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Truth Functions
Truth Functions
Beyond Validity
Beyond Validity
After translating statements into propositional logic, a number of assessment methods can be used to verify the potential truth values of those statements as well as the validity of arguments. As discussed previously, validity concerns the truth-preserving structure of a deductive argument if the premises are true.
After translating statements into propositional logic, a number of assessment methods can be used to verify the potential truth values of those statements as well as the validity of arguments. As discussed previously, validity concerns the truth-preserving structure of a deductive argument if the premises are true.
Given the often complex nature of arguments and their constituent parts, it can be difficult to determine just from the statements themselves whether or not they are true, let alone truth-preserving throughout. For instance, if we take the last example from the previous section, it quickly becomes apparent that the truth of the statement depends on a variety of factors, as in the example explored above.
Given the often complex nature of arguments and their constituent parts, it can be difficult to determine just from the statements themselves whether or not they are true, let alone truth-preserving throughout. For instance, if we take the last example from the previous section, it quickly becomes apparent that the truth of the statement depends on a variety of factors, as in the example explored above.
Challenge of Truth
Challenge of Truth
Even the truth of more simplistic conditional statements can be difficult to determine when we do not yet know whether any of the conditions hold. It becomes more challenging when there are so many conditions that need to be met, and even more so when we combine a compound statement like the one above with many others in an argument.
Even the truth of more simplistic conditional statements can be difficult to determine when we do not yet know whether any of the conditions hold. It becomes more challenging when there are so many conditions that need to be met, and even more so when we combine a compound statement like the one above with many others in an argument.
Luckily, the formal structure of propositional logic allows us to assess the potential truth values of even the most complex statements and arguments.
Luckily, the formal structure of propositional logic allows us to assess the potential truth values of even the most complex statements and arguments.
The next section will explore the basic rules for determining the potential truth values of statements, as well as the use of those truth values to determine the validity of arguments.
The next section will explore the basic rules for determining the potential truth values of statements, as well as the use of those truth values to determine the validity of arguments.
Truth Values & Tables
Truth Values & Tables
By truth value, we mean the attribution of truth (T) or falsity (F) to a given statement. Simple statements are assigned each possible truth value.
By truth value, we mean the attribution of truth (T) or falsity (F) to a given statement. Simple statements are assigned each possible truth value.
This resource will use the classical notion of truth value, with only two possible values. It is worth noting that in other logical systems (e.g., quantum theory and computer science), there may be more than two possible truth values which allow for different models of entailment, identity, and infinity.
This resource will use the classical notion of truth value, with only two possible values. It is worth noting that in other logical systems (e.g., quantum theory and computer science), there may be more than two possible truth values which allow for different models of entailment, identity, and infinity.
The truth value of compound statements will then be determined by the truth values of the simpler statements that make them up.
The truth value of compound statements will then be determined by the truth values of the simpler statements that make them up.
Truth tables are then constructed to determine the truth values of each statement and argument validity. A truth table is a representation of the ways that a statement can express truth values. In this sense, truth tables are mechanical and follow rules for their construction. Which is what we shall explore next.
Truth tables are then constructed to determine the truth values of each statement and argument validity. A truth table is a representation of the ways that a statement can express truth values. In this sense, truth tables are mechanical and follow rules for their construction. Which is what we shall explore next.
How to Construct Truth Tables for Statements
How to Construct Truth Tables for Statements
IN 2 EASY STEPS
IN 2 EASY STEPS
This section will examine the rules for constructing truth tables for simple and compound statements, as well as instructions for how to classify and compare statements.
This section will examine the rules for constructing truth tables for simple and compound statements, as well as instructions for how to classify and compare statements.
STEP 1: SETUP TABLE
STEP 1: SETUP TABLE
Columns = Variables
Columns = Variables
First, we need to identify the scope of the table being constructed. We do so by determining the number of simple statements present in the proposition. Remember that each simple statement will be represented by a variable. Each variable will get its own column in the table. For example:
First, we need to identify the scope of the table being constructed. We do so by determining the number of simple statements present in the proposition. Remember that each simple statement will be represented by a variable. Each variable will get its own column in the table. For example:
Rows = Truth Values
Rows = Truth Values
Next, we will sketch out the rows where we will place the truth values, once they are determined. We do this to ensure we explore every possible truth value to be assigned. Remember that each simple statement will be assigned each possible truth value (T and F). Each truth value will get its own row.
Next, we will sketch out the rows where we will place the truth values, once they are determined. We do this to ensure we explore every possible truth value to be assigned. Remember that each simple statement will be assigned each possible truth value (T and F). Each truth value will get its own row.
For compound statements with multiple variables, an easy trick for determining how many rows will be in the table is to multiply 2 (the number of possible truth values) to the power of the number of variables. For example:
For compound statements with multiple variables, an easy trick for determining how many rows will be in the table is to multiply 2 (the number of possible truth values) to the power of the number of variables. For example:
STEP 2: ASSIGN VALUES
STEP 2: ASSIGN VALUES
Simple Statements
Simple Statements
Finally, we will assign the appropriate truth values. In order to make sure all possible arrangements are accounted for in the table, begin on the far left column and divide the rows in half (assigning ‘T’ to the first half of the rows, and assigning ‘F’ the second half). Then move to the next column to the right and divide in half again, rotating equally between ‘T’ and ‘F’ assignments. For example:
Finally, we will assign the appropriate truth values. In order to make sure all possible arrangements are accounted for in the table, begin on the far left column and divide the rows in half (assigning ‘T’ to the first half of the rows, and assigning ‘F’ the second half). Then move to the next column to the right and divide in half again, rotating equally between ‘T’ and ‘F’ assignments. For example:
Compound Statements
Compound Statements
If you are determining the truth value of a simple statement (one variable), this will be all you need. If you are determining the truth value of compound statements and / or the validity of arguments, you will need to continue.
If you are determining the truth value of a simple statement (one variable), this will be all you need. If you are determining the truth value of compound statements and / or the validity of arguments, you will need to continue.
In order to determine the truth values for compound statements, it is worth noting the rules for each of the following tables. Each will correspond to the logical operator(s) present in each compound statement.
In order to determine the truth values for compound statements, it is worth noting the rules for each of the following tables. Each will correspond to the logical operator(s) present in each compound statement.
RULES: TRUTH VALUES
RULES: TRUTH VALUES
NEGATION
NEGATION
Notice that the truth values for P have simply been negated. All “not-P” truth values are just the opposite of what they would have been for P.
Notice that the truth values for P have simply been negated. All “not-P” truth values are just the opposite of what they would have been for P.
CONJUNCTION
CONJUNCTION
Notice that the conjunction of “P and Q” is only true if both components are true.
Notice that the conjunction of “P and Q” is only true if both components are true.
It is false if either or both components are false.
It is false if either or both components are false.
DISJUNCTION
DISJUNCTION
Notice that the disjunction of “either P or Q” is only false if both components are false.
Notice that the disjunction of “either P or Q” is only false if both components are false.
It is true if either or both components are true.
It is true if either or both components are true.
CONDITIONAL
CONDITIONAL
Notice that the conditional statement of “if P, then Q” is only false when the antecedent is true and the consequent is false.
Notice that the conditional statement of “if P, then Q” is only false when the antecedent is true and the consequent is false.
It is true in all other instances, even when the antecedent condition does not hold.
It is true in all other instances, even when the antecedent condition does not hold.
MATERIAL EQUIVALENCE
MATERIAL EQUIVALENCE
Notice that the material equivalence of “P if and only if Q” is only true when both components have the same truth value.
Notice that the material equivalence of “P if and only if Q” is only true when both components have the same truth value.
It is false if the truth values are different.
It is false if the truth values are different.
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