Introduction to Logic
Week Five: 29 Oct.., 2007
Putting Truth Tables to Work
Business Matters:
First Marked Homework Problem Set now available today; due Thursday, 4pm in the Philosophy UG office.
Review
A truth table is a logical calculating device. In particular, it is a table which specifies the truth value of a
compound formula for each and every assignment of truth values for the atomic formula that appear in it.
Exercise
Last time we learned the truth tables for the five basic connectives of propositional logic. Without consulting
your notes, complete the five truth tables below.
P
~P
P
Q
P&Q
P
Q
PvQ
P
Q
P→Q
P
Q
P ←→ Q
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USING TRUTH TABLES
OK, so what is the point of this? What is a truth table for? As we shall see, truth tables turn out to be quite a
powerful (if also somewhat clumsy) tool for a number of interrelated tasks in logic. Let’s consider them one at
a time.
A. Using Truth Tables to Define the Logical Connectives. Logical languages depend on perfect clarity and the
absence of ambiguity. While the logical connectives all have natural language correlates, they cannot be
defined by appeal to natural language without importing the ambiguities of natural language into the symbolic
systems. We have already seen one example of this with the ambiguity of the English word, “or,” which can be
used either inclusively or exclusively. Similar ambiguity infects the English word “and”. If someone says “I
got the money at the bank and I went to buy the car,” that would typically mean that they first got the money at
the bank and then went to buy the car. In short, the word “and” sometimes conveys temporal information. But
it need not. If I say that I have a bike and a scooter, I am not saying anything about which I got first. In order to
provide proper definitions of the logical connectives, therefore, we need a way of defining them more exactly
than is possible by simply providing natural language correlates. Truth tables provide the tool for this purpose.
In logic, the logical connectives are defined as truth-functions. (P&Q) is defined as the formula which is true if
and only if both of its constituent propostitions are true. (PvQ) is defined as the formula which is false if and
only if both of its constituent propositions are false. The truth tables for each connective spell out these
definitions exactly, and without circularity. That is, they specify the truth value of the compound formula given
any possible combination of truth value of its constituents. After all, that is exactly what truth tables do.
B. Using Truth Tables to Interpret Complex Compound Formulae. Once we have truth-functional definitions
of the five connectives, we can put truth tables to work for other purposes. As an example, consider this fairly
simple compound formula:
P v ~(P v R)
For the logician, an interpretation of this compound formulae must tell us its truth value (that is, whether it is
true or false) for every combination of the truth values of its constituent atomic propositions. But that is
exactly the job for which truth tables are designed.
P
T
T
F
F
Q
T
F
T
F
P
T
T
F
F
v
T
T
F
T
~
F
F
F
T
(P v Q)
T
T
T
F
mc
What this truth table shows is that this compound formula is false only in the case where P is false and Q is true
(that is the second row from the bottom of the table). It is true in every other case.
EXERCISES
Construct truth tables for the following formula
P & (Q → P)
(P & Q) → R
((P v Q) & ~P) → Q
C. Using Truth Tables to Assess for Validity: The Corresponding Conditional Method. Here we come to one
of the most important uses for Truth Tables. Recall that the central concern of logic is the validity of logical
form. An argument is valid if and only if the truth of its premises guarantees the truth of its conclusion. In
other words: a valid argument form is like a machine that guarantees true outputs when one puts in true
premises. (Remember that validity is different from soundness! A sound argument must have true premises
and valid form. But a valid argument might have false premises. To say that it is valid is simply to say that if
its premises are true then its conclusion must be true.) Given this characterization of valid form, and given what
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we now know of truth tables, it should be clear that we can use truth tables to prove validity of form. Here’s an
example.
Consider the following very primitive and basic argument (modus ponens):
Premise 1: P
Premise 2: P → Q
∴ Conclusion: Q
We want to know whether this argument has valid form. That is effectively to ask whether the truth of its
premises guarantees the truth of its conclusion. Put negatively, the question is this: Is there any circumstance
under which an argument of this form might have true premises but a false conclusion? If the answer is yes,
then the argument is invalid. If the answer is no, then the argument is valid.
Accordingly, we can test for validity as follows:
i) Formulate a complex compound formulae using → as the main connective.
note: Recall that a formula using → as its main connective is called a conditional. The “if” clause is called the
antecedent; the “then” clause is called the consequent.
ii) As the antecendent in this complex compound formula use a conjunction which compounds the two
premises of the argument into a conjunction.
note: recall that a conjunction is formed using the connective &; the two formulae which are conjoined are
called its conjuncts.
iii) As the antecedent in the formula use the conclusion of the argument.
iv) We now have a formula that looks like this (complete this section yourself):
v) Now construct a truth table for this compound formula.
vi) As always, the truth table for the compound formula gives us the truth values of the compound formula as a
function of the truth values of the atomic formulae from which it is constructed. In testing for valid form, we
are particularly interested in knowing whether there is any combination of truth values for the atomic
propositions whereby the premises of the argument are true but the conclusion is false. In terms of the formula
we have constructed, that means that we want to know whether there is any row of the truth table whereby the
compound antecedent (P & (P → Q)) is true but the consequent (Q) is false. But that is the same as asking
whether the overall formula comes out as true under any assignment of truth values for P and Q. (This is
because the truth table defining → specifies that a conditional is only false if the antecedent is true and the
consequent is false: “If Dice-K pitches then the Red Sox win” is false only if there is a case where Dice-K
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pitches but the Red Sox do not win.) Accordingly, if the corresponding conditional formula comes out as true
in every row of the truth table then the argument form is valid. If there is any row which shows the
compound conditional formula as false then the argument form is invalid.
Exercise: Use the Corresponding Conditional Method to proof that affirming the antecedent is a formal
fallacy. That is, use the truth table method to show that the following argument has invalid form.
Premise 1: If it is raining then the sidewalk is wet.
Premise 2: The sidewalk is wet.
∴ Conclusion: It is raining.
D. Using Truth Tables to Sort Formulae into Tautologies, Contingencies, and Inconsistencies (Contradictions):
A fourth use of Truth Tables will be particularly important for establishing a further set of tools of proof in the
propositional calculus. We can use truth tables to sort formulae into three different groups.
Tautologies are formulae that are always true, no matter what the truth value of their constituent
atomic propositions. “If today is Thursday then today is Thursday” is a simple tautology. But other
tautologies are more interesting and will provide us with a set of logical transformation rules. An
example is ((P v Q) & ~Q → P). A compound formula is a tautology if and only if the compound
formula comes out as true in every row of its truth table.
Contradictions (or Inconsistencies) are compound formulae that are never true, no matter what the
assignment of truth values to the atomic propositions that comprise them. An example of a primitive
contradiction is (P & ~P). A compound formula is a contradiction if and only if the compound formula
comes out as false in every row of its truth table.
Contingent formulae are all the rest. In order to know whether a contingent proposition is true or
false you need to know more than its logical form; you need to know the actual truth value of its
constituent atomic propositions. An example of a contingent proposition is “Tomorrow is Friday and I
am going down to the pub.” A compound formula is contingent if and only if the compound formula
comes out as false in some rows of its truth table and true in others.
E. Exercise:
Consider the following propositions. Without constructing a truth table, try to determine whether each one is a
tautology, a contradiction or a contingent proposition. After placing your bets we can split up the list and use
truth tables for each one.
a) (P → Q) & ~(P → Q)
b) (P v Q) & Q
c) (P → Q) & (~Q ←→ ~P)
d) ~P → (P → Q)
Homework
Read Tomassi, Logic, Ch IV, §§7-9 (pp. 141-163)
Exercises: The first assessed homework assignment is provided on a separate sheet. It would also be
useful to study some of the exercises in 4.3, 4.4, and 4.5
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