Outline

Contents

Truth Tables: An Introduction

Imagination test: an argument can be identified as deductively valid if and only if it was impossible to imagine a scenario where all of the premises were true and conclusion false.

• • If you can imagine such a scenario, then the argument is invalid.

• • If you cannot imagine such a scenario, then either the argument is valid.

Problems with the Imagination Test

The “imagination test” is a problematic test for a couple of reasons:

• 1. Some arguments consist of many premises and the state of affairs that these premises express can be difficult to picture in one’s mind.

• 2. People sometimes judge that certain arguments are “valid” because (1) they believe the conclusion and (2) are unwilling to consider scenarios where the premises are true and the conclusion is false.

• 3. Arguments about abstract topics can be difficult to imagine

A New Test

What we want then is a new test that

• 1. does not rely upon the limited imaginative powers of human beings

• 2. is immune to bias

• 3. can be formulated about any subject matter.

What is a truth table?

A truth table is a decision procedure

A truth table can be used to mechanically test sets of wffs and arguments.

Truth Tables: Step by Step

• • Step 1: Write down the wff.

• • Step 2: Write the truth value (\(T\) or \(F\)) under each propositional letter in the wff for each interpretation \(\mathscr {I}\).

• • Step 3: Assign T or F to subformulas in the order that the wff is constructed using (1) the truth values assigned to the wffs used to construct those subformulas and (2) the valuation function associated with the operator introduced in the construction.

Truth Table for Five Complex Wffs

Truth Table: An Example

Let’s determine the truth value for \(P\rightarrow \neg R\) under a single interpretation of \(P\) and \(R\): \(\mathscr {I}(P)=T\) and \(\mathscr {I}(R)=F\).

• • Step 1: Write out the formula or set of formulas you want to test.

Truth Table: An Example

Next, consider how \(P\rightarrow \neg R\) is constructed.

• 1. \(P\) is a wff

• 2. \(R\) is a wff

• 3. If \(R\) is a wff, then \(\neg R\) is a wff

• 4. If \(P\) is a wff and \(\neg R\) is a wff, then \(P\rightarrow \neg R\) is a wff.

Assign truth values to the subformulas of \(P\rightarrow \neg R\) in that order.

Truth Table: An Example

Start by writing truth values under the proposition letters. Since \(\mathscr {I}(P)=T\) and \(\mathscr {I}(R)=F\) write these values under the letters.

• 1. \(P\) is a wff

• 2. \(R\) is a wff

• 3. \(\neg R\) is a wff

• 4. \(P\rightarrow \neg R\) is a wff.

Truth Table: An Example

The next wff constructed is \(\neg R\) using \(R\). So, determine the truth value of \(\neg R\) using the truth value of \(R\).

• 1. \(P\) is a wff

• 2. \(R\) is a wff

• 3. \(\neg R\) is a wff

• 4. \(P\rightarrow \neg R\) is a wff.

Truth Table: An Example

The next wff constructed is \(P\rightarrow \neg R\) using \(P\) and \(\neg R\). So, determine the truth value of \(P\rightarrow \neg R\) using the truth value of \(P\) and \(\neg R\).

• 1. \(P\) is a wff

• 2. \(R\) is a wff

• 3. \(\neg R\) is a wff

• 4. \(P\rightarrow \neg R\) is a wff.

The Truth-Table Method

• • We have seen how the truth value of a complex wff \(\phi \) can be determined under a single interpretation of the propositional letters.

• • The truth-table method, however, is more general in that it allows for determining the truth value of wffs under all admissible interpretations of the propositional letters. For example, consider the following wff: \(\neg P\vee \neg R\).

Truth-Table Method for n-Interpretations

Step 1: Write out the wff \(\phi \) and all of the propositional letters in \(\phi \) all of the propositional letters to the left of \(\phi \).

Truth-Table Method for n-Interpretations

Step 2: Write all possible interpretations for the propositional letters in \(\phi \) under the propositional letters.

How many interpretations are there?

The number of possible interpretations (and therefore rows) is determined by the number of propositional letters in the set of wffs being considered. That is, the number of rows required for a truth table of any argument is determined by \(2^n\) where \(n\) is the number of propositional letters in the argument.

• 1. Since \(P\) involves 1 propositional letter, \((2^1 = 2)\) rows are needed.

• 2. Since \(P\wedge Q\) involves 2 propositional letters, \((2^2 = 4)\) rows are needed.

• 3. Since \((P\wedge Q)\wedge R\) involves 3 propositional letters, \((2^3 = 8)\) rows are needed.

Truth-Table Method for n-Interpretations

Step 3: For each row, write the truth values under the corresponding letter in the row.

Truth-Table Method for n-Interpretations

Step 4: Assign T or F to subformulas in the order that the wff is constructed using (1) the truth values assigned to the wffs used to construct those subformulas and (2) the valuation function associated with the operator introduced in the construction.

First determine the truth value of \(\neg P\) using the truth value of \(P\) and the truth values of \(\neg R\) using the truth value of \(R\).

The above shows the truth value of \(\neg P\vee \neg R\) under all of the different ways that \(P\) and \(R\) can be interpreted in \(PL\).

Truth-Table Analysis

So far, we see how truth tables can be used to:

• 1. determine the truth value of a complex wff under a single interpretation

• 2. determine the truth value of a complex wff under all interpretations

Truth-Table Analysis

Next, let’s use truth tables to test whether:

• 1. a wff \(\phi \) is a contingency, tautology, or contradiction.

• 2. a pair of wffs \(\phi , psi\) are equivalent

• 3. a collection of wffs \(\phi , \psi \) are consistent

• 4. an ”argument” is valid

Contingency, Tautology, Contradiction

Contradiction

Contingency, Tautology, Contradiction

Tautology

Contingency, Tautology, Contradiction

Contingency

Contingency, Tautology, Contradiction

How to Test

We can use a truth table to check whether a wff is a contradiction, tautology, or contingency by:

• 1. Constructing the truth table

• 2. Checking whether the wff is false under every interpretation (contradiction), true under every interpretation (tautology), or neither true nor false under every intrepretation (contingency).

Contingency, Tautology, Contradiction

Practice

Is \(\neg P\vee \neg R\) a PL-contingency, PL-tautology, or PL-contradiction?

It is a PL-contingency!

Contingency, Tautology, Contradiction

Practice

Is \(P\vee \neg P\) a PL-contingency, PL-tautology, or PL-contradiction?

This wff is a tautology.

Contingency, Tautology, Contradiction

Practice

Is \(P\wedge \neg P\) a PL-contingency, PL-tautology, or PL-contradiction?

This wff is a contradiction.

Contingency, Tautology, Contradiction

Practice

• 1. It is not the case that if Liz does not eat ice cream, then she does not eat cake.

• 2. Translate as \(\neg (\neg I\rightarrow \neg C)\)

• 3. Create the table for \(\neg (\neg I\rightarrow \neg C)\)

• 4. Check whether the wff is a PL-contingency, PL-tautology, or PL-contradiction.

The proposition ”It is not the case that if Liz does not eat ice cream, then she does not eat cake” is a contingency since \(\neg (\neg I\rightarrow \neg C)\) is a PL-contingency.

Consistency

So, if \(\phi \) and \(\psi \) are wffs, the set of wffs \(\{\phi , \psi \}\) is PL-consistent provided there is at least one interpretation where both \(\phi \) and \(\psi \) are true.

Inconsistency

Consistency

In many cases we can just see that a set of wffs is PL-consistent.

Consistency

In other cases, it isn’t obvious.

Consistency

A truth table offers a method for testing whether a set of propositions is consistent or inconsistent.

• 1. Write each wff down in a row.

• 2. Construct a single truth table.

• 3. Check for whether there is at least one row where all the wffs in the row are T.

• 4. If there is a row, then the test says the wffs are consistent.

• 5. If there is not a row, then the test says the wffs are inconsistent.

Consistency

Example

Let’s test whether \(P\rightarrow Q, P\wedge Q, P\vee \neg Q\) is PL-consistent.

Notice that in the first row, all three wffs are true. Therefore, there is at least one interpretation where all the wffs are true. Therefore, the set is PL-consistent.

Consistency

Example

Let’s test whether these wffs are PL-consistent: \((P\rightarrow Q), (\neg R\vee Q), R\wedge \neg Q\):

There is no row where all the wffs are T, so the set is PL-inconsistent.

Consistency

Application

Consistency

Practice

• 1. “If John is tall, then Mary is happy” and “John is not tall or Mary is happy.”

• 2. Translate as \(J\rightarrow M, \neg J\vee M\)

• 3. Create the table for \(J\rightarrow M, \neg J\vee M\)

• 4. Check whether the wff is a PL-consistent or PL-inconsistent.

There is at least one row where the wffs are true. So, \(J\rightarrow M, \neg J\vee M\) is PL-consistent, so the sentences are consistent.

Equivalence

In other words, if you have a set of wffs \(\phi , \psi \) and the truth values match for every interpretation, then the wffs are equivalent.

Equivalence

Equivalence

A truth table offers a method for testing whether the members in a set of wffs are equivalent.

• 1. Write each wff down in a row.

• 2. Construct a single truth table.

• 3. For each row, check for whether the truth values match

• 4. If they match for each row, then the wffs are equivalent.

• 5. If they do not match for each row, then the wffs are not equivalent.

Equivalence

Are \(P\leftrightarrow Q, P\rightarrow Q\) PL-equivalent?

No, they are not PL-equivalent.

Equivalence

Application

• 1. “If John is tall, then Mary is happy” and “John is not tall or Mary is happy.”

• 2. Translate as \(J\rightarrow M, \neg J\vee M\)

• 3. Create the table for \(J\rightarrow M, \neg J\vee M\)

• 4. Check whether the wffs are equivalent or not equivalent.

The wffs are PL-equivalent.

Validity

If \(A, B, C\) are the premises and \(D\) is the conclusion of an argument, \(A, B, C\) semantically entail \(D\) iff there is no interpretation where \(A, B, C\) are all true and \(D\) is false.

Semantic Entailment

• 1. We can say that \(\Gamma \) semantically entails \(\psi \) or \(\psi \) is a semantic consequence of \(\Gamma \).

• 2. We make use of the double turnstile “\(\models \)” to express entailment (“models” or “semantically entails”)

• 3. \(\Gamma \models \psi \) says “\(\Gamma \) semantically entails \(\psi \)”

• 4. If it is not the case that \(\Gamma \) semantically entails \(\psi \), then we write \(\Gamma \not \models \psi \).

Semantic Entailment

A truth table offers a method for testing whether \(\Gamma \models \psi \).

• 1. Write each wff down in a row.

• 2. Construct a single truth table.

• 3. For each row, check for a row where all the members of \(\Gamma \) are T and \(\psi \) is F.

• 4. If there is a row, then it is not the case that \(\Gamma \) semantically entails \(\psi \). So, we write \(\Gamma \not \models \psi \)

• 5. If there is no such row, then it is the case that \(\Gamma \) semantically entails \(\psi \). So, we write \(\Gamma \models \psi \)

Semantic Entailment

An Example

Does \(P\rightarrow Q\) and \(P\) semantically entail \(Q\)?

Yes. Notice that there is no row where \(P\rightarrow Q\) and \(P\) are true and \(Q\) is false. So, \(P\rightarrow Q, P\models Q\)

Semantic Entailment

An Example

Does \(P\vee Q\) and \(P\) semantically entail \(Q\)?

No. Notice that in row (2) \(P\vee Q\) and \(P\) are true and \(Q\) is false. So, \(P\vee Q, P\not \models Q\)

Semantic Entailment

Application

• 1. “If John is tall, then Mary is happy. Mary is not happy. Therefore, John is not tall.”

• 2. Translate as \(J\rightarrow M, \neg M\models \neg J\)

• 3. Create the table for \(J\rightarrow M, \neg M\models \neg J\)

• 4. Check whether \(J\rightarrow M, \neg M\) semantically entails \(\neg J\).

\(J\rightarrow M, \neg M\) semantically entails \(\neg J\)

Some Shortcuts

• 1. If \(\psi \) is a PL-tautology, then \(\Gamma \models \psi \).

• 2. Since \(\Gamma \not \models \psi \) only when \(\psi \) is false, check rows where \(\psi \) is false.

• 3. If \(\Gamma \) is PL-inconsistent, then \(\Gamma \models \psi \).

• 4. Since \(\Gamma \not \models \psi \) only when \(\Gamma \) is a PL-consistent, check rows where all the wffs in \(\Gamma \) are truie.

Limitations of Truth Tables

• • Problem 1: Provided an argument is capable of being fully expressed by a truth-functional language like \(PL\) , the truth-table method seemingly guarantees there is a way to determine whether that argument is “valid” or “invalid”, but not every English argument can be represented in a truth-functional language like PL. There are some arguments in English that are valid, but are not valid in PL.

• • Problem 2: The truth-table test’s complexity increases exponentially. For every new propositional letter introduced, the table grows: \(2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128\)