OPTIMIZED LOGIC STUDY GUIDE (CH 2.2-3.4)
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45 Terms
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Conditional (p→q)
False only when p is true and q is false.
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Truth table for p→q
Only false in the row where p is true and q is false.
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Vacuously true
Occurs when the hypothesis p is false.
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Key equivalence of p→q
p→q is equivalent to ~p∨q.
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Negation of p→q
~(p→q) is equivalent to p∧~q.
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Contrapositive
~q→~p, which is equivalent to the original conditional.
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Converse
q→p, which is not equivalent to the original conditional.
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Inverse
~p→~q, which is not equivalent to the original conditional.
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Meaning of 'p only if q'
Represents the conditional p→q.
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Biconditional (p↔q)
Defined as (p→q)∧(q→p).
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Necessary condition
Saying 'You can't have p without q' means p→q.
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Sufficient condition
Saying 'q automatically gives you p' means q→p.
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Modus Ponens
If p→q and p are true, then q must be true.
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Modus Tollens
If p→q and ~q are true, then ~p must be true.
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Transitivity
If p→q and q→r are true, then p→r must be true.
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Elimination
If p∨q and ~p are true, then q must be true.
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Division into Cases
If p∨q, p→r, and q→r are true, then r must be true.
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Contradiction Rule
If ~p leads to a contradiction, then p must be true.
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Converse Error
The fallacy where p→q and q leads to the invalid conclusion of p.
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Inverse Error
The fallacy where p→q and ~p leads to the invalid conclusion of ~q.
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Predicate P(x)
A statement that contains variables.
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Truth set
The set of values that make the predicate P(x) true.
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Universal quantifier (∀x)
Indicates 'for all x'.
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Existential quantifier (∃x)
Indicates 'there exists at least one x'.
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Universal statement form
∀x, P(x) or ∀x, if P(x) then Q(x).
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Negation of universal statements
~(∀x, P(x)) is equivalent to ∃x, ~P(x).
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Negation of existential statements
~(∃x, P(x)) is equivalent to ∀x, ~P(x).
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Negating Universal Conditional
~(∀x, P(x)→Q(x)) is equivalent to ∃x, P(x)∧~Q(x).
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Order of quantifiers
The order matters: ∀x, ∃y, P(x,y) is not the same as ∃y, ∀x, P(x,y).
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Negating multiple quantifiers
~(∀x, ∃y, P(x,y)) is equivalent to ∃x, ∀y, ~P(x,y).
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Universal Instantiation
If ∀x, P(x) is true, then P(a) is true for any specific a.
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Universal Modus Ponens
If ∀x, P(x)→Q(x) and P(a) are true, then Q(a) must be true.
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Universal Modus Tollens
If ∀x, P(x)→Q(x) and ~Q(a), then ~P(a) must be true.
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Venn diagram for 'All A are B'
Represented as A circle inside the B circle.
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Venn diagram for 'No A are B'
Represented as non-overlapping circles.
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'→' Symbol
Represents a conditional statement (if-then).
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'↔' Symbol
Represents a biconditional statement (if and only if).
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'∧' Symbol
Represents conjunction (and).
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'∨' Symbol
Represents disjunction (or).
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'~' Symbol
Represents negation (not).
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'∀' Symbol
Represents the universal quantifier.
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'∃' Symbol
Represents the existential quantifier.
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'∈' Symbol
Represents 'element of'.
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'∴' Symbol
Represents 'therefore'.
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Operation order for logical expressions
Negation (~), followed by conjunction (∧) and disjunction (∨), then conditional (→) and biconditional (↔).
Flashcards (43)