Truth Functional Connectives

Question 1

An argument’s validity is a function of ______

Answer 1

its form

Question 2

The form of an argument is a function of _______

Answer 2

the arrangement of its terms

Question 3

What is a statement connective?

Answer 3

Any word or collection of words that, when used with one or more statements, creates a new statement.

Question 4

What constitutes a logical term?

Answer 4

Depends on the level of logical analysis.

• In syllogistic logic, these are the words “all” “some” “no” “not” “is” and “are”
• In sentential logic, these are any truth-functional statement connective. Conjunction, disjunction, negation, conditional and biconditional.

Question 5

What is a compound statement?

Answer 5

A statement formed by using one or more statements and a connective.

Question 6

What is a simple statement?

Answer 6

A standalone statement not formed using a connective.

Question 7

How are syllogistic statements seen in sentential logic?

Answer 7

• considered simple statements with no internal structure
• the syllogistic logical terms (all, some, no, not, is, are) are not considered valid logical terms in sentential logic
• can be combined with connectives to create compound statements.

Question 8

What limits are there to the complexity of a compound statement?

Answer 8

• No theoretical limits. Multiple compound statements can be combined with connectives to form even larger, more complex compound statements.
• Practical limits in the amount of complexity a human can understand.

Question 9

In sentential logic, what is an atom?

Answer 9

A simple statement comprising part of a compound statement.
Note that the atom can be a full statement within syllogistic logic, but it is the smallest element of a sentential statement.

Question 10

In sentential logic, what is a molecule?

Answer 10

A compound statement made up of statements combined with statement connectives.

Question 11

True or False: All connectives are analyzed in sentential logic.

Answer 11

False: Sentential logic is only concerned with truth-functional connectives. That is, only connectives that determine the truth value of a compound statement when the truth value of the constituent statements is known.

Question 12

What is a truth value?

Answer 12

In sentential logic, a truth value is a value of True (T) or False (F) that can be assigned - or computed - for any statement.

Question 13

What is a truth function?

Answer 13

A function that uses statement truth values and truth-function operators (connectives) to determine the ultimate truth value of a compound statement.

Question 14

True or False: All connectives are truth-functional

Answer 14

False: Most connectives are not truth-functional.

Question 15

How can one determine whether a connective is truth-functional?

Answer 15

• If by using that connective, the truth value of a compound statement can be determined when the truth value of constituent statements is known.
• If a truth value for the compound statement cannot be determined, even when the truth values of the constituent statements is known, the connective is not truth functional

Question 16

What are the five primary truth-functional connectives?

Answer 16

• Conjunction
• Disjunction
• Negation
• Conditional (conditional promises/requests)
• Biconditional

Question 17

What connective is associated with Conjunction in sentential logic?
What symbol represents Conjunction in abbreviated notation?

Answer 17

• In sentential logic Conjunction is associated with the two-place connective “_____ and _____”
• Represented by the ampersand (&)

Question 18

What connective is associated with Disjunction in sentential logic?
What symbol represents Disjunction in abbreviated notation?

Answer 18

• In sentential logic Disjunction is associated with the two-place connective “_____ or _____”
• Represented by the logical and inclusive or (∨) (vel in latin)

Question 19

What connective is associated with Negation in sentential logic?
What symbol represents Negation in abbreviated notation?

Answer 19

• In sentential logic Negation is associated with the one-place connective “it is not true that _____”
• Represented by the tilde (~)
• “it is false that _____” “it is not the case that _____” are valid equivalents

Question 20

What connective is associated with the Conditional in sentential logic?
What symbol represents the Conditional in abbreviated notation?

Answer 20

• In sentential logic the Conditional is associated with the two-place connective “if _____, then _____.”
• Represented by an arrow pointing right (→)

Question 21

What connective is associated with the Biconditional in sentential logic?
What symbol represents the Biconditional in abbreviated notation?

Answer 21

• In sentential logic the Biconditional is associated with the two-place connective “_____ if and only if _____”
• Represented by a double arrow pointing left and right (↔)

Question 22

In sentential logic, how are statements abbreviated to simplify analysis?

Answer 22

A whole statement can be represented by a letter or other unique symbol. As long as it is clear for anyone analyzing which unique symbols correspond to which unique statements.

Question 23

How is the conjunction of R and S represented in sentential logic?

Answer 23

(R&S)

Question 24

What is a conjunct?

Answer 24

Either of the constituent statements in a conjunction may be referred to as a conjunct.

Question 25

Complete the following Conjunction Truth Value Table. Is this table true for all Conjunctions?

Answer 25

Yes, this table holds for all sentential truth-functional conjunctions.

Question 26

What is a truth-value table?

Answer 26

- Used to determine the truth-value (T or F) of a compound truth-functional statement. - A table can be constructed to compute the truth-value of a compound truth-functional statement for every possible combination of truth-values of the constituent statements.

Question 27

A conjunction is true if and only if \_\_\_\_\_\_\_\_

Answer 27

both conjuncts are true. If one or both conjuncts is false, the conjunction is false.

Question 28

What is a disjunct?

Answer 28

Either of the constituent statements in a disjunction may be referred to as a disjunct.

Question 29

What is the difference between the common English "or" and the logical "or" used in sentential logic?

Answer 29

* Common English "or" can either be exclusive or inclusive depending on context * Exclusive: Would you like a baked potato, OR french fries (choice between one or the other) * Inclusive: Would you like coffee or dessert (choosing one does not exclude the other) * Logical "or" is always inclusive * Represented by the wedge ∨, symbollic of Latin vel (inclusive or)

Question 30

Complete the following Disjunction Truth Value Table. Is this table true for all Disjunctions?

Answer 30

Yes, this table holds for all sentential truth-functional disjunctions.

Question 31

Under what circumstances is a disjunction false?

Answer 31

- A disjunction is false if and only if both disjuncts are false - If either (or both) disjuncts are true, the disjunction is true regardless of the truth value of the other disjunct

Question 32

Describe "because" as a connective. Is "because" truth-functional?

Answer 32

- A two-place connective - Not considered truth-functional - The only case where "A because B" might be true is when both constituents are true, however this is often not enough to establish the truth of the compound statement. More information is needed.

Question 33

Complete the following Negation Truth Table. Is this table true for all negations?

Answer 33

Yes, this table is true for all negations. Given the truth value of statement A, the negation of A (~A) will always be the opposite

Question 34

How are the constituents of a conditional statement identified? How do these differ from other compound statements?

Answer 34

- The antecedent is the first constituent of a conditional, immediately following the 'if' - The consequent is the second constituent in a conditional, immediately following the 'then' - These differ from other connectives in that the constituent statements do not have equal roles. The consequent is dependent on the antecedent. - Generally A→B is not equivalent to B→A

Question 35

True or False: All conditional "if \_\_\_\_\_, then \_\_\_\_\_" statements are truth-functional.

Answer 35

False: Any conditional that is stated in the subjunctive mood is not truth-functional. Subjunctive mood indicates a hypothetical scenario, a demand or suggestion.

Question 36

What versions of the conditional are considered truth-functional?

Answer 36

- Conditional promises and conditional requests - If condition A is met, then result B is promised

Question 37

Complete the truth table for a truth-functional conditional. Explain the result for each scenario.

Answer 37

1. The condition is met and the promised result is kept. Statement is True 2. The condition is met but the promised result is not kept. Statement is False 3. The condition is not met but the promised result is given anyway. The promise did not stipulate that consequent B would only occur once condition A was met. Presumably, other conditions may result in B. Statement is True 4. The condition is not met and the promised result is not given. The initial promise implies that not meeting condition A means result B is withheld, so the statement is True.

Question 38

A truth-functional conditional A→B is false if and only if...

Answer 38

...the antecedent A is true but the consequent B is false.

Question 39

A biconditional A↔B is true if and only if...

Answer 39

...both constituents share the same truth value.

Question 40

Complete the following Biconditional Truth Table

Answer 40

Question 41

What is a formula?

Answer 41

- A grammatically correct string of symbols that represents a compound statement

Question 42

What rules define grammatical correctness for a formula?

Answer 42

- Any upper case roman letter is a formula (these represent statements) - If A is a formula, then so is ~A - If A and B are formulas, then so is (A&B) - If A and B are formulas, then so is (A∨B) - If A and B are formulas, then so is (A→B) - If A and B are formulas, then so is (A↔B) - Nothing else is a formula

Question 43

What limits are there for the length of a formula?

Answer 43

- Just like with statements, a formula can theoretically be infinitely long, as long as everything is grammatically correct - Practically, there are limits to how long a formula can be and still be understandle by humans

Question 44

What is the difference between an official statement and an unofficial statement?

Answer 44

- Parantheses - Official statements use parantheses to distinguish one statement from another and can be easily combined with other statements - Unofficial statements have no parantheses. They can still be considered and analyzed as standalone statements, however parantheses must be added if you want to combine with other statements.

Question 45

Why are parantheses important when creating and analyzing complex formulas?

Answer 45

- Similar to algebraic expressions, parantheses isolate statements and determine order of operations for computing truth values.

Question 46

What is the order of operations for computing the truth value of a complex formula?

Answer 46

- Start with statements in the innermost parantheses of the formula - For whatever level of the formula you're in, start by computing any negations (~)