Deductive Logic/Truth Functions

Truth Functions
Consider the statement: (1)	Earth is larger than the moon. Astronomical observations show that this statement is true. Therefore, we can assign statement (1) a truth-value of true.

Now consider the statement: (2)	It is not the case that earth is larger than the moon. Certainly if statement (1) is true, this statement must be false. The truth-value of (2) can be determined by knowing only the truth-value of (1).

However, consider the statement: (3)	Harry told Sally the earth is larger than the moon. Even knowing that (1) is true, we cannot determine if (3) is true or false; we simply don’t know what Harry told Sally. This demonstrates that the phrase: “it is not the case that” performs a special operation on the statement that follows. The phrase inverts or negates the truth-value of the statement that follows. True statements become False and False statements become True.

If we
 * let p represent a statement such as (1),
 * let the symbol ¬ represent the phrase “it is not the case,”
 * use T to represent a statement with truth-value of true, and
 * let F represent a statement with truth-value false

then symbolically we have the following definition for the truth-function ¬ :

The truth-function operator ¬ is called “not” or “negation” or “inversion”. The table (known as a truth table) can be read as: “Whenever p is true then not p is false and whenever p is false, then not p is true.”

Deductive logic is an abstract conceptual model. Within deductive logic, statements are assigned a truth-value of either true or false; there is no maybe. We find this sensible for declarative statements such as “Earth is larger than the moon”, but troublesome or even nonsensical for complex or paradoxical statements such as “love is just a word”, or vague and general statements such “this is a great day” or statements of opinion such as “jazz is the greatest form of music”. The simplification caused by limiting the analysis to only two alternatives, true or false, can lead to a false dichotomy and often obscures many complexities present in our world. In fact, most natural language statements are uncertain and are only likely to be true, or only partially true, or can be interpreted in several ways so that they are true under certain interpretations and false otherwise. None-the-less, understanding the rigor of deductive logic provides an essential basis for clear thinking. “Essentially, all models are wrong,” George Box noted, “but some are useful.” Deductive logic is such a model. It introduces many tools and terms and provides an important and useful introduction to clear thinking.

Additional courses in the clear thinking curriculum explore the fascinating relationship between deductive logic and the logic used in real-world rhetoric. Start here and then continue learning with those courses.

Terminology summary
Here are brief definitions for the terms introduced so far:
 * Proposition—A statement that may be evaluated as being either true or false. Examples include: “Earth is larger than the moon”, “today is Tuesday”, “the cat is on the mat,” and “Paris is the capital of France”. Propositions can be assigned a truth-value.
 * Truth-value—Either the value true or the value false.
 * Assertion—A statement that assigns a truth value to a proposition. For example “It is true that earth is larger than the moon” is an assertion claiming the propulsion “earth is larger than the moon” is true (i.e. it has the truth-value of true).
 * Truth Function—A logical operation that derives a truth-value result from the truth value of its inputs in a prescribed way. The negation operation (represented by the ¬ symbol) introduced above is the first example of a truth function we will study in this course.
 * Negation operation—A truth function that returns the truth-value of true when given the input value false and returns the truth-value false when given the input value true. We represent the negation operation using the ¬ symbol which can be read as “not”.
 * Truth table—the results of a particular truth function tabulated over all possible truth-values of the inputs to that truth function. Truth tables are often used to define a particular truth function.

Assignment

 * 1) Review the definitions of each of the terms in the section above.
 * 2) Complete the following quiz:

Please select answers to each of the following questions: Press the "Submit" button after you have made your selections. {The statement “5 is a prime number” is an example of a: - Conjecture - Opinion + Proposition - Assertion - Truth-function
 * type=""}
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Correct
 * Incorrect. Please try again.
 * Incorrect. Please try again.

{The claim that “it is true that 5 is a prime number” is an example of a: - Conjecture - Opinion - Proposition + Assertion - Truth-function
 * type=""}
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Correct
 * Incorrect. Please try again.

{Within deductive logic, the only allowed truth-values are: - Perhaps - Maybe - True + True or False - Likely or unlikely - Uncertain - Unknown {The truth-values used within deductive logic correspond to those used in natural language. - True + False
 * type=""}
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Correct
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * type=""}
 * Incorrect. Please try again.
 * Correct

{The truth value of ¬true is: - True + False
 * type=""}
 * Incorrect. Please try again.
 * Correct

Conjunction
Recall the statement: (1)	Earth is larger than the moon Now consider the more complex statement: (4)	Earth is larger than the moon and mars. This means: (5)	Earth is larger than both the moon and mars Which is to say: (6)	Both Earth is larger than the moon and Earth is larger than mars. The truth value of (6) depends on the truth value of the two propositions that compose it. There are four possibilities:
 * 1) “Earth is larger than the moon” is true and “Earth is larger than mars is true.
 * 2) “Earth is larger than the moon” is true and “Earth is larger than mars is false.
 * 3) “Earth is larger than the moon” is false and “Earth is larger than mars is true.
 * 4) “Earth is larger than the moon” is false and “Earth is larger than mars is false.

Because (6) claims that both components of the statement are true, then (6) (and hence (4)) can only be true in the first case listed above. All other cases are false.

Statements such as (4) are called conjunctions. Conjunctions join two premises using the truth function “and” represented by the symbol ∧ (or alternatively by the symbol •)

The truth table for p ∧ q (also written as p • q, p & q, p ∩ q, or p AND q) is as follows:

In natural language terms, if both p and q are true, then the conjunction p ∧ q is true. For all other assignments of logical values to p and to q the conjunction p ∧ q is false.

Venn Diagram Notation
Venn diagrams provide a convenient and visual representation of logical relationships between a collection of sets. We introduce them here to provide an alternative representation of the truth functions we are studying.

The Logical conjunction truth function p ∧ q can be represented by this Venn diagram:

The circle on the left corresponds to “p” and the circle on the right corresponds to “q”. The overlapping lens-shaped region (shown red in the diagram) represents the conjunction of p and q. This region is part of p and part of q. The conjunction operation can also be interpreted as the intersection of two sets.

Disjunction
The dual of conjunction is disjunction, represented in natural language by “or”.

Consider the statement: (7)	Either the coin will turn up heads or the coin will turn up tails. and the statement: (8)	It may be cold today or it may be rainy. In natural language the word “or” has two distinct meanings. The statement (7) is an example of the exclusive use of the word “or”. In this case, if the coin turns up heads, it cannot turn up tails. The statement (8) is an example of the inclusive use of the word “or”. In this case it may happen that it is both cold and rainy. Deductive logic focuses on the inclusive form of “or” and generally ignores the exclusive or.

Disjunctions join two premises using the truth function “or” represented by the symbol ∨. Using (8) and following a line of thinking similar to that used above to introduce conjunctions the truth table for p ∨ q (also written as p | q, p ∪ q, or p OR q) is as follows:

In natural language terms, if both p and q are false, then the disjunction p ∨ q is false. For all other assignments of logical values to p and to q the disjunction p ∨ q is true. The logical disjunction truth function p ∨ q can be represented by this Venn diagram:

The circle on the left corresponds to “p” and the circle on the right corresponds to “q”. The area of both circles is colored red to represent the disjunction of p and q. This illustrates how the disjunction operation can also be interpreted as the union of two sets.

Assignment
Complete this quiz: Please select answers to each of the following questions: Press the "Submit" button after you have made your selections. {Given p is true and q is false, we can derive that p • q is: - True. + False
 * type=""}
 * Incorrect. Please try again.
 * Correct

{Given p is true and q is false, we can derive that p ∨ q is: + True. - False
 * type=""}
 * Correct
 * Incorrect. Please try again.

{Given that p • q is true, we can conclude that: - p is true - q is true + Both p and q are true - Both p and q are false
 * type=""}
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Correct
 * Incorrect. Please try again.

{Given that p ∨ q is false, we can conclude that: - p is true - q is true - Both p and q are true + Both p and q are false
 * type=""}
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Incorrect. Please try again.
 * Correct

Material Conditional
Consider the statement: (9)	If today is Saturday, then I get the day off. And also: (10)	 If I win the lottery then I’ll buy you a new car. These are examples of material conditional statements and are of the general form: if p then q These can be written as p → q, or alternatively 𝑝 ⇒ 𝑞, or 𝑝 ⊃ 𝑞. In such conditional constructions the p is called the antecedent and the q is called the consequent. What would be a helpful truth table for this function? If it is Saturday and I do get the day off, then statement (9) seems true. If it is Saturday and I don’t get the day off, the statement seems false. If it is not Saturday, then we don’t know how I will spend the day, so because the statement is not contradicted we can consider it true. I report sadly that I did not win the lottery so the antecedent of statement (10) is false. Since the consequent is conditional on the antecedent, and the antecedent is false, you can’t expect me to buy you a new car. The statement remains true, even if you don’t get your new car.

Following this reasoning, the truth table for the material conditional is:

In natural language terms, the material conditional is only false when the antecedent is true and the consequent is false. The material conditional truth function p → q can be represented by the Venn diagram shown on the right (red areas represent true). The diagram can be understood to correspond to the truth table, as follows. $$p\rightarrow q$$ is false only if $$p$$ is true and $$q$$ is false. This means that the set $$p\rightarrow q$$ excludes exactly those $$x$$ for which $$x\in p$$ and $$x\notin q$$, which corresponds to the white area.

Biconditional
Consider the statement: (11)	 I get the day off if and only if it is Saturday This is equivalent to these two material conditional statements: If it is Saturday then I get the day off And If I get the day off then it is Saturday. Statement (11) is an example of a material biconditional statement and has the general form p if and only if q These can be written as p iff q, or p ↔ q or p ≡ q. The expression is true only when the antecedent and consequence have the same truth value. The corresponding truth table is:

In natural language terms, the material biconditional is only true when the truth value of the antecedent is the same as the truth value of the consequent.



The material biconditional truth function p ↔ q can be represented by the Venn diagram shown on the right (red areas represent true). The diagram corresponds to the truth table as follows. The red lens region corresponds to the cases where both p and q are true. The external red area corresponds to the cases where both p and q are false.

Summary Truth Table
The definitions of the binary truth functions studied so far can be summarized in this truth table:

Key:
 * T = true, F = false
 * $$\land$$ = AND (logical conjunction)
 * $$\lor$$ = OR (logical disjunction)
 * $$\rightarrow$$ = conditional "if-then"
 * $$\iff$$ biconditional or "if-and-only-if"

Logical operators can also be visualized using Venn diagrams.

Assignment
Complete this quiz: Please select answers to each of the following questions: Press the "Submit" button after you have made your selections. {Given p is true and q is false, we can derive that p → q is: - True. + False
 * type=""}
 * Incorrect. Please try again.
 * Correct

{Given p is true and q is false, we can derive that p ↔ q is: - True. + False
 * type=""}
 * Incorrect. Please try again.
 * Correct

{Given p → q false, we can derive that q is: - True. + False
 * type=""}
 * Incorrect. Please try again.
 * Correct

{Given p ↔ q is false and p is true we can derive that q is: - True. + False
 * type=""}
 * Incorrect. Please try again.
 * Correct