2. The Logic of Declarative Statements
Chapter 17
The Logic of Declarative Statements

3. Learning Outcomes
Identify negations, conjunctions, disjunctions, and conditional declarative statements
Translate simple and compound natural language declarative statements to and from symbolic notation
Learning Outcomes
Chapter first identifies negations, conjunctions, disjunctions, and conditional declarative statements.
It further translates simple and compound natural language declarative statements to and from symbolic notation.

4. Learning Outcomes
Identify tautologies, inconsistent statements, and contingent declarative statements using truth tables
Test declarative statements for implication and equivalence using truth tables
Test arguments composed of declarative statements for validity using truth tables
Learning Outcomes
Chapter identifies tautologies, inconsistent statements, and contingent declarative statements using truth tables.
Then tests declarative statements for implication and equivalence using truth tables.
Finally tests arguments composed of declarative statements for validity using truth tables.

5. Chapter Opening Video Chapter Opening Video
The power of logic of declarative statements is being discussed in the video.
Drawing valid inferences and evaluating inferences for logical validity is important for success in today’s complex, legalistic world.

6. Declarative Statements
Simple statements
Negations
Statement compounds: And, or, if, then
Declarative Statements
Valid argument: Premises logically imply or entail its conclusion, which makes it impossible for all of its premises to be true and its conclusion to be false.
Simple statements
Sentences people use to make claims, to give reasons, to assert beliefs, and to express their opinions.
Negations
Sentences that make the claim that something is false.
Statement compounds: And, or, if, then.
Used to connect simpler statements and express logically significant relationships between the statements.

7. Simple Statements
Grammatically correct construction in a given language used to assert that an idea is true
Janet is going to the beach
Sally and Broderick met online
Evaluation is limited to only two values
True
False
Simple Statements
Grammatically correct construction in a given language used to assert that an idea is true.
Example - Janet is going to the beach.
Sally and Broderick met online.
Evaluation is limited to only two values - True/false.
Logic of statements provides for no other possibilities.

8. Negations
Grammatically correct construction used to assert that a statement is false
I do not like frozen yogurt
It is not true that the abacus is obsolete
Expressed by the symbol ~ (tilde)
Negations
Grammatically correct construction used to assert that a statement is false.
Example - I do not like frozen yogurt.
It is not true that the abacus is obsolete.
Expressed by the symbol ~ (tilde).
If p stands for a simple statement, ~p will represent the negation of the statement.

9. Statement Compounds: And, Or, If, Then
Conjunction: Grammatically correct construction used to assert that two statements are both true
Expressed by ampersand (&)
(p & q)
Statement Compounds: And, Or, If, Then
Conjunction: Grammatically correct construction used to assert that two statements are both true.
Expressed by ampersand (&).
Parentheses are used to mark the beginning and the end of a conjunction.
(p & q)
Given any two statements, p and q, the expression (p & q) is true only in the situation where p is true and q is true, otherwise (p & q) is false.

10. Statement Compounds: And, Or, If, Then
Disjunction: Grammatically correct construction used to assert that one or both of two statements are true
Expressed by a v-shaped wedge
(p v q)
Statement Compounds: And, Or, If, Then
Disjunction: Grammatically correct construction used to assert that one or both of two statements are true.
Expressed by a v-shaped wedge.
(p v q)
Given any two statements, p and q, the expression (p v q) is true in every situation except the one where p is false and q is false.

11. Statement Compounds: And, Or, If, Then
Conditional
Grammatically correct construction used to assert that if an antecedent statement is true, then a consequent statement is true
Expressed by an arrow (→) with opening and closing parentheses
Statement Compounds: And, Or, If, Then
Conditional: Grammatically correct construction used to assert that if an antecedent statement is true, then a consequent statement is true.
Expressed by an arrow (→) with opening and closing parentheses.
Example (p → q) means if p, then q.
Given any two statements, p and q, the expression (p S q) is false only in the situation where p is true and q is false, otherwise (p S q) is true.

12. Translating Between Symbolic Logic and a Natural Language
Grammatically correct expressions
Translation to English
Translating Between Symbolic Logic and a Natural Language
Language of the Logic of Statements - Relatively economical.
English - Rich, malleable, and nuanced natural language.
Grammatically correct expressions
Expressions in symbolic logic notation are easy to translate into a natural language.
Translation to English
Grammatically correct symbolic logic expressions can be translated into English using guidelines.

13. Grammatically Correct Expressions
Rules to form grammatically correct expressions in the language of symbolic logic
Statement letter is a grammatically correct expression
Placing ~ in front of any grammatically correct expression generates another grammatically correct expression
Grammatically Correct Expressions
Rules to form grammatically correct expressions in the language of symbolic logic.
Statement letter is a grammatically correct expression. p q
Placing ~ in front of any grammatically correct expression generates another grammatically correct expression. ~q

14. Grammatically Correct Expressions
Placing &, v, or → between any two grammatically correct expressions and enclosing it with a pair of parentheses generates another grammatically correct expression
Grammatically Correct Expressions
Rules to form grammatically correct expressions in the language of symbolic logic.
Placing &, v, or → between any two grammatically correct expressions and enclosing it with a pair of parentheses generates another grammatically correct expression.
((s v p) → ~q)

15. Translating to English
Rules
Render ~A as “It is not the case that A”
Render (A & B) as “A and B”
Render (A v B) as “Either A or B”
Render (A → B) as “If A, then B”
Translating to English
Rules
Render ~A as “It is not the case that A.”
Render (A & B) as “A and B.”
Render (A v B) as “Either A or B.”
Render (A → B) as “If A, then B.”
Grammatically correct symbolic logic expressions can be translated into English using a few guidelines, if we know what each statement letter stands for.

16. Translating to Symbolic Logic
Translating a telephone tree
Telephone tree instruction is an exercise in the Logic of Statements
Symbolic representation
((((p → q) v (r → s)) v (p1 → q1))v (r1 & s1))
Translating to Symbolic Logic
Translation to symbolic logic begins with the realization that natural languages like English offer a variety of grammatically correct ways to express negations, conjunctions, disjunctions, and conditionals.
Translating a telephone tree
Telephone tree instruction is an exercise in the Logic of Statements.
English statement - If you are calling for technical support, press 1.
If you want to know your account balance, press 2.
If you want to place an order, press 3.
Otherwise stay on the line and your call will be answered by the next available agent.
Symbolic representation - ((((p → q) v (r → s)) v (p1 → q1))v (r1 & s1))
p = You are calling for technical support.
q = [You should] press 1.
r = You want to know your account balance.
s = [You should] press 2.
p1 = You want to place an order.
q1 = [You should] press 3.
r1 = [You should] stay on the line.
s1 = Your call will be answered by the next available agent.

17. Translating to Symbolic Logic
Translation knowledge from telephone tree
Determine whether the sentence is a declarative statement
Sentences used to make assertions can be translated into symbolic logic
Declarative statements handle negations, conjunctions, disjunctions, conditionals, and simple assertions
Translating to Symbolic Logic
Translation knowledge from telephone tree
Determine whether the sentence is a declarative statement.
Sentences used to make assertions can be translated into symbolic logic.
Declarative statements handle negations, conjunctions, disjunctions, conditionals, and simple assertions.
Process of translating from English into symbolic logic requires the application of our critical thinking skills of analysis and interpretation.

18. Detecting the Logical Characteristics of Statements
Building truth tables
Tautologies, inconsistent statements, and contingent statements
Testing for implication and equivalence
Detecting the Logical Characteristics of Statements
Truth tables can be used to determine the truth value of any grammatically correct expression in the language of the Logic of Statements.
Building truth tables
Explains the steps involved in building a truth table for a grammatically correct expression.
Tautologies, inconsistent statements, and contingent statements
Concept of a tautology can be expressed in the Logic of Statements.
Testing for implication and equivalence
Implication is an important logical relationship between statements.

19. Building Truth Tables
Truth table for a grammatically correct expression in symbolic notation, A
Count how many different statement letters are used in A
Columns on the left-hand side of a truth table
Rows of a truth table are organized in a predictable order
Building Truth Tables
Truth table for a grammatically correct expression in symbolic notation, A.
Count how many different statement letters are used in A.
Determines the rows of the truth table.
Columns on the left-hand side of a truth table.
Display the possible assignments of truth values to each of the statement letters in A.
Statement letter columns are organized alphabetically.
Rows of a truth table are organized in a predictable order.
Top row assigns true to all the statement letters in A.
Bottom row assigns false to all the statement letters.
Middle rows follow a standard pattern.

20. Truth Table of ((p & q) → ~q)
Expression is a conditional statement with two distinct statement letters.
Antecedent is a conjunction.
Consequent is a negation.
Table is implemented using the steps mentioned in building truth tables.

21. Contingent Statement Grammatically correct expression
True under at least one possible assignment of truth values to its component simple statements
False under another possible assignment of truth values to its component simple statements
Contingent Statement
Grammatically correct expression.
True under at least one possible assignment of truth values to its component simple statements.
False under another possible assignment of truth values to its component simple statements.
(p & q), (p v q), and (p → q) are contingent statements.

22. Inconsistent Statement
Grammatically correct expression
False under every possible assignment of truth values to its component simple statements
Referred as self-contradictory
Inconsistent Statement
Grammatically correct expression.
False under every possible assignment of truth values to its component simple statements.
Referred as self-contradictory.
(p & ~p) is an inconsistent statement.

23. Tautology Grammatically correct expression
True under every possible assignment of truth values to its component simple statements
Logic of Statements does not contain all possible tautologies
Tautology
Grammatically correct expression.
True under every possible assignment of truth values to its component simple statements.
Logic of Statements does not contain all possible tautologies.
Internally inconsistent or tautological statements in English can be created in additional ways that become evident at levels beyond the Logic of Statements.

24. Testing for Implication and Equivalence
A implies B
If there is no interpretation of the statement letters of A and B such that A is true and B is false
If the grammatical expression generated by the structure (A → B) is a tautology
Equivalence
A and B, are equivalent: If, and only if, the biconditional (A ≡ B) is a tautology
Testing for Implication and Equivalence
Implication.
A implies B: If there is no interpretation of the statement letters of A and B such that A is true and B is false.
If the grammatical expression generated by the structure (A → B) is a tautology.
Important logical relationship between statements.
Can be checked by using truth tables.
Equivalence.
A and B, are equivalent: If, and only if, the biconditional (A ≡ B) is a tautology.
If A and B have the same truth value under every interpretation of their statement letters.

25. Evaluating Arguments for Validity
Testing symbolic arguments for validity
Testing natural language arguments for validity
Evaluating Arguments for Validity
Testing symbolic arguments for validity.
Argument is valid, if it is impossible for all of its premises to be true and its conclusion false.
Testing natural language arguments for validity.
Steps to test an argument for validity.
Form the conditional that uses as its antecedent the conjunction of the argument’s premises and as its consequent the conclusion of the argument.
Build the truth table for that conditional statement.
Examine the truth table to determine whether the conditional is a tautology.
If that conditional is a tautology, the argument is valid at the level of the Logic of Statements.

26. Testing Symbolic Arguments for Validity
Consider the example
Premise #1 (q v r)
Premise #2 ~r
Conclusion q
Form the conditional (((q v r) & ~r) → q)
Build the truth table
Conditional is a tautology then the argument is valid at this level of logic
Testing Symbolic Arguments for Validity
Consider the example.
Premise #1 (q v r)
Premise #2 ~r
Conclusion q
Form the conditional (((q v r) & ~r) → q)
Steps to form conditional.
Form the conditional that uses the conclusion as its consequent.
((_________) → q)
Add the conjunction of the two premises as its antecedent.
(((q v r) & ~r) → q)
Build the truth table.
Conditional is a tautology then the argument is valid at this level of logic.

27. Testing Natural Language Arguments for Validity
Translate natural language premises and conclusion into symbolic logic notation
Form the conditional ((conjunction of the premises) → conclusion)
Build the conditional’s truth table
If the conditional is a tautology, then the argument is valid
Testing Natural Language Arguments for Validity
Evaluation of natural language arguments for validity requires translating them into symbolic notation prior to applying the process.
Translate natural language premises and conclusion into symbolic logic notation.
Form the conditional ((conjunction of the premises) → conclusion).
Build the conditional’s truth table.
If the conditional is a tautology, then the argument is valid.

28. Discussion Question
What are the advantages that the careful analysis of language provides when trying to interpret exactly what is being said?
Answer by giving an example from your own experience and explain your example
Discussion Question
This chapter required one to translate between natural language declarative statements and symbolic logic.
What are the advantages that the careful analysis of language provides when trying to interpret exactly what is being said?
Answer by giving an example from your own experience and explain your example.

29. Sketchnote Video Sketchnote Video
The video summarizes the concepts of logic, translating symbolic notion into a natural language and truth tables.