MODERN LOGIC
Introduction. Modern Logic, sometimes called Symbolic Logic,
begins with the development of the Propositional Calculus.
This part of logic is also called sentential calculus.
Its foundations was established by Friedrich Ludwig Gottlob Frege (1848-1925)
and Charles Sanders Peirce (1839-1914). As its name implies it is a calculus
of "propositions." A proposition is a statement that may be either
true or false. For example, the following are propositions:
(1) The number 2 is prime. (True)
(2) 5 + 6 = 12 (False)
(3) Sunday is the first day of the week. (True)
(4) Ice is liquid. (False)
But the following are not propositions:
(5) Are you listening? (A question)
(6) Stop what you are doing! (A command)
(7) Wow, what a day! (An exclamation)
(8) The gobblegook spackled the whatach-you-call-it. (Meaningless)
This calculus of propositions is build on the formalization of the relations
"and", "or", "if-then", and
"if-and-only-if".
These relations are connections between "propositions"
and are called truth-functional connectives. The letters
"p", "q", "r", etc. are usually
used to stand for propositions to which the truth values, true
or false, may be assigned. For example, consider the compound
sentence: "It is raining today, and my car motor is running".
This sentence is composed of two simple sentences which are connected
by the conjunction "and". By definition this compound
sentence is true if and only if both of the parts making up the
compound sentence are true.
- Conjunction. The conjunction "and" is one of
the truth-functional connectives, and it is often represented
by the symbol "&" or a dot "·";
we shall use the ampersand "&". If the letter
"p" represents the truth value of the simple sentence
"It is raining today" and the letter "q" represents
the truth value of the other simple sentence "My car motor
is running", then the truth value of the compound sentence may
be represented by "p & q". The truth values assigned
to this truth value function connection may be defined by a definition
or by a truth table. The conjunction truth value function connection
may be defined by the following definition: the conjunction
of two constituent propositions is true if they are both true,
and it is false if either or both constituent are false. A truth
table is a matrix of all the possible truth values that can
be assigned to truth value function connection and its constituents.
The truth table for the conjunction is
| p |
q |
p & q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
F |
The other truth value functional connectives, "or",
"if-then", and "if-and-only-if" can also be
defined by a definition or by a truth table.
- Disjunction. The truth functional connective, "or",
is called disjunction, and in English usage has both a strong
or exclusive and a weak or inclusive meaning. For example, suppose
that you claimed as a youth that you were going to be either a
lawyer or a detective. Suppose that it turns out that you become
both a lawyer and a detective. Now were you as a youth making
a true statement when you claimed that you were going to either
a lawyer or a detective? In our ordinary use of the disjunction,
a disjunction is true if either part turns out to be true; the
disjunction is usually understood in the weak or inclusive meaning.
If the "or" in your statement is understood in this
weak or inclusive meaning, then your statement will be true.
In logic the weak or inclusive meaning, called the "inclusive
or", is taken as basic because its use allows for the analytical
development of logic as deductive system. But if this statement
is understood to be true only if either constituents are true
but not both, the disjunction has a strong or exclusive meaning.
In logic this strong or exclusive meaning of "or" is
called the "exclusive or", it is never the case where
both of the constituents are true, whereas in the weak or inclusive
meaning of "or" it might be the case that both of the
constituents are true. It is now clear that your claim "I
am going to be lawyer or a detective", will be false only
if the exclusive meaning of "or" is understood and both
the constituent sentences are false. If we let the letter "p"
represent the truth value of the simple sentence "I am going
to be a lawyer" and the letter "q" represent the
truth value of the other simple sentence "I am going to be
a detective", and the wedge shaped symbol "V" stand
for the disjunction in the inclusive sense of "or",
then the truth value of the disjunction of "I am going to
be a lawyer or a detective" may be represented by
"p V q". The following truth table defines disjunction in the
weak or inclusive meaning.
| p |
q |
p V q |
| T |
T |
T |
| T |
F |
T |
| F |
T |
T |
| F |
F |
F |
- Conditional. In ordinary discourse, we also use the
"if-then" connection between propositions. For example,
"If it rains, then the streets will be wet." This complex
connection is called a "conditional" and the "if" part
of this conditional statement is usually called the condition,
and the "then" part is called the conclusion.
Sometimes the "if" part is also called the antecedent, and
the "then" part is called the consequent. In mathematical
theorems of the form of a implication, the "if" part is called
the hypothesis and "then" part is called the
conclusion, and are so related that the implication can not be false;
that is, it is impossible for the hypothesis to be true, and the conclusion
to be false.
There are four cases of possible combinations for the truth values
of the condition and of the conclusion:
(1) the condition might be true and conclusion true;
(2) the condition might be true and the conclusion false;
(3) the condition might be false and the conclusion true; and
(4) the condition might be false and the conclusion false.
For the first two cases the truth value of their "if-then"
connection is obvious. In the first case, if it is true that it rains and
it is true that the streets are wet, then the "if-then"
connection does hold and is true. But in the second case where
it is true that it rains and the streets are not wet, then the
"if-then" connection does not hold and is false. That
is, if the occurrence of rain did not result in wet streets, we
would surely conclude that the total claim "if it rains,
then the streets will be wet" must be judged to be false.
But the last two cases are not that obvious. How are we to
judge the truth or falsity of these complex "if-then"
statements? In case three, when it has not rained but the streets
are wet, it is clear that the claim would not be false. But would
it be true? The common sense answer would be that the full assertion
is neither true or false but undetermined. In logic this ambiguity
is removed by considering the full assertion to be consider true,
since it not false. In the fourth case where both condition and
conclusion are false, common sense would suggest that the asertion
is, in this case, also undetermined. Cases three and four express
this ambiguity. In logic the ambiguity is again removed by considering
the full assertion to be true. A false statement may imply a
false statement just a false statement may imply a true statement.
In these cases the "if-then" relation between the two
statements are considered to hold or to be true. The right pointing
arrow "→" is usually used to stand for the "if-then"
connection, but we shall use the greater than sign ">"
and the following is its truth table.
| p |
q |
p > q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
T |
| F |
F |
T |
In logic this kind of conditional is called "material implication"
or just "implication". It is designedly a weak form
of implication, just as the form of the disjunction that was considered
appropriate for logic was the weak form of the disjunction. Indeed,
there exist aspects of material implication, relating to the last
two cases, which have gained the designation the "paradoxes
of material implication". There are three such paradoxes.
- The first paradoxial consequence is related to the first line
of the truth table. Material implication is considered as truth
functional connective. This means that any sentence can be substituted
for those in our example, and the truth of the whole expression
will be dtermined by the truth of its parts. It does seem odd
that the truth of if-then statement can be determined apart from
the subject matter of the constituent statements. The statement
"if it is raining outside, then 2 + 2 = 4" is a true
statement if it is true that "it is raining outside",
even though raining has nothing to do with addition.
- The second paradox relates to the first and third lines of
the truth table, and can be expressed by saying that a true statement
is implied by any statement whatever. As an example, the statement
"if it is raining outside, then 2 + 2 = 4" is true,
whether it is raining outside or not.
- The third paradox relates to the third and fourth lines in
the truth table, and can be expressed by saying a false statement
implies any statement whatever. Hence the statement "If
2 + 2 = 5, then the streets are wet" is true whether the
streets are wet or not.
Mathematicans often use another way of expressing implication.
For example, they speak of
"a necessary condition that p is true is that q is true."
That is, q is necessary for p. Eating is necessary for living.
(If one is to live, then one must eat.) Consider the truth table for
implication above. Note that in the rows where p imples q is true,
q must be true for p to be true, that is, among those rows there is
no case in which q is false and p is true (q is necssary for p.)
Mathematicans also speak of
"a sufficient condition that p be true is that q be true."
That is, p is sufficient for q. "To speed is sufficient
to be arrested." ("If one speeds, then one will be arrested.")
Consider again the true table above. In the rows where p imples q is true,
q is true if p is true. (p is sufficient for q.) Note also that in
those rows where p implies q is true, there is no case in which p is true
and q is false; that is, p is true only if q is true.
(Only if q, then p)
The following table summarizes this discussion.
| p > q |
if p, then q |
p is sufficient for q |
| p > q |
only if q, then p |
q is necessary for p |
| q > p |
only if p, then q |
p is necessary for q |
| q > p |
if q, then p |
q is sufficient for p |
p = q |
q if, and only if, p |
p is necessary and sufficient for q |
| p = q |
p if, and only if, q |
q is necessary and sufficient for p |
Entires in the same row are equivalent statements. The first two rows
represent the given implication; the third and fourth row represent its
converse, and the fifth and sixth rows represent a biconditional.
- Biconditional. This is a truth functional connective of
somewhat greater strength than material implication. Instead
of an "if-then" connective, it is an "if-and-only-if-then"
connective. Often the abbreviation iff is used for "if, and only
if". A biconditional is true when both propositions are true or both
are false. A biconditional is false when only one of its propositions are
true and the other is false. As an example consider the following sentence.
"If Louise is the wife of William, then William is the husband of
Louise." But this statement is equivalent to the following statement.
"If William is the husband of Louise, then Louise is the wife of
William." That is, "Louise is the wife of William is equivalent to
William is the husband of Louise."
This connective is called material equivalence, and one way of
symbolizing the material equivalence is with the double arrow
"↔", but we shall use the equal sign "=".
For any two sentences "p" and "q", p is material
equivalent to q, if both "p materially implies q" and "q
materially implies p"; that is,
(p = q) = [(p > q) & (q > p)].
Since material equivalence is by definition the conjunction of
two conditionals, it is also called a "biconditional".
The biconditional is defined by the following truth table.
| p |
q |
p = q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
T |
Note that in a biconditional the two statements need not be related. For
example, "2 + 2 = 5 iff July 4 is New Year's Day" is a true
biconditional, even though the two statements are unrelated as well as both
being false. This means that there is an equivalence between two propositions
when their truth values varies together. That is, if two propositions are
materially equivalent, then when one is true both are true, or when one is
false both are false,
- Negation. In addition to these four truth functional connections
there is a fifth truth-funcitonal called negation. Obviously,
the negation of a given statement is the denial of that statement,
and it is provided for in the English language by the appropriate
insertion in that statement of "not", "it is false
that ...", "it is not the case that ...", or a
similar phrase. In logic negation is widely expressed by the
use of the tilde, "~". By definition if a given statement
is true, its negation will be false; and if the given statement
is false, then its negation will be true. And negation can be
also defined by the following truth table.
The following truth table show the negative and
the four basic logical relations.
| p |
q |
~p |
~q |
p & q |
p V q |
p > q |
p = q |
| T |
T |
F |
F |
T |
T |
T |
T |
| T |
F |
F |
T |
F |
T |
F |
F |
| F |
T |
T |
F |
F |
T |
T |
F |
| F |
F |
T |
T |
F |
F |
T |
T |
From these logical functions, other formulae or theorems can be
derived, exhibiting the valid steps which may be taken in manipulating
propositions by means of these connections.
Definition: A logical statement that is always true is called a
tautology, and a logical statement that is always false is called
an absurdity, and one that is sometimes true and sometimes false is
called a contingency. For example, (p V ~p) is a tautology and
(p & ~p) is an absurdity. But ~(p & ~p), (p > p),
(p > ~~p) are tautologies, as can be seen by the following
truth table.
| p |
~p |
p V ~p |
p & ~p |
~(p & ~p) |
p > p |
~(~p) |
p > ~(~p) |
| T |
F |
T |
F |
T |
T |
T |
T |
| F |
T |
T |
F |
T |
T |
F |
T |