me that de-
spite all problems in short-term inferences like the
Wason Card Task, there was also the undeniable fact
that he had never met an experimental subject who
did not understand the logical solution when it was
explained to him, and then agreed that it was correct.
We suggest telling students about results on Wa-
son’s selection task or something similar, to make
them realize that their original intuition may lead
them astray. This may encourage them to learn, un-
derstand, trust and use the laws of propositional logic
3.2 The Debate on Carroll’s Paradox
In (Carroll, 1894), the following question was pre-
sented in this form and in the form of a story:
There are two Propositions, A and B. It is
given that
1. If C is true, then, if A is true, B is not true;
2. If A is true, B is true.
The question is, can C be true?
In the story there was a barbershop with three barbers,
Allen, Brown and Carr. At least one of them is not
out, giving rise to (1), where A denotes that Allen is
out, and similarly with B and C. Allen is very shy and
does not go out without Brown, hence (2).
The barbershop paradox is not a problem for mod-
ern propositional logic. It is possible that C is true and
A is false (and B may be either). So C can be true.
It is, however, worth noticing that it deﬁnitely was
a problem at its time. In the story, Uncle Joe reasoned
that if C is true, then we have simultaneously “if A
is true, B is not true” and “if A is true, B is true”,
which is a contradiction, because the same premise
A leads simultaneously to the conﬂicting conclusions
B and not-B. Since this contradiction was obtained
by assuming C, the principle of reductio ad absurdum
yields that C cannot be true. (Carroll, 1894) men-
CSEDU 2022 - 14th International Conference on Computer Supported Education
518
a year, with conﬂicting opinions by several practised
logicians. Indeed, John Cook Wilson, the Wykeham
Professor in Logic of the University of Oxford, held
Uncle Joe’s view (Mokteﬁ, 2007). The dispute con-
tinued for more than a decade (Jones, 1905).
Uncle Joe was wrong in claiming that “if A is true,
B is not true” and “if A is true, B is true” cannot hold
simultaneously. Instead, that happening means sim-
ply that A is not true. However, this forces us to ac-
cept that a false claim may simultaneously imply a
claim and its negation, because A, B and not-B pro-
the next subsection.
Because material implication was not easy for
professional logicians, we should not expect it to be
easy for students either.
The barbershop paradox can also be used to illus-
trate the power of formal manipulation. The premises
(1) and (2) can be represented as (C (A ¬B))
(A B). Replacing each P Q by ¬P Q results
in (¬C ¬A ¬B) (¬A B), which simpliﬁes to
¬A (B ¬C). So either Allen is in (and Brown and
Carr may be anywhere), or Brown is out and Carr is
in (and Allen may be anywhere).
3.3 Principle of Explosion
Material implication is truth-functional, that is, its
result is a truth value which depends on nothing
else than the incoming truth values. This sometimes
clashes with the feeling by many people that for “if
P then Q to be true, Q must somehow depend on
P. For instance, “if it is not sunny tomorrow, I will
stay at home” sounds sensible, while “if I do not stay
at home tomorrow, it will be sunny” sounds odd, be-
cause it seems to suggest that I could cause sunshine
by leaving home. However, as material implications,
they are logically equivalent.
This has led to a vast body of research; see (Egr
´
e
and Rott, 2021) for an up-to-date survey. Many ap-
proaches have rejected truth-functionality at the cost
of making the logic more complicated. However, fol-
lowing that path would take us far from everyday
mathematical practice.
Instead, we keep material implication, and sug-
gest to emphasize students that it does not capture
every aspect that our intuitive notion of implication
may cover. In particular, material implication does
not pay attention to what is the cause and what is
the effect. It only deals with what combinations of
truth values are possible and what are impossible. If
P Q, then P true and Q false is impossible, and
the remaining three combinations are possible (unless
something else makes them impossible). That is all.
It is also worth emphasizing students that within
its scope, material implication is reliable, while intu-
ition tends to give wrong results every now and then.
The students may also be told that there have been at-
tempts to ﬁnd notions of implication that match intu-
ition better, but the results have not been good enough
to replace material implication.
A related problem with intuition arises if P can-
not be true or Q cannot be false, because then P Q
yields T, although there is not necessarily any sensi-
ble connection between P and Q. For instance, the
following are true:
If Earth is ﬂat, then nobody likes coffee.
If Earth is ﬂat, then every natural number can be
expressed as a sum of three squares of natural
numbers.
They are true because the only way to make P Q
yield F is to make P true and Q false, but “Earth is
ﬂat” cannot be made true. On the other hand, “nobody
likes coffee” and “. . . sum of three squares . . . do
not depend on “Earth is ﬂat”. This makes the above
examples seem counter-intuitive.
Indeed, in standard logic, any false claim implies
just anything. This is called the principle of explo-
sion. It underlies proof by contradiction and proof by
contrapositive, which are central in everyday mathe-
matics. In the experience of the present author, this
principle is difﬁcult for many students. Apparently it
was difﬁcult also for John Cook Wilson. The barber-
shop paradox illustrates that it is necessary to accept
that a false claim may simultaneously imply a claim
and its negation. Although accepting it does not nec-
essarily mean accepting the principle of explosion in
its full generality, it is at least a long step in that di-
rection.
Now assume that a person who is held in a secure
prison says “if it rains tomorrow, I will stay in prison”.
The sentence is true, but sounds ironic, because the
prisoner cannot leave the prison, no matter what the
weather is. This is an example of material implica-
tion that is true not because of a sensible connection
between P and Q, but because Q cannot be made false.
Indeed, in standard logic, any true claim is implied by
just anything. This principle can be thought of as a
dual to the principle of explosion.
We recommend teaching students to rely on the
idea that a claim is true if and only if it has no counter-
examples. Similarly, a reasoning rule is incorrect
if and only if it has a counter-example. A counter-
example consist of a combination of values of vari-
ables that is allowed by the assumptions made in the
context where the claim or reasoning rule is stated,
and makes the claim false or rule incorrect. For in-
stance, in the case of real numbers, x =
1
2
and y = 0
Adapting Formal Logic for Everyday Mathematics
519
is a counter-example to x > y x y + 1. How-
ever, in the case of integers, x cannot be
1
2
. Indeed,
x > y x y + 1 is true on integers.
If P cannot be true, then we cannot have P true and
Q false. So P Q has no counter-examples. That is,
the principle of explosion agrees with the idea that a
claim is true if and only if it has no counter-examples.
3.4 Confusion with Reasoning Operator
This example has been modiﬁed from the Wikipedia
page on Paradoxes of material implication. The fol-
lowing is assumed to hold:
3. If John is in London then he is in England.
If “if . . . then . . . is interpreted as material impli-
cation, then the following can be proven, although it
seems plain wrong:
4. If John is in London then he is in France, or
if he is in Paris then he is in England.
To prove it, let (5) and (6) denote “if John is in London
then he is in France” and “if he is in Paris then he is
in England”, respectively. Now (4) can be rewritten
as ((5) or (6)). If “John is in London” is true, then
by (3) also “he is in England” is true. Then by the
dual to the principle of explosion, also (6) is true. If
“John is in London” is not true, then by the principle
of explosion, (5) is true. So no matter where John is,
at least one of (5) and (6) is true. Therefore, ((5) or
(6)) is true. Q.E.D.
A perhaps even more striking example is “if John
is in London then he is in Paris, or if he is in Paris then
he is in Brussels. Namely, if John is in Paris then the
ﬁrst implication holds, and if he is not in Paris then
the second implication holds. Indeed, no matter what
P, Q and R are, (P Q) (Q R) yields T.
Hammack’s idea in Section 2 chases this paradox
away. That is, each material implication P Q gives
rise to the reasoning rule P Q. The rule P Q
is correct if and only if P Q yields T for all inter-
pretations of P and Q that are possible in the context.
In other words, P Q is incorrect if and only if the
context allows at least one interpretation that makes
P true and Q false. This is similar to mathematics,
where “if x 1 then x > 1” is doomed incorrect by
the fact that it fails when x = 1, although it works
okay for all other values of x.
When understood as a reasoning rule, (3) is cor-
rect by assumption. However, (5) and (6) are incor-
rect, because the real Europe is a counter-example
that is allowed by (3). Furthermore, (4) must be in-
terpreted as saying that (5) is a correct reasoning rule
or (6) is a correct reasoning rule. Under this interpre-
tation, (4) is indeed wrong, matching intuition.
More formally, let x L denote that John is in
London, x E that he is in England, and so on. The
reasoning rule interpretation of (4) yields (x : (x L
x F)) (x : (x P x E)), while the ma-
terial implication interpretation and the proof of (4)
only provide x : ((x L x F)(x P x E)).
This resolution of the paradox suggests that peo-
ple do not think of each “if John is in x then he is in y
as material implication but as a reasoning rule. It re-
sembles the strict implication in Section 3.1 of (Egr
´
e
and Rott, 2021), but aims at less deviation from stan-
dard logic.
Unfortunately, we saw in Section 2 that this idea
does not always work. Therefore, we recommend to
teach that “if . . . then . . . may translate to material
implication or to a reasoning rule, and the student has
to choose the right one. If neither one matches intu-
ition, then the student should perhaps ask for clariﬁ-
cation on the intended meaning of the sentence. If it
translates to a reasoning rule but is part of a bigger
formula like ((5) or (6)) above, then may have to be
added as was illustrated above. It is a task of the stu-
dent to choose how many are added and where they
are added to capture the intended meaning. No simple
general rule can be given, because different choices
are needed by (4) and by the “if x is non-negative,
then if x
2
1 then x 1” example in Section 2.
4 REASONING OPERATORS
In this section we describe the meaning of , and
as reasoning operators. The ideas have been devel-
oped from (Valmari and Hella, 2017).
A reasoning chain is of the form P
0
1
P
1
2
.. .
n
P
n
, where n 1, each P
i
is a formula and each
i
is from the set {⇒, , ⇐}. The same reasoning
chain may not contain both and . Each P
i1
i
P
i
is a reasoning step.
Each reasoning chain occurs in a context. It speci-
ﬁes those properties of proposition, constant, variable,
function and relation symbols that are allowed to be
used in writing formulas and reasoning. To write for-
mulas one needs to know such thing as + denotes a
function from a pair of real numbers to a real number;
variable x contains a real number; and variable ~v con-
tains a 3-dimensional vector. To reason one needs to
know that if Allen is out then Brown is out as well;
and for every x we have x + 0 = x. The speciﬁcation
need not be exhaustive. For instance, the barbershop
paradox allows 5 and rules out only 3 combinations of
locations of persons. As another example, the group
axioms have inﬁnitely many different models.
The context is not a concrete piece of text but an
CSEDU 2022 - 14th International Conference on Computer Supported Education
520
abstract entity. Properties that the reader may be ex-
pected to already know, need not be mentioned ex-
plicitly. For instance, if it is said that the domain of
discourse is real numbers, then the familiar properties
of 0, π, +, cos and so on are automatically available.
Each proposition, constant, function and relation
symbol has the same meaning in every formula to
which the same context applies, and and cannot
be applied to them. The values of variables are not
speciﬁed by the context. They may vary between for-
mulas, and within a formula as determined by or .
Information may be temporarily added to the context
by such phrases as: “to derive a contradiction assume
that p is not a prime number”, “consider ﬁrst the case
that p is even” and “so there is an integer k 2 such
that p = 2k”.
The difference between reasoning steps and ma-
terial implication is easier to keep in mind, if we do
not think of the former returning a truth value, but be-
ing correct or incorrect. A reasoning step is correct
if and only if it has no counter-examples. A counter-
example to P Q is any combination of variable val-
ues that is allowed by the context such that P yields T
but Q does not yield T. The step P Q is correct if
and only if Q P is correct. The step P Q is cor-
rect if and only if both P Q and P Q are correct.
That is, for each value combination of variables that
is allowed by the context, either both P and Q yield
T, or neither of them yields T. A reasoning chain is
correct if and only if every step in it is correct.
A reasoning rule is a reasoning step. Typically
“rule” is used for steps whose instantiations (for ex-
ample, 2n + 1 in place of x) are intended for later use.
By the above deﬁnition, P Q R is correct if
and only if both P Q and Q R are correct, and
similarly with P Q R, and so on. Figure 1 illus-
trates that this is how is used by many. Because
reasoning operators occur between, not in, formulas,
P (Q R) is a syntax error and thus means noth-
ing. This is analogous to common practice with re-
lation symbols in mathematics, where, for instance,
0 x < 1 means 0 x x < 1 and x A B means
x AA B; and 0 (x < 1) means nothing. It saves
us from having to decide whether ¬(x 0 x 1)
means 0 x < 1 or that x 0 x 1 is an incorrect
reasoning step.
It would be analogous to interpret P Q R as
(P Q) (Q R), but few, if any, do so. Instead,
many interpret it as P (Q R), some interpret it
as (P Q) R, and many reject it as ambiguous
because of lacking ( and ). All these interpretations
are different. For instance, if P is x 0, Q is x
2
1
and R is x 1, then
formula yields T
(P Q) (Q R) when 1 < x < 0 x 1
P (Q R) for every x
(P Q) R when x 0
That is, is treated like + and in mathemat-
ics. It is also how other binary logical connectives are
treated. Indeed, P Q R deﬁnitely does not mean
the same as (P Q) (Q R)!
We observe that in literature, when used as a log-
ical symbol, almost always denotes material im-
plication, while the use of is somewhat fuzzy but
has aspects of a reasoning operator. This is why we
recommend teaching and not as the material
implication, making it clear what reasoning operators
are, and teaching as a reasoning operator.
We recall from Section 2 that as a reasoning
operator is different from logical consequence, be-
cause it uses the standard meaning of 0, +, and
so on, while logical consequence uses every meaning
allowed by Γ. Therefore, to students whose interest
is in applying logic, is both more useful and much
easier to teach than logical consequence.
In this framework, solving an equation, inequation
or a system of them consists of deriving a formula that
is mathematically equivalent to the original system
and shows explicitly the value combinations of vari-
ables that make the formula yield T. We have already
started an example by deriving 2x+2y = 83x2y =
7 x = 3. We may continue 3x 2y = 7 x = 3
9 2y = 7 y = 1 and 2x + 2y = 8 3x 2y = 7
x = 3 y = 1. That there are no roots is expressed by
F, and that every value combination is a root by T.
For instance, with real numbers, x
2
+ 1 = 0 F.
We believe that many students learn solving (sys-
tems of) (in)equations ﬁrst as a mechanical procedure
rect roots. Seeing that it is actually application of
more general logical reasoning may widen their un-
derstanding and promote high-level thinking skills.
5 DEALING WITH UNDEFINED
EXPRESSIONS
In Section 2 we presented a fake proof that
1
0
= 0. It
was based on ignoring the fact that
1
0
is undeﬁned.
Standard systems of logic assume that every applica-
tion of each function symbol is deﬁned, and are thus
unable to deal with undeﬁned operations. Textbooks
on mathematics may warn about them, but it is hard to
ﬁnd a systematic discussion. Compared to the amount
of effort devoted to teaching the zero product property
or the truth table of , little is done to give students
tools for avoiding such errors as in our proof of
1
0
= 0.
Adapting Formal Logic for Everyday Mathematics
521
3
p
|x|1 = x + 1
(x < 0 3
x 1 = x + 1) (x 0 3
x 1 = x + 1) x < 0 and |x| = x or x 0 and |x| = x
(x < 0 x = 1) (x 0 (x = 2 x = 5)) solve each equation elsewhere
x = 1 x = 2 x = 5 combine roots
Figure 2: A solution via analysis by cases presented in logical notation.
Some mathematicians, logicians and computer
scientists have taken this problem seriously. As a mat-
ter of fact, there is a debate on how undeﬁned ex-
pressions should be treated. Unfortunately, none of
the earlier solutions seems to match everyday math-
ematical thinking. Therefore, (Valmari and Hella,
2017) presented an initial version of a system of our
own. Since then it has been given a sound and G
¨
odel-
complete proof system and compared to many other
solutions, please see (Valmari and Hella, 2021).
Perhaps the ﬁrst (insufﬁcient) idea is that when us-
ing
1
x
, the domain of discourse is not R but R \{0};
and with
x, it is {x R | x 0}. It runs into trouble
with the reasoning in Figure 2. It shows the split-
ting of an equation to two cases and the combination
of the results of the cases, with the cases solved via
other reasoning chains that are not shown. The idea
makes the domain of discourse of the second line be
the empty set.
Despite this, the idea works well and is widely
used with arithmetic comparison chains such as
x
2
9
x
2
+x6
=
(x+3)(x3)
(x+3)(x2)
=
x3
x2
. In the absence of a re-
mark to the effect that x 6= 3, the second = is widely
considered incorrect, but the ﬁrst = is often accepted.
The unconscious rule seems to be that when both
sides of a comparison are undeﬁned for precisely the
same combinations of values of variables, then the re-
mark need not be made. This does not mean, however,
treating undeﬁned as equal to itself, since mathemati-
cians do not accept 0 as a root of
1
2x
+ x =
1
x
.
In logic, a term is an expression whose value is in
the domain of discourse (as opposed to a truth value).
Many computer scientists entertain the idea that ev-
ery undeﬁned term yields a value from the domain
of discourse, but we do not necessarily know what
value (Gries and Schneider, 1995). This approach is
not G
¨
odel-complete. It makes it unknown whether 0
is a root of
1
x
= x, while in everyday mathematics it is
deﬁnitely not a root. Furthermore, it makes
1
0
R and
thus
1
0
+ 0 =
1
0
, so 0 becomes a root of
1
2x
+ x =
1
x
.
In our fake proof that
1
0
= 0, the error took place
in the step from “if
1
x
> 0, then x > 0” to “if x 0,
then
1
x
0”. There
1
0
0 was derived from ¬(
1
0
> 0).
However, mathematicians consider both
1
0
> 0 and
1
0
0 as not true. This suggest that an undeﬁned for-
mula is neither false nor true. Indeed, many people
developers were asked how various undeﬁned situa-
tions should be interpreted, between 74 % and 91 %
chose “error/exception” instead of “true”, “false”, and
“other (provide details)” (Chalin, 2005).
Therefore, we introduce a third truth value U (un-
deﬁned) and have that ¬U yields U. At least three
options for and have been suggested in the liter-
ature. The one that matches the intention in Figure 2
is obtained by thinking of F less true than U which is
less true than T, and letting and pick the mini-
mum and and the maximum of the truth values of
their arguments. A relation yields U if and only if at
least one of its arguments is undeﬁned.
For 3
p
|x|1 = x + 1 x = 1 x = 2 x = 5
to hold when x = 0, we have to accept that U F. To
not lose the symmetry of , we also have F U. To
avoid reasoning U ¬U ¬F T, we no longer let
P Q give permission to replace P by Q in a bigger
formula R(P) of which P is a part.
Instead, we introduce a new reasoning operator
that does not treat U as equivalent to F. That is, P Q
if and only if for every value combination of variables
that is allowed by the context, P and Q yield the same
truth value. If P Q, then P Q and R(P) R(Q),
and thus R(P) R(Q).
We let every function symbol f have a corre-
sponding formula bf e that speciﬁes when f is de-
ﬁned. For instance, with real numbers, b
x e is x 0,
b
x
y
e is y 6= 0 and bx + ye is T, denoting that x + y is
always deﬁned. To ensure that bf e itself is always
deﬁned, we require that if it mentions any function
symbol g, then bge is T.
This idea extends naturally to terms and formulas.
For instance,
f
g
is deﬁned if and only if both f and
g are deﬁned and g does not yield 0. That is, b
f
g
e is
bf ebgeg 6= 0. Furthermore, bf ge is bf ebge,
Pe is bPe, and bP Qe is (bPe bQe) (bPe
¬P) (bQe¬Q). For any formula P and combi-
nation of values of variables, precisely one of P, ¬P
and ¬bPe yields T.
Assume that P(Q) is a formula that contains no
other logical connectives or quantiﬁers than ¬, , ,
and , and has Q as a sub-formula. If Q is within
the scope of an even number of ¬, then P(Q)
P(bQeQ), and otherwise P(Q) P(¬bQe Q).
CSEDU 2022 - 14th International Conference on Computer Supported Education
522
This facilitates reduction of problems to a form where
nothing is undeﬁned, and then solving them using
classical two-valued logic. For instance,
x
2
9
x
2
+x6
= 2
x
2
+ x 6 6= 0 x
2
9 = 2(x
2
+ x 6)
x
2
+ x 6 6= 0 (x = 3 x = 1)
x = 1.
It is also correct to reason
x
2
9
x
2
+x6
= 2 x
2
9 =
2(x
2
+ x 6) x = 3 x = 1 and then check the
roots, rejecting 3 and accepting 1.
This idea has been used in computer science, and
also many mathematicians seem to use it. Its correct-
ness is not trivial to prove. Furthermore, it fails if a
connective is allowed such that F T T and
U F. Please notice that b e is not such a connec-
tive, because it is not a connective, because it acts on
Familiar laws on the domain of discourse remain
correct, such as x x = 0 and 0x = 0; but an is-
sue arises regarding their use: clearly
1
x
1
x
= 0 and
0 ·
1
x
= 0 are not correct when x = 0. Therefore, we
have to teach students to distinguish between values
and terms. Constant and variable symbols (such as
1 and x) denote values and are always deﬁned. How-
ever, when a more complicated term (such as
1
x
) is put
in the place of x (this is called instantiation of x with
1
x
), it is not necessarily always deﬁned.
Fortunately, there are special cases where every-
thing works like in classical two-valued logic, and
there are theorems that tell how to deal with many of
the remaining cases. We cannot cover here the topic
in full, but we try to give a feeling.
Let f and g be terms and P(x) a formula. If for
each variable value combination that is allowed by
the context either f = g or both f and g are unde-
ﬁned, then P( f ) P(g). Therefore, if f = g has
been promised under the convention mentioned ear-
lier without any remark about the domain, we need
not worry about the domain. For instance, we may
replace
(x+3)(x3)
(x+3)(x2)
for
x
2
9
x
2
+x6
even if x may be 3.
In particular, the condition holds automatically
with instantiations of =-laws whose both sides con-
tain precisely the same variables. So the law may
be instantiated without worrying whether what goes
in the place of the variables is deﬁned. For instance,
x + y = y + x is such, but x x = 0 is not.
Unfortunately, the same does not apply to - and
-laws. For instance, the rule that a product is 0 if
and only if at least one of its factors is 0, is vulnerable
to mis-reasoning x ·
1
x
= 0 x = 0
1
x
= 0 x =
0. We recommend teaching that a product is 0 if and
only if at least one of its factors is 0 and all factors
are deﬁned. This is different from the law x > y
¬(x y), which does not need a similar remark. Even
x > y ¬(x y) holds unconditionally.
To deal with such cases as
x
x
= 1 or
1
x
1
x
= 0,
there is a theorem that assumes that the logic has a
property known as regularity. A sufﬁcient but not
necessary condition for regularity is that there are no
other connectives and quantiﬁers than ¬, , , and
. Then bf e f = g implies P( f ) P(g).
In addition to the above, the presence of U affects
propositional reasoning. For instance, the law of con-
traposition now has many forms, including “if P Q,
then ¬Q ¬bQe ¬P ¬bPe and “if P ¬Q
¬bQe, then Q ¬P ¬bPe”. Of course, P¬P T
must be replaced by P ¬P ¬bPe T. The law
P ¬P F still holds, but only because U F; we
do not have P ¬P F. We have P ¬P bPe F.
These effects are mostly easy to take into account in
reasoning. Furthermore, almost all widely mentioned
propositional laws that do not use or are valid
also in the presence of U.
An interesting exception is that P (¬P Q) and
P (¬P Q) are no longer equivalent to P Q and
PQ. If P yields F, U and T, then P(¬PQ) yields
F, U and the same as Q yields, respectively. This
is exactly how the “and” operator of many progam-
ming languages behaves, if U is interpreted as pro-
gram crash. Three-valued logic thus naturally repre-
sents a phenomenon that is important in progamming
but ignored by classical two-valued logic.
There are two main conventions of what should
mean in three-valued logic: one by Jan Łukasiewicz
and another by Stephen Cole Kleene. Modus po-
nens holds for both, but Deduction theorem for nei-
ther. Neither of them matches , because U F.
Łukasiewicz’s convention is not regular but Kleene’s
is, since it makes P Q equivalent to ¬P Q.
Let f be free for x in P(x). The law x : P(x)
P( f ) must be replaced by bf ex : P(x) P( f ). If
the logic is not regular, then P( f ) x : P(x) must
be replaced by bf eP( f ) x : P(x).
There is strong evidence that the phenomena we
discussed above, cover all deviations from familiar
reasoning rules. Namely, so-called ﬁrst-order logic
covers much of the use of predicate logic. Unlike
higher-order logics, it is complete in the sense that
all logical consequences can be proven. (Valmari and
Hella, 2021) proved that also our three-valued ﬁrst-
order logic is complete. Furthermore, only ﬁve proof
rules of our complete proof system differ from the
corresponding system for classical two-valued logic:
the above-mentioned quantiﬁer rules,
/
0 ` P ¬P
¬bPe, {P} ` bPe, and {bf e} ` f = f .
Adapting Formal Logic for Everyday Mathematics
523
6 CONCLUSIONS
The success of teaching of formal logic as a practical
tool has been mediocre. We pointed out issues that
may be problematic for students.
We suggested showing students examples where
intuition has led people astray, such as Wason’s selec-
help them reject incorrect intuitive rules of reasoning
in favour of rules of formal logic.
We suggested teaching students the following:
Trust on the principle that “if . . . then . . . holds if
and only if it has no counter-examples. The principle
of explosion follows from this, so accept it, although
it may seem counter-intuitive at ﬁrst. “If . . . then . . .
may mean material implication or a reasoning rule.
Use appropriately to capture the intended meaning.
Denote material implication with , and the similar
reasoning rule with . Use the phrases “mathemati-
cal consequence” and so on instead of “logical conse-
quence” and so on, because the latter do not assume
that 0, +, and so on have their standard meaning.
We presented conventions for reasoning operators
and the treatment of undeﬁned operations. They have
proven well-deﬁned and rigorous enough to be used in
educational software written by us (Valmari, 2021).
REFERENCES
Association for Computing Machinery (ACM) and IEEE
Computer Society (2013). Computer Science Curric-
ula 2013: Curriculum Guidelines for Undergraduate
Degree Programs in Computer Science. ACM, New
York, NY, USA.
Bronkhorst, H., Roorda, G., Suhre, C., and Goedhart, M.
(2020). Logical reasoning in formal and everyday rea-
soning tasks. International Journal of Science and
Mathematics Education, 18:1673–1694.
Carroll, L. (1894). A logical paradox. Mind, 3(11):436–
438.
Chalin, P. (2005). Logical foundations of program as-
sertions: What do practitioners want? In Aich-
ernig, B. K. and Beckert, B., editors, Third IEEE In-
ternational Conference on Software Engineering and
Formal Methods (SEFM 2005), 7-9 September 2005,
Koblenz, Germany, pages 383–393. IEEE Computer
Society.
Egr
´
e, P. and Rott, H. (2021). The Logic of Conditionals. In
Zalta, E. N., editor, The Stanford Encyclopedia of Phi-
losophy. Metaphysics Research Lab, Stanford Univer-
sity, Winter 2021 edition.
Gries, D. and Schneider, F. B. (1995). Avoiding the un-
deﬁned by underspeciﬁcation. In Computer science
today, volume 1000 of Lecture Notes in Comput. Sci.,
pages 366–373. Springer, Berlin.
Hammack, R. (2018). Book Of Proof. Self-published, 3rd
edition. Approved by the American Institute of Math-
ematics’ Open Textbook Initiative.
Jones, E. E. C. (1905). Lewis Carroll’s logical paradox.
Mind, 14(56):576–578.
Kahneman, D. (2011). Thinking, fast and slow. Farrar,
Straus and Giroux, New York.
Lethbridge, T. (2000). What knowledge is important to a
software professional? IEEE Computer, 33(5):44–50.
Mathieu-Soucy, S. (2016). Should university students know
about formal logic: an example of nonroutine prob-
lem. In INDRUM 2016: First conference of the Inter-
national Network for Didactic Research in University
Mathematics. hal-01337943.
Mokteﬁ, A. (2007). Lewis Carroll and the British
nineteenth-century logicians on the barber shop prob-
lem. In Cupillari, A., editor, Proceedings of The Cana-
dian Society for the History and Philosophy of Math-
ematics’ Annual Meeting, pages 189–199.
Niemel
¨
a, P., Valmari, A., and Ali-L
¨
oytty, S. (2018). Al-
gorithms and logic as programming primers. In
McLaren, B. M., Reilly, R., Zvacek, S., and Uho-
moibhi, J., editors, Computer Supported Education
- 10th International Conference, CSEDU 2018, Fun-
chal, Madeira, Portugal, March 15-17, 2018, Revised
Selected Papers, volume 1022 of Communications in
Computer and Information Science, pages 357–383.
Springer.
Ragni, M., Kola, I., and Johnson-Laird, P. (2017). The Wa-
son selection task: A meta-analysis. In Proceedings
of the 39th Annual Meeting of the Cognitive Science
Society, pages 980–985.
Reeves, S. and Clarke, M. (1990). Logic for Computer Sci-
Stefanowicz, A. (2014). Proofs and Mathematical Reason-
ing. University of Birmingham, Mathematics Support
Centre.
Valmari, A. (2021). Automated checking of ﬂexible
mathematical reasoning in the case of systems of
(in)equations and the absolute value operator. In
Csap
´
o, B. and Uhomoibhi, J., editors, Proceedings
of the 13th International Conference on Computer
Supported Education, CSEDU 2021, Online Stream-
ing, April 23-25, 2021, Volume 2, pages 324–331.
SCITEPRESS.
Valmari, A. and Hella, L. (2017). The logics taught and used
at high schools are not the same. In Karhum
¨
aki, J.,
Matiyasevich, Y., and Saarela, A., editors, Proc. of the
Fourth Russian Finnish Symposium on Discrete Math-
ematics, number 26 in TUCS Lecture Notes, pages
172–186. Turku Centre for Computer Science.
Valmari, A. and Hella, L. (2021). A completeness proof for
a regular predicate logic with undeﬁned truth value.
arXiv:2112.04436.
van Benthem, J. (2008). Logic and reasoning: Do the facts
matter? Studia Logica, 88(1):67–84.
Wason, P. C. (1968). Reasoning about a rule. Quarterly
Journal of Experimental Psychology, 20(3):273–281.
CSEDU 2022 - 14th International Conference on Computer Supported Education
524