Term
| Term | Description |
|---|---|
| Proposition | a declarative sentence, declare a fact (T or F) |
| Compound Proposition | Proposition with logical connectives (the logic operators) |
| converse | = converse |
| contrapositive | = contrapositive |
| inverse | = inverse |
| Commutative Law | 交換律 |
| Associated Law | 結合律 |
| Distributive Law | |
| Absorption Law | |
| satisfiable | any value for any variable can make it T |
| unsatisfiable | no value for any variable can make it T |
| taotology | Always T |
| contradiction | Always F |
| contingency | neither tautology nor contradiction |
| Predicates | a proposition with parameter |
| Propositional function | generalized proposition / proposition as function with parameter ( P(x) ) |
| Compound Expression | propositional logic with predicates |
Symbol
| Column1 | Column2 |
|---|---|
| U | Domain |
| conjunction/and | |
| disjunction/or | |
| not | |
| implication / | |
| if and only if / iff / xnor / | |
| for all | |
| exists/for some |
Precendence
Implication
Expression
is equivalent to
- if p then q
- if p, q
- q unless
- q if p
- q whenever p
- q follows from p
- p implies q
- p only if q
- q when p
- p is sufficient for q
- q is neccessary for p
- a neccessary condition for p is q
- a sufficient condition for q is p
Equivalent
(de morgan)
Logically Equivalent
Prop p,q
is tautology
written as
Quantifier / Predicate
Universal
for all x, P(x)
All natural number is greater than 0
Existential
for some x, P(x) / there exists
There is an element in natural number and it is 1
Uniqueness (Optional)
There is a unique x, one and only one
Examples
U = all people
S(x) = student
H(x) = study hard
G(x) = High score
No one studies hard
All students study hard
Some students get high score
If any student studies hard then he will get a high score
No student gets a high score
Examples 2
- All lions are fierce
- Some lions do not drink coffee
- Some fierce creatures do not drink coffee
logics:
U=creature
is lion
is fierce
drink coffee
Logically Equivalent
- Same truth value(iff)
- Valid in every domain of discourse
check
De Morgan for Quantifier
The Argument
is equivalent to
Rule of Inference
Modus Ponens
Modus Tollens
Hypothetical Syllogism
Disjunctive Syllogism
Addition
Simplification
Resolution
Universal Instantiation (UI)
Universal Generalization (UG)
Existential Instantiation (EI)
Existential Generalization (EG)
Valid Argument
example: