introduction to matheatic logic

key concepts

 

proposition: a statement that is true of false

propostional variable: a variable that represents a proposition, e.g., use p describe proposition Tom is a boy

negation of p: the proposition with truth value opposite to the truth value of p

logic operator: operator used to combine propositions

compound propostion: a proposition constructed by combining propositions using logical operators

truth table:  a table displaying ALL the truth values of propositions

disjunction of p and q: p or q

conjunction of p and q： p and q

exclusive or of p and q: p xor q, exactly one of p and q is true

p implies q: p->q

contrapositive of p->q: negation(q) -> negation(p)

converse of p->q: q->p

inverse of p->q: negation(p)->negation(q)

p<->q biconditional: p->q and q->p

tautology: a compound proposition that is always true

contradiction: a compound proposition that is always false

contingency: a compound proposition that is sometime true and sometimes false

consistent compound propostions: compound propostions for which there is an assignment for truth values to the variables that makes all these propositions true

logically equivalent compound propositions: compound propositions always have the same truth values

 

predicate: part of a sentence that attributes a property to the subject

propositional function: a statement containing one of more variables that becomes a proposition when each of its variables is assigned a value or is bound by a quantifier

domain (or unniverse) of discourse: the values a variable in a propositional funciton may take

existential quantification of p(x): there is x such that p(x) is true

universal quantification of p(x): for every x, p(x) is true

logically equivalent expressions: expressions that have the same truth value no matter what proposition functions and domains are used

bound variable: a variable that is quantified

free variable: a variable that is not bound in a propositional function

scope of a quantifier: portion of a statement where the quantifier binds its variable

 

argument: a sequence of statements

argument form: a sequence of compound propositions involving propositional variables

premise: a satement in an argument, or argument form, other than then final conclusion

conclusion: the final statement in an argument or argument form

valid argument form: the truth of all the premises imply the truth of the conclusion

valid argument: an argument whose argument form is valid

rule of inference: a valid argument form that can be used in the demostration that arguments are valid

fallacy: an invalid argument form often used incorrectly as a rule of inference

circular reasoning or begging the question: reasoning where one or more steps are based on the truth of the statement being proved

theorem: a mathematical assertion that can be shown to be true

conjecture: a mathematical assertion proposed to be true, but that not been proved

proof: a demonstration that a theorem is true

axiom: a statement that is assumed to be true and that can be used as a basis for proving theorems

lemma: a theorem used to prove other theorems

vacuous proof: a proof that p->q is true is based on p is false

trival proof: a proof that p->q is trur is based on q is true

direct proof: a proof that p->q is true proceeds by showing that q must true when p is true

proof by contraposition: a proof that p->q is true proceeds by showing that negation(q)->negation(p)

proof by contradiction: a proof that p is true proceeds by showing that negation(p)->q, where is a contradiction

forward reasoning: direct proof, or proof by contraposition, or proof by contradiction

backward reasoning: when facing p->q, thinking r->q, where r is the last step leads to q

proof by case: (p1 or p2 or p3)->q is equivalent to (p1->q) and (p2->q) and (p3->q)

exhaustive proof: a proof that establishes a result by checking a list of all cases

without loss of generality: an assumption in a proof that makes it possible to prove a theorem by reducing the number of cases needed in the proof

counterexample: an element x such that p(x) is false

constructive existence proof: a proof that an element with a specified property exists that explicitly finds such an element

noconstructive existence proof: a proof that an element with a specified property exists that doesn't explicitly find such an element, usually by proof by contradiction, or claim exists in a small set

uniqueness proof: a proof that there is exactly one element satisfying a specified propery.

 

Results

logical equivalence (laws about and, or, negation)

De Morgan's law for quantifiers (negation of quantifiers)

rules of inference for proposition (8 laws)

rules of inference for quantified statement (unversal instantiation, unversal generalization, existential instantiation, existential generalization)

 

others

A collection of logical operators is called functionally complete if every compound proposition is logically equivalent to a compound proposition involving only these logical operators.

and, or, negation forms a functionally complete collection of logic operator, since a compound proposition with its truth table can be formed by taking the disjunction of conjunctions of variables or their negations, with one conjunction included for each combination of values for which the compound proposition is true.