LCR Truth Table Analysis

Logic & Critical Reasoning
Truth-Table Analysis
What is a Truth-Table
A truth-table is a table that shows the distribution of truth-values, T and F, over a

set of compound formulas. The distribution is determined by a fixed set of truth-
values as inputs, the definitions of the truth functions involved in the compound
formulas, and the determination of dependency.
The fixed set of inputs are under ‘P’ and ‘Q’ at the left. The final value of the
formula is under the main connective in bold.
Computation goes from P Q (P  Q)  (Q  P)

subordinate operators T T T T T
to the main connective
T F F T F
Rules for Constructing a Truth-Table
The number of rows that a truth-table needs is determined by the number of basic
statement letters involved in the set of formulas that will be involved in the
computation. The formula for the rows is 2 n where n = the number of basic
statement letters involved. The most common instances are where n = 1, 2, and 3;
in those cases one needs 2, 4, and 8 rows respectively.
The number of columns that a truth-table needs is determined by the number of

basic statement letters and formulas involved. There should be one column for
each basic statement letter and formula.
For each formula the computation proceeds from weakest scope to widest scope.
The operation in the widest scope is computed last. And if two sub-formulas of a
a formula have no greater weight in scope, then it does not matter which one is
computed first.
Distributing Basic Truth-Values in the Rows
A truth-table is a computation based on an initial set of input truth-values. These

values are fixed. The number of basic rows is given by the number of basic
statement letters, each of which can be either true or false. Thus, in making a truth-
table, one needs to account for every possible combination of how T and F can be
distributed amongst the basic statement letters prior to computing the compound
cases via the definition.
The easiest way to make sure all possible distributions of T and F have been
accounted for, one can use the procedure of halving and alternating Ts and Fs,
explained by the case in which one has 3 basic statement letters, P, Q, and R, and
thus 8 rows, by the formula 2n.
Set the first column under P to 4Ts, followed by 4Fs.
Set the second column under Q to 2Ts, 2Fs, 2Ts, 2Fs
Set the third column under R to 1T, 1F, 1T, 1F, 1T, 1F, 1T, 1F
The Case of Eight Rows
Since the formula has 3 basic statement letters, we need 8 rows,

following the procedure we have every possible combination of T
and F that applies to P, Q, and R.
P Q R (P  ( Q  R)
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
What can a truth-table be used for?
A truth-table can be used to determine the answer to a number of

questions that logicians care about.
Where ϕ and φ are arbitrary formulas of PL, a truth-table can be used to

answer any of the following questions:
Is ϕ a tautology, contingent, or a contradiction?
Are ϕ and φ logically consistent or inconsistent?
Are ϕ and φ logically equivalent?
Is φ a valid consequence of ϕ?
Definitions of Key Logical Concepts
Let the truth-profile of a formula be the column
underneath the main connective of the formula.
Logical Representation on a Truth-Table
Concept
Tautology The truth-profile is all T.
Contingent The truth-profile is some T, and some F.
Contradiction The truth-profile is all F.
Consistent At least one row has T in the truth-profile for all formulas involved.
Inconsistent There is no row that has T in the truth-profile for all formulas
involved.
Equivalent The truth-profiles are identical for all formulas involved.
Valid There is no row where the truth-profile for all the premises has a T,
and the the truth-profile for the conclusion has an F.
Invalid There is some row where the truth-profile for all the premises has a
T, and the truth-profile for the conclusion has an F.
Examples: Tautology
A statement or formula is a tautology when it is always true. On a truth-

table this means that the truth-profile contains only Ts.
P Q (P  (Q  P))
T T T T
T F T T
F T T F
F F T T
Examples: Contingent
A statement or formula is contingent when it is true sometimes, and

false other times. On a truth-table this means that the truth-profile
contains some Ts and some Fs.
P Q (P  Q)
T T T F
T F T T
F T F F
F F T T
Examples: Contradiction
A statement or formula is a contradiction when it is always false. On a

truth-table this means that the truth-profile contains only Fs.
P (P  P)
T F F
F F T
Examples: Consistency
Two or more statements or formulas are consistent when they can be

true at the same time. On a truth-table this means that there is at least
one row where there is a T in the truth-profile of each formula involved.
P Q (P  Q) (P  Q)
T T T T
T F F T
F T F T
F F F F
Examples: Inconsistency
Two or more statements or formulas are inconsistent when they can

never be true at the same time. On a truth-table this means that there is
no row where there is a T in the truth-profile of all of the formulas
involved.
P Q  (P  Q) (P  Q)
T T F T F T
T F T F T F
F T F T F T
F F F T T T
Examples: Equivalence
Two or more statements or formulas are logically equivalent when they

mean the same thing from the point of view of logic. On a truth-table
this means that the the truth-profile of all the formulas involved is
identical.
P Q (P  Q)  (P  Q)
T T T T F F
T F F F T T
F T T T F F
F T F T F T
Examples: Validity
An argument is valid when it is impossible for the premises to be true

and the conclusion false. On a truth-table this means that there is no row
where there is a T in the truth-profile of each premise, and an F in the
truth-profile of the conclusion.
Let (P  Q) and P be the premises, and Q the conclusion
P Q (P  Q) P Q
T T T T T ok
T F F T F ok
F T T F T ok
F F F F F ok
Examples: Invalidity
An argument is invalid when it is possible for the premises to be true

and the conclusion false. On a truth-table this means that there is at least
one row where there is a T in the truth-profile of each premise, and an F
in the truth-profile of the conclusion.
Let (P  Q) and P be the premises, and Q the conclusion
P Q (P  Q) P Q
T T T T T ok
T F T T F !!!!!
F T T F T ok
F F T F F ok

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Logic & Critical Reasoning

A truth-table is a table that shows the distribution of truth-values, T and F, over a

Computation goes from P Q (P  Q)  (Q  P)

to the main connective

The number of columns that a truth-table needs is determined by the number of

A truth-table is a computation based on an initial set of input truth-values. These

Set the first column under P to 4Ts, followed by 4Fs.

Set the second column under Q to 2Ts, 2Fs, 2Ts, 2Fs

Since the formula has 3 basic statement letters, we need 8 rows,

A truth-table can be used to determine the answer to a number of

Where ϕ and φ are arbitrary formulas of PL, a truth-table can be used to

Is ϕ a tautology, contingent, or a contradiction?

Are ϕ and φ logically consistent or inconsistent?

Are ϕ and φ logically equivalent?

A statement or formula is a tautology when it is always true. On a truth-

A statement or formula is contingent when it is true sometimes, and

A statement or formula is a contradiction when it is always false. On a

Two or more statements or formulas are consistent when they can be

Two or more statements or formulas are inconsistent when they can

Two or more statements or formulas are logically equivalent when they

An argument is valid when it is impossible for the premises to be true

Let (P  Q) and P be the premises, and Q the conclusion

An argument is invalid when it is possible for the premises to be true

Let (P  Q) and P be the premises, and Q the conclusion

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