Introduction to Logic
Week Four: 22 Oct. 2007
Truth Functions and Truth Tables
0. Business Matters: The first marked homework will be due on Thursday, 8 November. The
problem set will be available two weeks in advance, i.e., on this Thursday, from the Philosophy UG
office, or in class next Monday (29 October).
1. Review
-- Upper case letters (P, Q and R) are sentential constants, or sentence letters.
-- When translating a sentence from English to Propositional Logic, we require a key.
-- There are five logical connectives in Propositional Logic: Negation (~), Conjunction (&),
Disjunction (v), Conditional (→), and Biconditional (←→).
-- The formulas connected by a conjunction are conjuncts. The formulas connected by a disjunction
are disjuncts. In a conditional, the formula after the ‘if’ is the antecedent, the formula after the
‘then’ is the consequent’.
-- Negation is a unary (one-place) connective, the others are all binary (two-place).
-- Logical connectives connect either atomic or compound formulas.
-- Brackets (parentheses) are used to specify the scope of a connective.
2. Truth and Falsity
-- Arguments can be valid or invalid, sound or unsound.
-- Sentences, on the other hand, are either true or false. As logicians say, there are two truth-values:
true and false.
-- This is known as the Principle of Bivalence, which states: every sentence is either true or false but
not both and not neither.
3. Scope
Every connective has a specific scope. The scope of the tilda (~) is the formula that is being negated.
In the simplest case this will be a simple atomic proposition, but in other cases it will be some more
complex wff (well-formed formula). To determine the scope of a tilda, then, we simply ask: exactly
what is being negated here? Consider these examples:
~P
~ (P & Q)
~(P & Q) → R
~((P & Q) → R)
scope of negation (what is being negated): P
scope of negatation: (P & Q)
scope of negation: (P & Q)
scope of negation: ((P & Q) → R)
For a two place connective, the scope of the connective are the TWO wffs that are being connected. In
the following examples I have used underlining to indicate the scope of the ampersands.
P&Q
P & ~Q
(P v Q) & (~P → Q)
4. Exercise: Using underlining, show the scope of the indicated connective.
Indicate the scope of the tilda
Indicate the scope of the arrow
Indicate the scope of the second arrow
~(P v ~P)
P → (P & Q)
~(((P → Q) → (Q → R)) & ~(P → R))
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5. SKILL: Find the Main Connective
The main connective in a formula of propositional calculus is the connective whose scope encompasses
the whole formula. The main connective tells us what kind of compound formula we are dealing with:
whether a negation, a conjunction, a disjunction, a conditional, or a biconditional. The simplest
compound formulae have only a single connective, which accordingly serves as the main connective.
Hence the ampersand is the main connective in (P&Q); the whole formula is accordingly a
conjunction. A crucial skill in interpreting the propositional calculus is the skill of finding the main
connective in moderately complex formulae. This is a bit like finding the main verb in an English
sentence that has various subordinate and relative clauses. Read the following formulae (aloud if you
can) and find the main connective. Specify what kind of formula is being expressed:
a) P → (P & Q)
b) P & ~P
c) P v ~P
d) ~(P v ~P)
e) P → (Q → P)
f) (P & Q) ←→ (Q & P)
g) ((P → Q) → (Q → R)) → (P → R)
h) ~(((P → Q) → (Q → R)) & ~(P → R))
i) (P → Q) v (Q → P)
6. Truth Tables
A truth table is a logical calculating device. In order to understand its distinctive utility, we need to
start by recalling some crucial background.
1.. In the formal symbolic language of the propositional calculus, a sentence is called a formula (or a
‘well-formed formula’ – a WFF).
2. As we have seen, there are two classes of formulae in the propositional calculus: atomic and
compound. An atomic formula is represented by a single sentential constant (e.g., ‘P’) and represents a
simple declarative sentence such as “Socrates is wise.” A compound formula is formed by combining
atomic formulae using the logical connectives (~, &, v, →, ←→). Hence the sentence “If Socrates is
not wise then Plato was a fool” can be expressed as a compound formula of the propositional calculus:
(~P → Q), where P represents the proposition that Socrates is wise and Q represents the proposition
that Plato is a fool.
3. Now here is a key point: the truth or falsity of a compound formula is a function of the truth value
of the atomic formulae that figure in it. Don’t be put off by the abstract talk of functions here. The
point is really as simple as this: Suppose I want to know whether the following claim is true or false:
“Today is Monday and the weather is fine.” In order to figure out whether this is true I need to know
whether each of its constituent claims is true. That is, Is today Monday? And is the weather fine?
Once I know the answers to these questions, and the meaning of the logical term “and” I can determine
the truth value of the compound sentence. That is: the truth value of the compound formula is a
function of the truth value is its atomic sentences.
4. Now truth tables are nothing more than a device for spelling this out exactly and in full detail. A
truth table is divided into two sides, left and right. On the left hand side is a specification of all the
possible combinations of truth values for the atomic propositions that figure in a compound formulae.
The right side specifies the truth value of the compound formula as a function of the truth values of the
atomic formulae. Hence for instance the truth table for conjunction shows under what circumstances
the compound formula is true and under what circumstances false.
P
T
T
F
F
Q
T
F
T
F
P&Q
T
F
F
F
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Starting from the left side and reading one row at a time, we can see the following:
If both atomic propositions are true then the conjunction is true. (That is the first row of under the
horizontal line.) But for any other combination of truth values, the conjunction is false. (That is
represented in the last three rows of the table.)
6. Using Truth Tables to Define the Logical Connectives. 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. Our first use will be to provide rigorous definitions of
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 truthfunctions. (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.
7. Truth-Tables for the five connectives (LEARN THESE!)
a) Negation
-- The truth functional nature of each of the logical connectives is represented by a truth-table.
-- The truth-table for negation is written as follows:
P
~P
T
F
F
T
-- Negation is a truth-function that reverses truth values.
b) Conjunction
-- The truth-table for conjunction is written as follows:
P
Q
P&Q
T
T
T
T
F
F
F
T
F
F
F
F
-- A conjunction is true only when both conjuncts are true. The sense of temporal direction that exists
in the English ‘and’ is not present in &.
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c) Disjunction
-- The truth-table for disjunction is as follows:
P
T
T
F
F
Q
T
F
T
F
PvQ
T
T
T
F
-- A disjunction is true when either or both of the disjuncts are true. It is false only when both are
false.
-- Disjunction in Propositional Logic is inclusive (it is true when both disjuncts are true), rather than
the exclusive ‘or’ that we often find in natural language.
d) Conditional
-- The truth-table for conditional is written as follows:
P
T
T
F
F
Q
T
F
T
F
P→Q
T
F
T
T
-- There only way in which a conditional comes out false is if its antecedent is true and its
consequent is false.
-- The conditional is, intuitively, the least close to its natural language equivalent ‘if…then…’.
e) Biconditional
-- The truth-table for biconditional is written as follows:
P
T
T
F
F
Q
T
F
T
F
P←→Q
T
F
F
T
-- A biconditional is true when both sides have the same truth value.
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 that figured in Willem’s Hypothesis:
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.
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5. Fallacies Catalogue
Last time we made a start on a catalogue of informal fallacies. We continue this week …
Fallacies of Relevance:
Straw Man: An argument attacks a straw man when it is directed at some unduly weakened or
vulnerable version of the proposition or proposal actually under dispute.
The white paper proposal to check the power of Local Education Authorities is
fundamentally flawed. We must not abandon the principle that local representatives
maintain local control over local schools. After all, it is local people who understand local
conditions, and can be called to account for local policies.
False Dilemma: An argument turns on a false dilemma when it misleadingly suggests that only
two (usually extreme) possibile positions are possible on a particular question.
Either we stick it out in Iraq or we effectively surrender to International Terrorism. To
surrender to terrorism will only make matters worse, so we simply have to stick it out and
resist the instinct – however natural it may be – to cut and run when the going gets tough.
Ad Hominem: An argument is ad hominem (against the man) when it is directed against the
person or institution advancing a thesis or proposal rather than against the thesis or proposal itself.
Who is opposing this legislation? The record is clear. The money to fund the opposition to
the smoking ban comes from big multinational tobacco companies, who have a vested
interest in preserving the status quo – regardless of its health consequences. Our democracy
is not for sale at any price, and neither is the health of our citizenry. I urge the House to
stand firmly with the Government in supporting this bill to keep smoke out of public places –
including public houses.
Homework for the Next Session
Reading: Tomassi, P. Logic, Ch. IV, §§5-6
Exercises: Tomassi, Exercises 4.1 & 4.2.
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