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Logic > Truth Functional Connectives > Flashcards

Truth Functional Connectives Flashcards

1
Q

An argument’s validity is a function of ______

A

its form

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2
Q

The form of an argument is a function of _______

A

the arrangement of its terms

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3
Q

What is a statement connective?

A

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

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4
Q

What constitutes a logical term?

A

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.
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5
Q

What is a compound statement?

A

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

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6
Q

What is a simple statement?

A

A standalone statement not formed using a connective.

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7
Q

How are syllogistic statements seen in sentential logic?

A
  • 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.
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8
Q

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

A
  • 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.
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9
Q

In sentential logic, what is an atom?

A

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.

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10
Q

In sentential logic, what is a molecule?

A

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

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11
Q

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

A

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.

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12
Q

What is a truth value?

A

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

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13
Q

What is a truth function?

A

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

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14
Q

True or False: All connectives are truth-functional

A

False: Most connectives are not truth-functional.

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15
Q

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

A
  • 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
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16
Q

What are the five primary truth-functional connectives?

A
  • Conjunction
  • Disjunction
  • Negation
  • Conditional (conditional promises/requests)
  • Biconditional
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17
Q

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

A
  • In sentential logic Conjunction is associated with the two-place connective “_____ and _____”
  • Represented by the ampersand (&)
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18
Q

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

A
  • In sentential logic Disjunction is associated with the two-place connective “_____ or _____”
  • Represented by the logical and inclusive or (∨) (vel in latin)
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19
Q

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

A
  • 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
20
Q

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

A
  • In sentential logic the Conditional is associated with the two-place connective “if _____, then _____.”
  • Represented by an arrow pointing right (→)
21
Q

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

A
  • 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 (↔)
22
Q

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

A

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.

23
Q

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

A

(R&S)

24
Q

What is a conjunct?

A

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

25
Q

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

A

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

26
Q

What is a truth-value table?

A
  • 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.
27
Q

A conjunction is true if and only if ________

A

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

28
Q

What is a disjunct?

A

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

29
Q

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

A
  • 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)
30
Q

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

A

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

31
Q

Under what circumstances is a disjunction false?

A
  • 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
32
Q

Describe “because” as a connective. Is “because” truth-functional?

A
  • 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.
33
Q

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

A

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

34
Q

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

A
  • 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
35
Q

True or False: All conditional “if _____, then _____” statements are truth-functional.

A

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

36
Q

What versions of the conditional are considered truth-functional?

A
  • Conditional promises and conditional requests
  • If condition A is met, then result B is promised
37
Q

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

A
  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.
38
Q

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

A

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

39
Q

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

A

…both constituents share the same truth value.

40
Q

Complete the following Biconditional Truth Table

A
41
Q

What is a formula?

A
  • A grammatically correct string of symbols that represents a compound statement
42
Q

What rules define grammatical correctness for a formula?

A
  • 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
43
Q

What limits are there for the length of a formula?

A
  • 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
44
Q

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

A
  • 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.
45
Q

Why are parantheses important when creating and analyzing complex formulas?

A
  • Similar to algebraic expressions, parantheses isolate statements and determine order of operations for computing truth values.
46
Q

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

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

Logic (2 decks)

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