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A STANDARDIZED LOGIC NOTATION FOR EVERYDAY CLASSROOM USE

Abstract: There are two main purposes of this paper. The first is to describe a simple and intuitive standardization of

logic notation, both for foundational courses which develop symbolic logic and proof styles, and for general everyday

use in undergraduate and even high school mathematics courses, particularly calculus though there is no reason

students cannot learn it and benefit from it in their algebra and geometry courses as well. This leads to the second

purpose of this paper, which is to argue that courses which do not usually include logic notation formally could

nonetheless benefit from widespread use of it, particularly if it is standard and thus truly portable. Several examples

are given. (The author assumes the reader has seen elementary treatments of logic symbols and truth tables, for which

there are many simple texts and online sources, including the author’s own homepage listed at the end of the paper.)

For students studying undergraduate mathematics, it is common and proper to encounter

logical symbols, such as connectives , Λ, V, →, and ↔, in courses specifically

containing logic and truth tables as topics, in other courses relying on the ability to

manipulate these symbols, or just incidentally as an instructor uses them to help

communicate mathematical ideas in lectures for any course. Some introduction to these

symbols, along with truth tables, historically occurred in public education in other

countries such as Taiwan as a standard topic by the end of secondary school. The

usefulness and beauty of symbolic logic often motivate U.S. instructors to introduce the

subject at some low level to students, even in the secondary schools, though it is rare for

the actual classroom texts to discuss abstract, symbolic logic except in courses involving

proofs, and then only well into the university level. In particular, algebra and calculus

texts almost completely avoid the issue, apparently appealing instead to the students’

“common sense.”

Done well, the use of symbolic logic can bring much needed clarity to a topic or course,

while done casually or incoherently it can deepen confusion. To further confound

matters, there is no universal standard for notation or its use, so what little a student

absorbs in one course might not be as portable as he or she would like. I address all of

these problems with my proposed standard, but I first address the problems themselves in

more detail.

Most authors and instructors who employ symbolic logic in their teaching do use the

connectives , Λ, V or minor variations similarly. However, some then use ⟹ and ⟺ where others use → and ↔. To indicate logical equivalence of compound statements,

some authors use the symbol ≡ (used elsewhere to indicate function identities or equality

by definition), while others use ↔, ⟺, or “iff” meaning “if and only if.” (Other

notations can also be found but are less common.) In lecture notes, it is not unheard of to

see ⟹ when ⟺ is also correct and therefore more precise (or even intended), and when

it would be very instructive to the students to see the differences between the two;

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similarly if ↔ and → are used instead. Indeed, often in some computational argument a

string of equivalence is broken, as in f (x) = x ∙ x ⟺ f (x) = x2 ⟹ f ′(x) = 2x , where of

course we cannot replace ⟹ with ⟺. In a calculus context it seems inadvisable to use

the symbol → in-line because of its ubiquitous use in limit arguments. Even when we

restrict the discussion to the use of logic symbols and their meanings, we have contexts

where P → Q is supposed to be understood to be a tautology (always true), and others

where it is meant to have truth values that may be T or F, depending upon the context.

Similarly with P ↔ Q.

The simple notational proposal put forward here is this:

That we use → and ↔ as connectives, i.e., logical operations, which can

return either truth values T or F, and ⟹ and ⟺ only when the

corresponding connectives return tautologies.

This is not wholly original (see for instance Finite Mathematics and Its Applications,

Goldstein, Schneider, and Siegel, Prentice Hall), though I developed the convention

independently (being later pointed to the earlier development), arguably because it seems

like the next reasonable step in everyday logic notation that can be used throughout

mathematical presentation and writing within the classroom. However, while it has been

tested and used in my Calculus 1 course repeatedly, it has clearly not yet gained the

promulgation to or acceptance in the classroom advocated here. This seems to be because

of its present obscurity, perhaps due in part to the fact that most professional

mathematicians and mathematics educators who have had some exposure to logic and

truth tables understand an author’s intent in using logic symbols from the context, and so

one standard (or even no standard) might seem as legitimate as another. It is often only

after attempting to use them coherently in the classroom that one discovers we mean, for

instance, different things by “implies” in different contexts. It is my hope that some

demonstrations of this standard’s usefulness in calculus, and even algebra, as seen here

will motivate instructors to consider adopting it in the future.

The connectives will be referred to here as “single-line arrows” and the other symbols as

“double-line arrows.” Thus one may write

P ↔ Q ⟺ (P → Q) Λ (Q → P) (1)

P ↔ Q ⟹ P → Q. (2)

While the first of these (1) would normally be proved using truth tables by showing that

P ↔ Q and (P → Q) Λ (Q → P) have the same truth values for each of the 22 = 4 possible

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combinations of truth values of the P and Q, it should be noted that it can also be proved

by showing that [P ↔ Q] ↔ [(P → Q) Λ (Q → P)] is a tautology:

P Q P ↔ Q P → Q Q → P (P → Q)

Λ (Q → P)

[P ↔ Q]

↔ [(P → Q)

Λ (Q → P)]

T

T

F

F

T

F

T

F

T

F

F

T

T

F

T

T

T

T

F

T

T

F

F

T

T

T

T

T

Of course it is easier, and time saving, to note simply that the third and sixth columns of

the truth table have the same value, and to spare the effort of producing the final column.

However it is worth noting that showing two compound statements have the same truth

values in all cases is equivalent to showing that the connective ↔ operating on the

compound statements will always return T.

To rephrase, the spirit of the longer method of showing equivalence is the notion that ⟺ means that if its place is taken by ↔ we get a tautology, after adding necessary grouping

symbols. This can be extended to implications, in which ⟹ is taken to mean that, if replaced by →, we get a tautology. So to prove (2) we construct a truth table as follows

(note how the second and third cases have ( )→( ) returning T “vacuously”):

P Q P ↔ Q P → Q (P ↔ Q) → (P → Q)

T

T

F

F

T

F

T

F

T

F

F

T

T

F

T

T

T

T

T

T

When verbalizing → versus ⟹ in the classroom, I usually simply use “implies” if there

is no ambiguity, and “single-line implies” versus “double-line implies” if both are

present. Perhaps better diction can be developed eventually, such as “implies” versus

“strongly implies.” For ↔ versus ⟺, one could use “bi-implies” versus “if and only if,”

while I am careful to reserve “is equivalent to” for ⟺. Another acceptable terminology

could use “logically implies” and “is logically equivalent to” for ⟹ and ⟺, respectively,

though no doubt there are other good terminologies possible. The symbols , Λ, and V

of course are verbalized as “not,” “and” and “or,” respectively.

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Now consider compound statements P = P (P1, P2..., Pn) and Q = Q (P1, P2..., Pn), where

the P1, P2..., Pn are assumed to be independent component statements, with 2n possible

truth value combinations of T and F. When it is appropriate to write P ⟺ Q, or P ⟹ Q

for compound statements P , Q, I call these, respectively, a valid equivalence and a valid

implication. Otherwise they would be fallacies, and perhaps, unlike statements using the

connectives ↔ and →, we would usually avoid writing the fallacies (just as we avoid

writing 2 = 3) except for demonstration purposes. We can use this notation to display

valid argument styles in-line. For instance consider modus tollens, i.e.,

which becomes (P → Q) Λ ( Q) ⟹ P. A check with a truth table construction shows

[(P → Q) Λ ( Q)] → ( P) is indeed a tautology. (I have found that using extra

“grouping” characters ( ) and [ ] expedites the learning of logic by my calculus students,

and thus defers the whole discussion of an “order of operations” until they are more

comfortable with the notation. Note that in written English the standard grouping

symbols are single and double quotation marks, which limits the practical level of nested

grouping available. Of course spoken English is often quite ambiguous.)

The symbols ⟺ and ⟹ are robust enough to use in many contexts, including algebraic

contexts, in which they can very much enhance student understanding of the logical

nature of simple mathematics. For example,

Students witness where we “broke” our equivalence, allowing us (at this point) only to

conclude that , and thus we have a need to

“check” our candidate answers, discovering that will work in the original while

will not, so we finally conclude . Of course this kind

of thing happens all the time, and in fact we can prove (as above) that P ⟹ P V Q, as for

example .

We can also point to cases in algebra and elsewhere for which there is no need to check

solutions (except for actual errors), as for instance when we have a polynomial equation

that we solve by factoring, or a linear equation we solve by steps which only yield

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equivalent statements. Knowing when we do not need to check the solutions except for

errors can be a useful skill in its own right.

At this point in the discussion it is interesting to show students how, if we have P ⟹ Q

and Q ⟹ P, then we can write P ⟺ Q. (It is reasonable on its face, and can also be

shown with creative use of truth tables.) This fact is also often useful, such as when we

show how the three “row operations” in Gauss-Jordan elimination produce systems with

the same solutions, because the operations are reversible. More specifically it is easy to

show that a solution of the original system is contained in the solution of the transformed

system, and that “going backwards” we see the inverse operation of the transformation

gives that the solution of the transformed system is contained in the original system’s

solution. However, differentiation–much like squaring both sides of an equation–is not

quite reversible.

Similarly, . This simple example reminds

students of two important, often forgotten ideas: that squaring both sides of the equation

can lose important information about the signs of the original expressions (the

technicality which can be buried in the process, as in our original algebraic example (3)),

and that . Students are also interested to note that we do not have

the same difficulty when cubing both sides: , assuming

.

More simple examples where we lose equivalence can be found in solving logarithmic

equations, such as the following:

A quick check shows is not a solution while is a solution. We lost

equivalence in combining the logarithms, since when we have

while . In fact, the first equivalence is only valid because (for the

direction ).

When dealing with quantifiers, it also seems appropriate to keep the single-line-arrow

connectives “inside” the quantified statements. For instance, since ,

we can write

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.

Perhaps less clear at first is the one-way nature of the implication

.

Once students learn about negating quantified statements and implications P → Q the

following is relatively easy, assuming f is defined:

In fact, to be precise we should note that is actually a statement about

f being defined at x and a, as well as the inequality holding true. Thus we can replace

in the continuity with the equivalent statement

or simply understand that it can be so replaced, and then the negation will include

common cases where is undefined, is undefined or . This

uses , and the obvious extension to three statements:

, which are unambiguous because of the

associative natures of and .

Of course, quantified statements do not conform to the usual truth table based analysis.

Indeed, a quantified statement is either true or false, i.e., a tautology or a contradiction. In

this standard the symbol is appropriate between any two “true” quantified statements,

or between any two “false” quantified statements.

It can also be useful to define a symbol to represent a generic tautology, and another for a

generic contradiction. For these purposes I define “script-T” and “script-F,” or and ,

respectively. For instance, ,

and so on. These allow for some interesting computations. For instance, if we wish to

reason one level above truth tables, we can use well known equivalences to prove say,

by showing :

Of course there are some results which surprise students at first, such as

, and . However some are quite reasonable, such as

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, which is perhaps best shown with a truth table, which will have

possible truth value combinations to check, since can have two truth values while

has only one:

T

F

F

F

F

T

F

T

T

T

In fact, a quick glance at the truth table reveals that , another interesting

computational result for student meditation.

I have already begun to make my case for the second proposition offered here, which is

this:

That secondary or early undergraduate mathematics education include a

standard and coherent introduction to symbolic logic, to improve the

clarity of the material and general discussion within textbooks, lecture

discussions and homework.

I have already pointed out how some confusing topics in algebra and calculus can be

clarified by the arrows showing the flow of implication, whether it be two way or

one way . In my own calculus courses over the last four years, I have seen enough

benefits to justify spending four complete lectures on an outline of symbolic logic

notation and its use. The outline I have used for those four lectures is as follows:

1. the operators and truth tables with examples from everyday life;

2. logical equivalence with several examples which are mostly intuitive upon

reflection;

3. valid implications and arguments along with symbols for generic tautologies

and contradictions ;

4. and finally a short introduction to quantified statements and their negations, some

from mathematics and some from everyday life, such as how to negate “for every

man there is a woman who loves him,” using quantifiers. (Students seem to

benefit from such contextual examples in all four lectures.)

With these and a few other innovations which work well with the notation, much clarity

can be gained. For instance, the definitions of continuity and a finite limit at a point are

less mysterious after some brief experience with quantifiers:1

1 One could also argue that we should make some distinction between equivalence by definition and regular logical equivalence. This suggests a possible third standard, that be only used for definitions, which is similar in spirit to its use when we say, for instance, in other contexts:

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f continuous at

.

From here students can see how the limit definition is silent at the point (since

means , and thus is vacuously

true), while continuity includes the case . They can also see how the part of in the

limit definition is played by in the continuity definition. Both definitions cut

through the seemingly ad hoc nature of the explanations one often needs to demonstrate

the idea of limits in all the various cases without using - definitions. Many questions

can be answered by referring back to the definitions, thus achieving coherence across

examples, at least for the cases of a finite limit at a point.

In fact I have used symbolic logic in - proofs for continuity at points with some

additional success (compared to not using it) in Calculus 1. For instance, suppose I wish

my students to prove that the function is continuous at the point .

The exercise is broken into two parts: the “scratch work” and the “proof.”

Scratch-work: We want to follow from our choice of . We work

backwards from that statement, with .

Now many texts explain that because we have equivalences, we found our . However

this is not a good strategy if we look ahead to nonlinear functions, so instead it is useful

to write a stand-alone proof.

.

Of course, where we have “ ” as a logic symbol, we also have “ .”

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Proof: For any choose . Then exists and satisfies

Within the proof, students can trace the definition of continuity piece-by-piece, though

some “pieces” are longer than others.

Suppose instead we wish to prove that is continuous at . We can attempt

the same strategy but it quickly falls short, until we remember which direction of “ ”

we actually need.

Scratch-work: Here and . We therefore want to choose such that

Working backwards as before we get

Of course is not constant, so we are stuck until we realize the direction we really

need is in the above i.e., We can also have

statements before and after our antecedent if the arrows continue to flow the

correct direction. Of course we accomplish our implication by employing some a priori

assumption, for instance that . Then

If this last quantity is less than when , we will have our proof, and that can

happen if , and this last part is true if . We note that we keep

thinking, “that can happen if,” and so we are in fact working backwards again from

, except that (1) we do not have equivalence, but our “arrows” are

pointing towards , and (2) we in fact require two things to be present

to accomplish . Indeed, we need to accomplish both

, and we do this by taking . From there we have our proof.

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Proof: For , choose . Then exists and satisfies

The level of sophistication required of calculus students to understand such a proof, and

moreover reproduce such a proof, is less remarkable when the mathematics is put

precisely and concisely with the aid of such a system of symbolic logic. Unfortunately

topics such as - proofs are more and more commonly excluded in high school calculus

since the Advanced Placement calculus exams no longer cover these. In both high school

and college, this loss dulls the aura of sophistication of the calculus. I am left to conclude

that the successful re-introduction of these topics, particularly with the aid of this system

of symbolic logic, has the potential to improve the mathematical sophistication of our

students and to prepare them better for more challenging mathematics in future studies.

However, there are pitfalls to avoid when introducing symbolic logic to students for the

first time in, say, a first-semester engineering calculus or a senior high school calculus

course. First, it should be done with some completeness; a piecemeal introduction,

particularly “on the fly,” will likely cause more confusion with most students than it is

worth. Second, while most students will enthusiastically embrace symbolic logic, they

will also need some correction, as the students often write “ ” where what is meant is

“=,” for instance, so some patience is necessary, but in that sense symbolic logic is no

different from other topics. Third, some theorems and arguments instructors take for

granted can have surprisingly sophisticated logical structure when forced into symbolic

form, and the instructors will need to work these things out carefully before presenting

them. While students can learn to produce an - proof of continuity for a linear case

reasonably quickly (though not immediately), it becomes much more interesting for the

nonlinear cases, where must be restricted a priori. Part of the solution is to keep the

“scratch-work” seeking and the actual - proof of continuity separate, which

is less necessary but still a good practice for the linear case. (This practice of separating

scratch-work from the final proof is hardly unique.) Fourth, some rewriting of lecture

notes and some reshuffling of order and priorities may be required to take full advantage

of the notation. I have had reasonable success by introducing the more intuitive (in this

context) topic of continuity before the less intuitive limits, requiring some - proofs of

continuity at points (the hardest part of the entire semester for some), then defining

continuity on intervals and giving the usual theorems there (including the Intermediate

Value Theorem and using it to solve inequalities), and then “breaking” the continuity and

introducing limits to describe the behavior of discontinuous functions, as well as the

continuous ones, but mostly excluding - proofs for limits in the exercises. Finally,

some time is lost initially by including a primer on logic, so some consolidation may be

required. For instance, I often introduce some easily grasped calculus facts in earlier

contexts than usual--such as the fact that on an interval implies is (strictly)

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increasing on that interval while implies decreasing--and use these for some

simple graphing problems long before the Mean Value Theorem is introduced to prove

them. In other words, introducing logic does not have to turn calculus into rigorous

analysis, though it naturally lends itself to increasing the rigor somewhat. An instructor

can judge the level of rigor appropriate for individual topics and classes.

These pitfalls are also arguments for introducing symbolic logic earlier, perhaps in

geometry but one could argue it is not inappropriate even within Algebra 1. Indeed, it is

a topic which stands alone, as many who teach mathematics for liberal arts students

discover when teaching college mathematics. Students who are afraid of algebra are not

necessarily afraid of symbolic logic. While exposure to symbolic logic might not be a

sufficient cure for this fear of algebra, it can help students to think mathematically, which

bodes well for their future success.

In the end, an introduction to logic symbols and their manipulations can help students to

clarify their own thinking, help illuminate the flow of a mathematical argument, and

better illustrate the idea that mathematics is a language of sorts. This is assuming the

logic is introduced in a somewhat complete and coherent fashion. It would also help if it

were “portable,” i.e., standardized, as well. Simply using as connectives, and

when the respective connectives would return tautologies as well as between

quantified statements where appropriate, can do much towards standardizing the

classroom-level use of symbolic logic, with the potential to make symbolic logic a

powerful tool for mathematics instruction.

One criticism of this approach is that we cannot expect students to perform the

calculations better it they are constantly worrying whether they should connect them with

. This criticism has some merit. A possible relaxation of the approach would be

to use throughout the process, until the problem is more or less complete, and then to

step back and notice where we in fact have . Even if is never used in a particular

classroom, using can give direction on the road map of a given argument (though then

the opportunity to explain symbolically why we sometimes must “check” the apparent

solutions, and other times we need not except for errors, is lost). When useful, the

standard presented here gives the instructor some structure to fall back upon, even if the

symbols are only resorted to when a worked solution can use some clarification.

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I would like to thank the reviewers for their comments and suggestions, one of which was

inserted almost verbatim into the text. This feedback certainly improved the content of

the manuscript. I would also like to thank Jacob Pool for typesetting my LaTeX

manuscript into Microsoft Word format for this publication. Readers wishing to see this

standard written more for student consumption are referred to my calculus textbook

project, Chapter 1, which can be found from my homepage

http://faculty.swosu.edu/michael.dougherty.