A Comparison of Techniques for Introducing Material Implication
Matthew C. Clarke - March 1996
Senior Lecturer
Department of Computer Science and Information Systems
University of Natal, Pietermaritzburg
Department of Computer Science and Information Systems
University of Natal
Private Bag X01
Scottsville, 3209
South Africa
Phone (27) 331 260 5641
=46ax (27) 331 260 5966
E-mail clarkem@unpsun1.cc.unp.ac.za
Abstract
--------
A large volume of research shows that humans reason poorly
about conditional statements and that the formal notion of
material implication is difficult to learn. Textbooks on
logic have used a variety of approaches to the introduction
and justification of a truth-functional definition of
material implication. This presentation surveys seven such
techniques - definition by truth table, definitions based on
other logical operators, the use of examples, ways of
avoiding the need for a definition, an adaptation of
Peirce's notation, the analogy with contractual reasoning,
and finally I suggest an alternative based on elementary set
theory.
--------------------------------------------------------------------
Motivation
----------
Material implication is an attempt to capture the essence of
conditional statements: that is (in English at least), statements of
the form "if ... then ...". However, statements of this form have a
variety of intentions, not all of which are truth-functional. Thus,
no truth-functional definition of implication will be able to fully
capture the diversity of meanings in conditional statements.
A large volume of research shows that people do not naturally reason
well about conditional statements. For instance, studies with late
high-school students, college students and medical students by
O'Brien and Shapiro indicate that less than 10% of subjects reason
correctly about conditional statements [OBRI72, OBRI73, SHAP73].
=46urther support for this claim comes from the innumerable variations
of Wason's Four-Card Selection Problem [see, for instance GILH88
pp113-123, EVAN82 Chapter 9 and WASO83].
=46urthermore, it is difficult for many students to accept the truth-
functional definition of material implication. This is at least
partly because it violates their intuition about conditional
statements with false antecedents. To the student, it seems
ludicrous to suggest that when a statement of the form
"if ... then ..." has a false antecedent, the overall statement
should be considered to be true. The truth table for material
implication is not at all self-evident and must be accompanied by
some explanation.
Approaches to Explaining Material Implication
---------------------------------------------
Given the difficulties students encounter with conditional
statements and with understanding the definition of material
implication as a truth function, it is interesting to compare the
methods used to justify this definition. There are many approaches
to this - some indicate a primary concern with technical precision
while others show varying degrees of concern for avoiding or
alleviating student distress. After reading as many logic textbooks
as I could find, I have gathered these approaches into six
categories which are described and critiqued below. A seventh
approach is also suggested which I have not seen in any textbook.
1. Definition by Truth Table
----------------------------
Most commonly, material implication is defined by truth table or
some verbal equivalent such as "X-->Y is always true if X is false and
also if Y is true" [HILB50 p4] or "A conditional sentence is false
if the antecedent is true and the consequent is false; otherwise it
is true" [SUPP57 p6].
This definition is justified by the authors in various ways, though
frequently no justification is given at all [HILB50, COOL42, WERK48,
BELL77, ROBI79, MANN85(1), PAUL87, REEV90]. Quine claims that the
given truth table "constitutes the nearest truth-functional
approximation to the conditional of ordinary discourse" [QUIN40 p15]
and adds that this definition dates back to Philo of Megara(2).
[AHO92], [DOWS86] and [BASS70] take the same approach, admitting
that this truth function does not always match English usage. Suppes
takes a bold approach and simply states that in maths and logic this
is the way it is done! [SUPP57]
Shoenfield defines material implication as a function rather than as
a truth table, though the effect is the same. He claims that this
definition follows the "mathematical meaning of if ... then" [SHOE67
p11].
A variant on the truth table approach, shown in Figure 1, starts by
showing that sixteen distinct truth tables may be constructed for
two variables. After columns 1 (tautology), 16 (inconsistency), 2
(disjunction), 8 (conjunction) are discussed and named, the author
draws the reader's attention to column 5 and says in effect "this is
an interesting and useful column so let's give it a name as well".
This approach is taken by [JEFF67 p49] and [KORF74 p254]. Jeffrey
also comments "Except in odd cases the truth conditions for the
indicative English conditional are accurately given by the usual
truth table [i.e. Column 5 in Figure 1]" [JEFF67 pviii].
P Q | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
------+--------------------------------------------------
T T | T T T T T T T T F F F F F F F F
T F | T T T T F F F F T T T T F F F F
=46 T | T T F F T T F F T T F F T T F F
=46 F | T F T F T F T F T F T F T F T F
Figure 1 - Sixteen Possible Truth Tables
2. Definition in Terms of Other Operators
-----------------------------------------
Other authors define material implication as an abbreviation of some
other boolean expression. The normal form of this definition is
(P=3D=3D>Q) =3Ddef (~PvQ) [WHIT25, STEB46, STEB52, EATO31] while others use
(P=3D=3D>Q) =3Ddef ~(P&~Q) [QUIN41, COPI67, MITC62, CARN80, KIRW78, HOCU79].
Virtually all books show these equivalences at some point. Several
texts explicitly note that the two definitions are interchangeable,
and Ambrose and Lazerowitz make a major point of showing that not
only can material implication be defined in terms of negation and
conjunction, but one could equally well define material implication
as the primitive operation and then define disjunction and
conjunction in terms of negation and material implication(3) [AMBR48].
They prefer the definition (P=3D=3D>Q) =3Ddef ~(P&~Q) over
(P=3D=3D>Q) =3Ddef (~PvQ), saying that while the first makes good English
sense, the second looks problematic, even though the two are
provably equivalent [AMBR48 p75].
When Davis introduces the horseshoe operator ("=8A") he defines it to
be equivalent to both ~(P&~Q) and (~PvQ). Long before this, however,
he discusses non-truth-functional conditional statements (using the
symbols "=3D=3D>" and "-->") and presents the idea of material implication
in syllogistic arguments. [DAVI86]
Like the explicit definition by truth table, this form of definition
is sometimes not accompanied by any justification [WHIT25, STEB46].
In her later work, Stebbing notes that what ~PvQ defines is material
implication, which is not necessarily the same as the English "if
... then ..." structure [STEB52 p139].
3. Definitions Relying on Examples
----------------------------------
Several authors use selected examples (either in mathematics,
science or conversational English) to justify their definition of
material implication. Rosser uses some verbal trickery to show that
English phrases of the form "if A then B" are the same as "we cannot
have both 'A' true and 'B' false" [ROSS53, p15]. He also cites a
number of mathematical examples. Massey uses the example that "If my
memory is correct, then I owe you a dollar" means the same as
"Either it is false that my memory is correct or else I owe you a
dollar". This example suggests that P=3D=3D>Q is simply an abbreviation
for ~PvQ [MASS70 p52]. [See also GUTT71, KELL90, SAIN91 and PRIO55.]
Hermes claims that the structure ~(P&~Q) occurs frequently in
mathematics and that mathematicians express this as "if P then Q".
With this as justification, he defines the two to be equivalent
[HERM73]. Hamilton, writing specifically for mathematicians,
justifies his truth table definition with the example "if n>2 then
n^2>4" which, he says, is still a true statement even when n happens
to be less than two [HAMI78 p5].
Quine draws on an English example to convince the reader of his
claim that "if P then Q" is equivalent to ~(P&~Q), though he does
this without explicitly constructing truth tables [QUIN41 p20]. In a
later work, Quine writes -
An affirmation of the form "if P then Q" is commonly felt
less as an affirmation of a conditional than as a
conditional affirmation of the consequent. If, after we have
made such an affirmation, the antecedent turns out true,
then we consider ourselves committed to the consequent, and
are ready to acknowledge error if it proves false. If on the
other hand the antecedent turns out to have been false, our
conditional affirmation is as if it had never been made.
[QUIN82 p21]
Consequently, he claims that the choice of declaring a conditional
to be true whenever the antecedent is false is arbitrary. Kneebone,
who relies on truth tables to define material implication rather
than examples, makes a similar point -
The truth-values that are to be ascribed to f-->j in cases in
which f is false are unimportant, since we do not draw
conclusions from premises unless these are known to be true,
or at least assumed to be true for the sake of the argument;
but it greatly simplifies the formal logic of propositions
if we define the truth-value of f-->j in all cases, taking it
as T whenever f has the truth-value F (compare with such
conventional definitions in mathematics as a^0=3D1 and 0!=3D1).
[KNEE63 p31, emphasis mine]
Mendelson also claims that the truth-functional definition is simply
a convention, though it is also justified by the desire that (P&Q)=3D=3D>P
should be a tautology [MEND64 p13].
Korfhage uses a pedagogically fascinating, though technically
misguided analogy with computer programming to show why a
conditional should be treated as true if the antecedent is false. He
writes that given P=3D=3D>Q, where P is known to be false, it is true that
we can't deduce anything about Q, however, we don't want the
argument to stop there. Compare this with a fortran program
containing the statement "if alpha .gt. x+7 goto 13". Even if "alpha
.gt. x+7" is false, the overall statement is a good piece of fortran
and we want the program to continue running. Hence, so as not to
disrupt an argument, we assign P=3D=3D>Q the value true whenever P is
false. [KORF74]
This approach may appeal to students who already understand the
fortran "if" statement. However, there is some evidence that
children who have been taught the "if ... then ... else" programming
construct misconstrue conditional statements as bi-conditionals
[SEID89]. That is, after being exposed to the program language
interpretation of an "if" statement, they infer an incorrect truth
table for material implication. Korfhage's approach is technically
misguided since it confuses a form of conditional in which the
antecedent and consequent are causally connected ("if the condition
alpha .gt. x+7 is true then the next thing to do is execute the
instruction at line 13") with the truth functional form which
requires no causal connectivity.
The method of choosing or contriving an example which fits the
author's intention is rather artificial. A more sophisticated
approach is to note that the English "if ... then ..." structure is
used in a variety of senses and that since we need to use symbolic
operators unambiguously, we must choose just one of those senses.
According to Church, "we select the one use of the words 'if ...
then' ... in which they may be construed as denoting a relation
between truth-values" [CHUR56 p38].
Reichenbach, who uses a truth table to define the horseshoe
operator, makes a useful distinction between adjunctive implication
and connective implication.
It recently happened in Los Angeles that, while the screen
of a movie theatre was showing a blasting of lumber jammed
in a river, an earthquake shook the theatre. The implication
"the blasting of lumber on the screen implied the shaking of
the theatre" was then true in the adjunctive sense whereas
it was false in the connective interpretation. ... We
realise that the word "implies" here has not the same
meaning as in conversational language; the implication in
this case simply adjoins one statement to the other without
connecting the statements. Adjunctive implication has a
wider meaning than connective implication; if a connective
implication holds, there also exists an adjunctive
implication, but not vice versa. [REIC47 pp29,30]
Copi's "Introduction to Logic" has the most extensive variant of
this approach. After explaining how we choose one of the two senses
of the English "or" (the inclusive rather than exclusive sense), he
lists four senses of implication and then chooses the one which can
be written symbolically as ~(P&~Q). Choosing this interpretation
over the other three is not arbitrary. Rather, he shows that this
definition specifies the common ground between the four senses.
[COPI67 pp245-252]
Galton argues that the truth table definition is the minimal truth-
functional definition which will apply to all conditional
statements.
Even though --> may not capture everything that is implied by
"if ... then ...", at least we can say that a statement of
the form A-->B will be true whenever (if not more often than)
"if A then B" is true, so that the English inference with
statements of this form amongst its premises will be valid
so long as the propositional calculus translation is.
[GALT90 p57]
Georgacarakos and Smith devote many pages to this same point [GEOR79
p53ff].
4. Avoiding any Truth-Functional Definition
-------------------------------------------
Another way to tackle the problematic definition of material
implication is to avoid explicit definition altogether [FITC52,
LEMM71, POSP84]. Fitch uses a natural deduction system in which
conditional expressions may be manipulated by Modus Ponens and
Distribution(4). No mention is made of truth tables and (~PvQ)=3D=3D>(P=3D=
=3D>Q)
is left as an exercise for the reader [FITC52]. Lemmon follows a
similar path using the symbol =3D=3D> in proofs long before defining it as
a truth function. He uses a natural deduction system with ten Rules
of Derivation to prove that P=3D=3D>Q, ~PvQ and ~(P&~Q) are all
interderivable. When he eventually gets around to discussing truth
tables, it is then clear that P=3D=3D>Q should be defined to have the same
truth table as both ~PvQ and ~(P&~Q). Even so, he admits that the
truth table definition of material implication "seems rather
arbitrary" [LEMM71 pp67-68].
5. Using Peirce's Notation(5)
--------------------------
Another approach relates material implication to the mathematical
concept of less-than-or-equal-to. This is inspired by Peirce, who
used a modified "<" sign to stand for material implication in 1885
[HART74 paragraph 3.373]. The implication P=3D=3D>Q can be explained by
showing that the truth of Q is at least as certain as the truth of
P, because Q must be true whenever P is true, but Q can also be true
on other grounds independent of P. If the value "true" is
interpreted to be greater than the value "false", then the truth
table for material implication will be identical to the truth table
for less-than-or-equal-to.
This explanation may be especially useful for computer science
students since they will already have a mental correspondence
between "true" and "false" and the binary values 0 and 1.
6. Definition Based on the Idea of Contracts
--------------------------------------------
In [NISB87], Nisbett et. al. claim that an effective way to teach
conditional logic is to draw on pre-existing concepts rather than to
define an entirely new concept. The pre-existing concepts they
suggest are those of permission and obligation, both of which are
forms of contract.
The statement "In order to do some action P you must have permission
Q" follows precisely the same truth-function as P=3D=3D>Q. A contract of
permission is violated only when the action P is performed without
the required permission Q. If P is performed with permission Q, or
if P is not performed at all, then the permission contract stands
unviolated.
The statement "If you perform some action P then you are obligated
to do Q" follows the same pattern. Such an obligation is violated
only when P occurs but not Q, and hence an obligation schema behaves
the same as material implication.
Note that permission and obligation are not presented simply as
examples as described in the third category above. Rather, the aim
is to proffer these to students as inference schema with which they
are already well acquainted, and to indicate that processing
conditional statements should be undertaken using those same schema.
7. Definition Based on Elementary Set Theory
--------------------------------------------
If one can assume that students have an understanding of the basic
concepts of set theory, then those concepts can be matched with
parallel concepts in boolean logic. Negation can be explained as the
logical counterpart to set complementation; conjunction as the
counterpart to intersection; disjunction as the counterpart to
union(6); material implication as the counterpart to the subset
relation; and the bi-conditional as the counterpart to set
equivalence. This approach does not imply that there is a formal
equivalence between set expressions and propositional expressions,
but rather that set operators invoke the same reasoning schema as
propositional operators.
Students familiar with set concepts will understand the assertion
P[subset]Q to mean that every element of P is also an element of Q. But
this is a disguised form of the assertion "if x is an element of P
then x is an element of Q", or symbolically, x[member]P=3D=3D>x[member]Q.=
Thus, an
appropriate explanation of the subset relationship can provide a
very clear justification of a truth-functional material implication.
Such an explanation may proceed as follows. Consider four arbitrary
members of the universe x1,x2,x3 and x4 such that x1[member]P and x1[member]=
Q,
x2[member]P and x2[not member]Q, x3[not member]P and x3[member]Q, x4[not
member]P and x4[not member]Q. Three of these (x1,x3
and x4) are consistent with the assertion P[subset]Q, but the fourth (x2)
is inconsistent with that assertion. The claim P[subset]Q can only be shown
to be false by finding an object which is a member of P but is not a
member of Q. By analogy, the claim P=3D=3D>Q is only false in the
circumstance where P is true but Q is not true. The truth table for
P=3D=3D>Q will have three rows (corresponding to x1,x3 and x4) marked as
true and one row (corresponding to x2) marked as false.
I have not seen a textbook which explicitly links the definition of
logical operators to set operators in this way, but I have found
this approach effective in the classroom. The approach has several
advantages -
* Since students already have a grounding in set theory, the
approach defines new concepts in terms of familiar concepts.
* The visual model of set structures given by Venn Diagrams can
immediately be transferred as a tool to aid understanding of
logical expressions.
* The definition of material implication is no longer "arbitrary",
as some of the authors quoted above apologetically assert. The
truth table for material implication can be shown to follow
naturally from the subset relationship P[subset]Q which unambiguously
disallows the situation where membership of P is true but
membership of Q is false, and allows the other three possible
situations.
Conclusion
----------
When introducing the concept of material implication to students, a
teacher may choose from a number of techniques. Seven such
techniques have been described in this paper and we have noted that
some emphasise technical correctness while others more directly
promote student understanding.
Our suggestion is that defining material implication by truth table
provides insufficient explanation or justification for many
students. Such definitions may enable students to manipulate logical
expressions correctly, but with only a surface-level understanding
of what they are doing. Techniques such as definition based on the
subset relationship and definition based on the idea of a contract
should be used to promote deeper conceptual understanding.
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------------
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End Notes
---------
1. Manna and Waldinger have the added novelty of defining a truth
table for "if ... then ... else"! [MANN85, p13]
2. Peirce provides a summary of the debate between Philo and Diodorus
on whether hypothetical propositions (i.e. conditionals) are at all
different from categorical propositions. Philo claims (and Peirce
agrees) that the forms "If P then Q" and "Every P is Q" are
identical, but Diodorus (supported by Peirce's contemporary Shr=F6der)
claims they have different meanings. [HART74, paragraph 3.439ff,
written in 1896]
3. There is at least one book which actually takes this approach -
[BELL77].
4. The rule of Distribution may be symbolised as P=3D=3D>(Q=3D=3D>R) [there=
fore]
(P=3D=3D>Q)=3D=3D>(P=3D=3D>R).
5. Suggested by John Sowa pers.
comm.
6. The English word "or" is sometimes interpreted inclusively and
sometimes exclusively, and when introducing logical connectives one
must somehow address the issue of which way to interpret
disjunction. Various approaches are used in textbooks, but a key
advantage of presenting disjunction as the counterpart to union is
that it provides a very natural rationale for making disjunction
inclusive - just as the set A[union]B includes elements which are common
to both sets, so AvB is true even when both disjuncts are true.