Transcript 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.

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 truthvalues 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
T F
F
T
F
F T
T
F
F
F F
F
T
F
to the main connective
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 2n 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 truthtable, 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
T T T
T T F
T F
T
T F
F
F T T
F T F
F F
T
F F
F
(P  ( Q  R)
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 Concept
Representation on a Truth-Table
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 truthtable 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)
T T
T
T
T F
F
T
F
T
F
T
F
F
F
F
P
Q
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)


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
(P
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
T T
T
T T
ok
T F
F
T F
ok
F
T
T
F T
ok
F
F
F
F F
ok
P
Q
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