Logic Flashcards
Argument
An ARGUMENT is any collection of premises together with a conclusion
Validity
An argument is VALID if and only if it is impossible for all of its premises to be true and the conclusion false
Soundness
An argument is SOUND if and only if it is both valid and all of its premises are true
Jointly Consistent
Sentences are JOINTLY CONSISTENT if and only if it is possible for them all to be true together
Jointly Inconsistent
Sentences are JOINTLY INCOSISTENT if and only if it is not possible for them all to be true together
Necessary Truth
A sentence is a NECESSARY TRUTH if and only if it is true and it is not possible for it to be false
Necessary Falsehood
A sentence is a NECESSARY FALSEHOOD if and only if it is false and not possible for it to be true
Contingent
A sentence is CONTINGENT* if and only if* it is possible for it to be true and possible for it to be false
Symbolization Key (TFL)
A SYMBOLIZATION KEY specifies which atomic sentence each letter represents in the relevant case
CONNECTIVES, also known as logical operators, are symbols used to combine simple statements into more complex ones
Negation (¬)
Flips the truth value of a statement
Example: ¬P means “It is not the case that P”
Conjunction (∧)
True only when both conjuncts are true
Example: P ∧ Q means “P and Q”
Disjunction (∨)
True when at least one disjunct is true
Example: P ∨ Q means “P or Q” (inclusive or)
Conditional (→)
False only when the antecedent is true and the consequent false
Example: P → Q means “If P, then Q”
Biconditional (↔)
True when both components have the same truth value
Example: P ↔ Q means “P if and only if Q”
Main Logical Operator
The MAIN LOGICAL OPERATOR in a formula is the operator that was introduced last when that formula was constructed using the recursive rules
Scope of a Connective (TFL)
The SCOPE of a connective (in a sentence) is the subsentence for which that connective is the main logical operator
Truth-Functional Connective
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)
Valutation
A VALUATION is any assignment of truth values to particular atomic sentences of TFL
Complete Truth Table
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
Tautology
𝒜 is a TAUTOLOGY if and only if it is true on every valuation
Contradiction
𝒜 is a CONTRADICTION if and only if it is false on every valuation
Tautologically Equivalent
𝒜 and 𝐵 are TAUTOLOGICALLY EQUIVALENT if and only if they have the same truth value on every valuation
Jointly Tautologically Consistent
𝒜1, 𝒜2, … 𝒜ₙ are JOINTLY TAUTOLOGOCALLY CONSISTENT if and only if there is some valuation that makes them all true
Jointly Tautologically Inconsistent
𝒜1, 𝒜2, … 𝒜ₙ are JOINTLY TAUTOLOGOCALLY INCONSISTENT if and only if there is no valuation that makes them all true
Tautologically Entail
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
If 𝒜1, 𝒜2, … 𝒜ₙ tautologically entail 𝒞 , then 𝒜1, 𝒜2, … 𝒜ₙ , ∴ 𝒞 is valid
Vacuously True
When F is an empty predicate, every sentence of the form Ɐx(Fx → …) will be VACUOUSLY TRUE
Term
A TERM is any name or variable
Scope
The SCOPE of a logical operator in a formula is the sub-formula for which that operator is the main logical operator
Bound Variable
A BOUND VARIABLE is an occurrence of a variable x that is within the scope of either Ɐx or ∃x
Free Variable
A FREE VARIABLE is any variable that is not bound
Interpretation
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
