With the help of the word "not" one forms the NEGATION of any sentence; two sentences, one of which is the negation of the other, are called CONTRADICTORY SENTENCES. In sentential calculus, the word "not" is put in front of the whole sentence, while in everyday language it is customary to place it with the verb, or should it be desirable to have it at the beginning of the sentence, it has to be replaced by the phrase "it is not the case that". For example, the negation of the sentence:
Whenever we utter the negation of a sentence, we intend to express the idea that the sentence is false; if the sentence is actually false, its negation is true, while otherwise its negation is false.
When two (or more) sentences are joined by the word "and" , the result is their so-called CONJUNCTION, or, LOGICAL PRODUCT; the sentences which are joined in this manner are called the MEMBERS OF THE CONJUNCTION or the FACTORS OF THE LOGICAL PRODUCT. If, for instance, the sentences:
are joined in this way, we obtain the conjunction:
When we assert the conjunction of two this is tantamount to stating that both sentences of which the conjunction is formed are true. If this is actually the case, then the conjunction is true, but if at least one of its members is false, then the whole conjunction is false.
When joining sentences by means of the word "or", one obtains the DISJUNCTION of those sentences, which is also called the LOGICAL SUM: the sentences forming the disjunction are called the MEMBERS OF THE DISJUNCTION or the SUMMANDS OF THE LOGICAL SUM. Now, in everyday language, the word "or" has at least two different meanings. Taken in the so-called NON-EXCLUSIVE MEANING, the disjunction of two sentences expresses only that at least one of these sentences is true, without saying anything as to whether or not both sentences may be true; taken in another meaning, known as the EXCLUSIVE one, the disjunction of two sentences asserts that one of the sentences is true but that the other is false. To illustrate, let us suppose we see the following notice put up in a bookstore:
Customers who are teachers or college students are entitled to a special reduction.
Here the word "or" is undoubtedly used in the first meaning, since one would not refuse the reduction to a teacher who is at the same time a college student. On the other hand, if a child has asked to be taken on a hike in the morning and to a theater in the afternoon, and we reply:
no, we shall go on a hike or we shall go to the theater,
then our usage of the word "or" is obviously of the second kind, since we intend to comply with only one of the two requests. In logic and in mathematics the word "or" is used always in the first, non-exclusive meaning; the disjunction of two sentences is considered true if at least one of its members is true, and otherwise false. For instance, we may assert:
every number is positive or less than 3,
although there are numbers which are both positive and less than 3. In order to avoid misunderstandings it would be expedient, in everyday as well as in scientific language, to use the word "or" by itself only in the first meaning, and to replace it by the compound expression "either ... or ... " whenever the second meaning is intended.
Even if we confine ourselves to those cases in which the word "or" occurs in its first meaning, we find quite noticeable differences between its usage in everyday language and that in logic. In common language two sentences are joined by the word "or" only when they are in some way connected in form and content. (The same applies, though perhaps to a lesser degree, to the usage of the word "and".) It is not altogether clear what kinds of connections would be appropriate here, and any attempt at their detailed analysis and description would lead to considerable difficulties. As we shall see, such connections are disregarded in contemporary logic, where consequently one has to allow some strange examples; and indeed, anybody unfamiliar with its language would presumably be little inclined to consider a phrase such as:
2. 2 = 5 or New York is a large city
as a meaningful expression, and even less so to accept it as a true sentence.
Moreover, the usage of the word "or" in everyday speech is influenced
by certain psychological factors. Usually we state a disjunction of two
sentences only if we believe that one of them is true but wonder which
one. For example, if we look upon a lawn in normal light, it will not enter
our mind to say that the lawn is green or blue, since we are able to affirm
something simpler, and at the same time stronger, namely that the lawn
is green. Sometimes we even take the utterance of a disjunction as an
admission by the speaker that he or she does not know which member of
the disjunction is true, and which is false. And if we later arrive at the
conviction that the speaker knew at the time that one - an, specifically,
which - of the members was false, we are inclined to look upon the whole
disjunction as a false sentence, even though the other member should be
undoubtedly true. For instance, let us imagine that a friend of ours, upon
being asked when he will leave town, answers that he is going to do so
today, tomorrow, or the day after. Should we later ascertain that, at that
time, he had already decided to leave the same day, we shall probably get
the impression that we were deliberately misled and that he told us a lie.
When creators of contemporary logic were introducing the word "or"
into their considerations, they desired, perhaps subconsciously , to simplify
its meaning; in particular, they endeavored to render this meaning clearer
and independent of all psychological factors, especially, of the presence or
absence of knowledge. Consequently, they decided to extend the usage of
the word "or", and to consider the disjunction of any two sentences as
a meaningful whole, even when no connection between their contents or forms
should exist: and they also decided to make the truth of a disjunction
like that of a negation or a conjunction - dependent only on the truth
of its members. Therefore, a person using the word "or" according to
contemporary logic will consider the expression given above:
2. 2 = 5 or New York is a large city
as meaningful and in fact a true sentence, since its second part is surely true. Similarly, if we assure that our friend, who was asked about the date of his departure, used the word "or" in its strict logical meaning, we shall be, compelled to regard his answer as true, independently of our opinion as to his intentions.
As in the case of disjunctions, considerable differences between the usage
of implications in logic and in everyday language manifest themselves.
Again, in ordinary language, we tend to join two sentences by the words
"if ... then ..." only when there is some connection between their forms
and contents. Connections which would be suitable for this purpose are
hard to characterize in a general way, and only at times is their nature
relatively clear. In particular, we often expect a given implication to be
such that the consequent follows necessarily from the antecedent, that is
to say, that if the antecedent is true under certain circumstances, then
the consequent must also be true under the same circumstances (and that
possibly we can even deduce the consequent from the antecedent on the
basis of some general laws which we might not always be able to quote
explicitly). And here, as before, an additional psychological factor manifests
itself; usually we formulate and assert an implication only if we have no
exact knowledge as to whether or not the antecedent and the consequent
are true. Otherwise the use of an implication seems unnatural, and doubts
may arise with regard to its meaningfulness and its truth.
The following example may serve as an illustration. Let us consider the
law of physics:
every metal is malleable,
and let us put it in the form of an implication containing variables:
if x is a metal, then x is malleable.
If we believe in the truth of this universal law, we must also believe in the
truth of all of its particular cases, that is, of all implications obtainable
by replacing "x" by names of arbitrary materials such as iron, clay, or
wood. And, indeed, it turns out that all sentences which are obtained in
this way satisfy the conditions given above for a true implication; it never
happens that the antecedent is true while the consequent is false. In every
implication of this kind, moreover, there is a close connection between the
antecedent and the consequent, which depends on their having the same
subject. We notice, finally, that by assuming the antecedent of any of such
implications to be true, we can deduce the consequent (for instance, from
"iron is a metal" we can deduce "iron is malleable") by referring to
the general law that every metal is malleable.
Nevertheless, some of the sentences which we have been led to consider
seem artificial and doubtful from the point of view of common language.
One would not hesitate to use the universal law given above nor to use any
of the particular cases which are obtained by replacing "x" by the name
of a material, as long as one does not know whether it is a metal or whether
it is malleable. But if we replace "x" by "iron", we are
confronted with a
case in which the antecedent and the consequent are certainly true; then
the implication seems indeed artificial, and we prefer to use, instead, an
expression such as:
since iron is a metal, it is malleable.
Similarly, if for "x" we substitute "clay", we obtain an implication with a false antecedent and a true consequent, and we are inclined to replace it by the expression:
although clay is not a metal, it is malleable.
And finally, replacing "x" by "wood" results in an implication with a false antecedent and a false consequent; in this case, if we want to retain the structure of an implication, we should alter the grammatical form of the verbs:
if wood were a metal, it would be malleable.
The logicians, with due regard to the needs of scientific languages, adopted the same procedure with respect to the phrase "if ... , then ..." as they had done in the case of the word "or" . They decided to simplify and to clarify the meaning of this phrase and to free it, from extraneous factors. For this purpose they extended the usage of this phrase, by considering an implication as a meaningful sentence, even if no connection whatsoever exists between its two members, and they made the truth or falsity of an implication dependent exclusively upon the truth or falsity of the antecedent and of the consequent. To characterize this situation briefly, we say that contemporary logic uses IMPLICATIONS IN THE MATERIAL MEANING, or simply, MATERIAL IMPLICATIONS; this is opposed to the usage of the concept of IMPLICATION IN THE FORMAL MEANING, or, of FORMAL IMPLICATION, in which case the presence of a certain formal connection between the antecedent and the consequent is an indispensable condition for the meaningfulness and the truth of the implication. The concept of formal implication perhaps has not been made completely clear, but at any rate, it is narrower than that of material implication; every meaningful and true formal implication is at the same time a meaningful and true material implication, but not vice versa.
In order to illustrate the foregoing remarks, let us consider the following four sentences:
if 2.2 = 4, then New York is a large city;
if 2.2 = 5, then New York is a large city;
if 2.2 = 4, then New York is a small city;
if 2.2 = 5, then New York is a small city.
In everyday language, these sentences would hardly be regarded as meaningful, much less as true. From the point of view of mathematical logic, on the other hand, they are all meaningful, the third sentence being false, while the remaining three are true. Of course, we do not thereby suggest that sentences like these are particularly relevant from any viewpoint whatever, or that we want to apply them as premisses in our arguments.
It would be a mistake to think that the difference between everyday language and the language of logic, which has been brought to light here, is fixed and definite, and that the customs outlined above, regarding the usage of the words "if ... then ..." in common language, do not admit any exceptions. Actually, the usage of these words fluctuates to some extent, and if we look around, we can find cases which trespass the sense of formal implication. Let us imagine that a friend of ours is confronted with a very difficult problem and that we do not believe that he will ever solve it. We can then express our disbelief in a jocular form by saying:
if you solve this problem, I shall eat my hat.
The sense of this utterance is quite clear. We have here an implication whose consequent is undoubtedly false; and since we affirm the truth of the whole implication, we therefore, at, the same time, reject the truth of the antecedent; that is to say, we express our conviction that our friend will fail to solve the problem in which he is interested. But it is also quite clear that the antecedent, and the consequent of our implication are not connected in any way, so that we have a typical case of a material and not of a formal implication.
The divergency between the usage of the phrase "if ... , then ..." in
ordinary language and its usage in mathematical logic has been at the root
of lengthy and even passionate discussions, - in which, by the way,
professional logicians took only a minor part.(3)
(It is perhaps surprising, that considerably less attention was paid to
the analogous divergency in the case of the word "or".)
It has been objected that logicians, on account of
their adoption of the concept of material implication, arrived at paradoxes
and even at plain nonsense. This has resulted in an outcry for a reform of
logic, and in particular, for bringing about a far-reaching rapprochement
between logic and ordinary language with regard to the use of implication.
It, would be hard to grant that these criticisms are well founded. There is
no phrase in ordinary language which has a precisely determined meaning.
It would scarcely be possible to find two people who would use every word
with exactly the same meaning, and even in the language of a single person
the meaning of a given word may vary from one period of the person's life
to another. Moreover, the meaning of words of everyday language is usually
very complicated; it depends not only on the external form of the word,
but also on the circumstances in which it is uttered, and sometimes even on
subjective psychological factors. If a scientist wants to transfer a concept
from everyday life into a science and to establish general laws concerning
this concept, he (or she) must always make its content clearer, more precise,
and simpler, and free it from inessential attributes. it does not matter here
whether he is a logician who is concerned with the phrase "if ... then ..."
or, for instance, a physicist wanting to establish the exact meaning of
the word "metal". In whatever way the scientist realizes his task, the
resulting
usage of the term will deviate more or less from the practice of everyday
language. If, however, he states explicitly in what sense he decides to use
the term, and if afterwards he acts always in accordance with this decision,
then nobody will be in a position to object, or to argue that his procedure
leads to nonsensical results.
Nevertheless, in connection with the discussions that have taken place,
some logicians attempted to reform the theory of implication. Generally
they do not deny a place in logic to material implication, but they are
anxious to find also a place for another concept of implication, for instance,
of such a kind that one must be able to deduce the consequent from the
antecedent in order to have a true implication; they even desire, so it
seems, to place the new concept in the foreground. These attempts are
of a relatively recent date, and it is too early to pass a final judgment as
to their value.(4)
But today it appears certain that the theory of material
implication will surpass all other theories in simplicity; and, in any case,
it must not be forgotten that logic, which has been founded upon this simple
concept, turned out to be a satisfactory basis for the most complicated and
most subtle of mathematical reasonings.
The use of implications in mathematics
The phrase "if ... then ..." is among those expressions of logic which
are the most widely used in other sciences and, especially, in mathematics.
Mathematical theorems, particularly those of universal character, tend to
have the form of implications; the antecedent is called in mathematics the
HYPOTHESIS, and the consequent is called the CONCLUSION.
As a simple example of a theorem of arithmetic having the form of an
implication, we may quote the following sentence (5):
if x is a positive number, then 2x is a positive number
in which "x is a positive number" is the hypothesis, while
"2x is a positive number" is the conclusion.
Apart from this classical form of mathematical theorems (so to speak),
there are various alternative formulations, in which the hypothesis and the
conclusion are connected through some other means than by the phrase
"if ... then ...". The theorem just mentioned, for instance, can be
paraphrased in any of the following ways:
Therefore, instead of asserting a conditional sentence, it is usually just as appropriate to say that the hypothesis IMPLIES the conclusion or HAS it AS A CONSEQUENCE, Or that it is a SUFFICIENT CONDITION for the conclusion. or one can express the same idea by saying that the conclusion FOLLOWS from the hypothesis, or that it is a NECESSARY CONDITION for the latter. A logician may raise certain objections against some of the formulations given above, but they are in general use in mathematics.
The objections which might be raised here concern those of the above
statements in which any of the words "hypothesis", "conclusion",
"consequence", "follows", "implies" occur.
In order to understand the essential points in these objections, we
observe first that, strictly speaking, those statements differ in content from
the one originally given. While in the "classical form" of the theorem
we talk about numbers, properties of numbers, operations upon numbers, and
so on - in short, about objects with which mathematics is concerned - , in
the formulations now under discussion we talk about hypotheses, conclusions,
conditions, that is, about sentences or sentential functions occurring
in mathematics. It might be noted on this occasion that, in general,
people do not distinguish clearly enough the terms which denote the objects
dealt with in a given science, from those terms which denote various kinds
of expressions occurring within it. This can he observed, in particular,
in the domain of mathematics, especially on the elementary level. It appears
that only relatively few individuals are aware of the fact that such terms
as "equation", "inequality", "polynomial", or "algebraic fraction"
, which are met at every turn in textbooks of elementary algebra, do not
belong to the domain of mathematics or of logic (in the strict sense of the
words), since they do not denote objects which are considered in these
domains; equations and inequalities are certain special sentential functions,
while polynomials and algebraic fractions - especially as they are treated
in elementary textbooks - are particular instances of designatory functions
(cf. [ ... ]). The confusion on this point is reinforced by the fact that
terms of this kind are frequently used in the statements of mathematical
theorems. This has become a very common usage, and perhaps it is not
worth our while to put up a stand against it, since it does not present any
particular danger; but it might be useful to recognize that, for many theorems
formulated with the help of such terms, there are other formulations
in which those terms do not occur at all, and which are therefore logically
more satisfactory. For instance, the theorem:
the equation x^2 + ax + b = 0 has at most two roots
can be expressed in a more appropriate manner as follows:
there are at most two numbers x such that x^2 + ax + b = 0.
Let us return to the controversial formulations of an implication in order to emphasize one further point, which is particularly important. In these formulations we assert that one sentence, namely the antecedent of the implication, has another - the consequent of the implication - as a consequence, that is, that the second follows from the first. Ordinarily when we express ourselves in this way, we have in mind that the assumption of the first sentence being true leads us, so to speak, necessarily to the same assumption concerning the second sentence (and that perhaps we are even able to derive the second sentence from the first). As we know already from [ the above ], however, in contemporary logic the meaningfulness of an implication does not depend on whether its consequent has any such connection with its antecedent; and it is interesting to look again at the examples which were given there. Anyone who was shocked by the fact that the expression:
if 2.2 = 4, then New York is a large city
is considered in logic as a meaningful and even true sentence, will find it still harder to reconcile himself with a transformed phrase such as:
the hypothesis that 2.2 = 4 has as a consequence that New York is a large city.
We see that the alternate ways of formulating or transforming a conditional sentence lead to paradoxical-sounding utterances, and so they intensify the discrepancies between common language and mathematical logic. It is therefore not surprising that such reformulations repeatedly brought about various misunderstandings, and have been one of the causes of those passionate and frequently sterile discussions which we mentioned above.
From the purely logical point of view, we can obviously avoid all objections which are raised here by stating explicitly once and for all, that we shall disregard the literal meaning of the formulations in question, and that we shall attribute to them exactly the same contents as to ordinary conditional sentences. But this would be inconvenient in another respect; for there are situations - though not in logic itself, but in a field closely related to it, namely, in the methodology of deductive sciences (cf. [ later ] ) - where we talk about sentences and about the relation of consequence among them, and where we use such terms as "implies" and "follows" in a meaning which is different, and which is more closely akin to the ordinary one. It would therefore be better to avoid such formulations altogether, especially since we have several others at our disposal which are not open to any of these objections.
[ Rest deleted ]
(1) Logic was created by ARISTOTLE, the illustrious Greek thinker of the
4th century B.C. (384-322); his logical writings are collected in the work
Organon. As the creator of mathematical logic we have to look upon
the great German philosopher and mathematician of the 17th century G.W.
LEIBNIZ (1646-1716). However, Leibniz's works on logic failed to have much
influence upon the subsequent development of logical investigations; there
was even a period during which they sank into oblivion. A continuous
development of mathematical logic began near the middle of the 19th
century, at about the time when the logical system of the English
mathematician G. BOOLE was published (1815-1864; principal work: "An
Inivestigation of the Laws of Thought, London 1854). Later the new
logic found its full expression in the epochal work of the great English
logicians A.N. WHITEHEAD (1861-1947) and B. RUSSELL (1872-1970):
"Principia Mathematica" (Cambridge 1910-1913).
(2) Historically, the first systematic exposition of sentential calculus is
contained in the work Begriffsschrift (Halle 1879) of the German
logician G. FREGE (1848-1925) who, without doubt, was the greatest logician
of the 19th century. The eminent Polish logician and historian of logic
J. LUKASIEWICZ (1878-1956) succeeded in giving the sentential calculus a
particularly simple and precise form, and prompted extensive
investigations concerning this calculus.
(3) It is interesting to notice that the beginning of these discussions
dates back to antiquity. It was the Greek philosopher PHILO OF MEGARA
(in the 4th century B.C.) who presumably was the first in the history of
logic to advocate the usage of material implication, this was in opposition
to the views of his master, DIODORUS CRONUS, who proposed the use of
implication in a narrower sense, rather related to what is called here the
formal meaning. Somewhat later (in the 3d century B.C.) - and probably
under the influence of PHILO - various possible conceptions of implication
were considered by the Greek philosophers and logicians of the Stoic school
(whose writings included the first discussions of sentential calculus).
(4) The first attempt to develop a systematic theory of implication of this
kind was made by the American philospher and logician C.I. LEWIS (1883-1964).
(5) Strictly speaking, the expression which follows is a sentential function
rather than a sentence. We recall that the term "theorem" was first introduced
for proved sentences, but it is convenient and natural to extend the scope of
this term also to proved sentential functions. (Note that, if we were to adjoin
the universal quantifier to this sentential function, or to any other, then we
would not have an implication in the strict sense.)