Logic

Question 1

Argument

Answer 1

An ARGUMENT is any collection of premises together with a conclusion

Question 2

Validity

Answer 2

An argument is VALID if and only if it is impossible for all of its premises to be true and the conclusion false

Question 3

Soundness

Answer 3

An argument is SOUND if and only if it is both valid and all of its premises are true

Question 4

Jointly Consistent

Answer 4

Sentences are JOINTLY CONSISTENT if and only if it is possible for them all to be true together

Question 5

Jointly Inconsistent

Answer 5

Sentences are JOINTLY INCOSISTENT if and only if it is not possible for them all to be true together

Question 6

Necessary Truth

Answer 6

A sentence is a NECESSARY TRUTH if and only if it is true and it is not possible for it to be false

Question 7

Necessary Falsehood

Answer 7

A sentence is a NECESSARY FALSEHOOD if and only if it is false and not possible for it to be true

Question 8

Contingent

Answer 8

A sentence is CONTINGENT* if and only if* it is possible for it to be true and possible for it to be false

Question 9

Symbolization Key (TFL)

Answer 9

A SYMBOLIZATION KEY specifies which atomic sentence each letter represents in the relevant case

Question 10

Expression of TFL

Answer 10

An EXPRESSION OF TFL is any string of symbols in TFL

Question 11

Connectives

Answer 11

CONNECTIVES, also known as logical operators, are symbols used to combine simple statements into more complex ones

Question 12

Negation (¬)

Answer 12

Flips the truth value of a statement

Example: ¬P means “It is not the case that P”

Question 13

Conjunction (∧)

Answer 13

True only when both conjuncts are true

Example: P ∧ Q means “P and Q”

Question 14

Disjunction (∨)

Answer 14

True when at least one disjunct is true

Example: P ∨ Q means “P or Q” (inclusive or)

Question 15

Conditional (→)

Answer 15

False only when the antecedent is true and the consequent false

Example: P → Q means “If P, then Q”

Question 16

Biconditional (↔)

Answer 16

True when both components have the same truth value

Example: P ↔ Q means “P if and only if Q”

Question 17

Main Logical Operator

Answer 17

The MAIN LOGICAL OPERATOR in a formula is the operator that was introduced last when that formula was constructed using the recursive rules

Question 18

Scope of a Connective (TFL)

Answer 18

The SCOPE of a connective (in a sentence) is the subsentence for which that connective is the main logical operator

Question 19

Truth-Functional Connective

Answer 19

A connective is TRUTH-FUNCTIONAL if and only if the truth value of a sentence featuring that connective as its main logical operator is uniquely determined by the truth value(s) of the constituent sentence(s)

Question 20

Valutation

Answer 20

A VALUATION is any assignment of truth values to particular atomic sentences of TFL

Question 21

Complete Truth Table

Answer 21

A COMPLETE TRUTH TABLE has a line for every possible assignment of True and False to the relevant atomic sentences

Each line represents a valuation, and a complete truth table has a line for all possible different valuations

Question 22

Tautology

Answer 22

𝒜 is a TAUTOLOGY if and only if it is true on every valuation

Question 23

Contradiction

Answer 23

𝒜 is a CONTRADICTION if and only if it is false on every valuation

Question 24

Tautologically Equivalent

Answer 24

𝒜 and 𝐵 are TAUTOLOGICALLY EQUIVALENT if and only if they have the same truth value on every valuation

Question 25

Jointly Tautologically Consistent

Answer 25

𝒜1, 𝒜2, ... 𝒜ₙ are **JOINTLY TAUTOLOGOCALLY CONSISTENT** *if and only if* there is some valuation that makes them all **true**

Question 26

Jointly Tautologically Inconsistent

Answer 26

𝒜1, 𝒜2, ... 𝒜ₙ are **JOINTLY TAUTOLOGOCALLY INCONSISTENT** *if and only if* there is no valuation that makes them all **true**

Question 27

Tautologically Entail

Answer 27

The sentences 𝒜1, 𝒜2, ... 𝒜ₙ **TAUTOLOGOCALLY ENTAIL** the sentence 𝒞 if there is **no** valuation of the atomic sentences which makes all of 𝒜1, 𝒜2, ... 𝒜ₙ **true** and 𝒞 **false** ## Footnote If 𝒜1, 𝒜2, ... 𝒜ₙ tautologically entail 𝒞 , then 𝒜1, 𝒜2, ... 𝒜ₙ , ∴ 𝒞 is valid

Question 28

Vacuously True

Answer 28

When F is an empty predicate, every sentence of the form Ɐx(Fx → ...) will be **VACUOUSLY TRUE**

Question 29

Expression of [FOL](https://obsidian.nuallan-ai.com/Introduction+to+Logic/Terms/First-Order+Logic)

Answer 29

An **EXPRESSION of [FOL](https://obsidian.nuallan-ai.com/Introduction+to+Logic/Terms/First-Order+Logic)** is any string of symbols of [FOL](https://obsidian.nuallan-ai.com/Introduction+to+Logic/Terms/First-Order+Logic)

Question 30

Term

Answer 30

A **TERM** is any name or variable

Question 31

Scope

Answer 31

The **SCOPE** of a *logical operator* in a formula is the sub-formula for which that operator is the **main logical operator**

Question 32

Bound Variable

Answer 32

A **BOUND VARIABLE** is an occurrence of a variable x that is within the scope of either Ɐx or ∃x

Question 33

Free Variable

Answer 33

A **FREE VARIABLE** is any variable that is not bound

Question 34

Sentence of [FOL](https://obsidian.nuallan-ai.com/Introduction+to+Logic/Terms/First-Order+Logic)

Answer 34

A **SENTENCE of FOL** is any formula of [FOL](https://obsidian.nuallan-ai.com/Introduction+to+Logic/Terms/First-Order+Logic) that contains no free variables

Question 35

Interpretation

Answer 35

An **INTERPRETATION** consists of *three* things: 1. The specification of a domain 2. An assignment of exactly one object within the domain for each name that we care to consider 3. For each predicate (except ‘=‘) that we are considering, a specification of what things (in what order) the predicate is to be true of