Logic & Critical ReasoningTruth-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

subordinate operators

to the main connective

P Q (P Q) (Q P)

T T

T T T

T F

F T F

F T T F F

F 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 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 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

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

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.

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

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.

Let the truth-profile of a formula be the column underneath the main connective of the formula.

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