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36 Cards in this Set

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  • Back
commutative laws
p and q == q and p;
p or q == q or p;
order changes
associative laws
(p and q) and r == p and (q and r);
(p or q) or r == p or (q or r);
requires all same operators
distributive laws
p and (q or r) == (p and q) or (p and r);
p or (q and r) == (p or q) and (p or r);
like multiplication
identity laws
p and t == p; p or c == p;
negation laws
p or ~p == t;
p and ~p = c;
double negative law
~(~p) == p
idempotent laws
p or p == p;
p and p == p;
universal bounds laws
p or t == t;
p and c == c;
DeMorgan's laws
~(p or q) == ~p and ~q;
~(p and q) == ~p or ~q;
absorption laws
p or (p and q) == p;
p and (p or q) == p;
negations of t and c
~t == c;
~c == t;
Define "Statement"
A sentence that is either true or false, and therefore not both.
Define "Statement Form"
An expression of statement variables and logical connectives that becomes a statement when the statement variables are replaced with statements.
Define "Tautology"
A statement form that is true regardless of the statements subsistuted for the statement variables.
Define "Contradiction"
A statement form that is false regardless of the statements substituted for the statement variables.
Define "Logical Equivalence"
Two statement forms with the same truth tables.
Define "Conditional Statement"
If p then q, or p implies q, where p & q are statement variables, denoted p->q, where p is the hypothesis and q is the conclusion. Example: "If 6 divides 12, then 3 divides 12."
Law of division into cases
(p or q)->r == (p->r) and (q->r)
Unnamed law for conditional statement
p->q == ~p or q;
"If you don't turn in your homework you will fail." or "Turn in your homework or fail."
Define "Contrapositive of p->q" (can be used as a transformative law)
Contrapositive of p->q is ~q->~p (see, it's reversed!)
Define "Converse of p->q"
Converse of p->q is q->p, and is a common logical error, NOT a law.
Define "Inverse of p->q"
Inverse of p->q is ~p->~q, and is a common logical error, NOT a law.
Define "Biconditional Statement"
"p if and only if q", denoted p<->q, also "p is necessary and sufficient for q." TT looks like ~XOR. AKA (p->q) and (q->p).
Define "Necessary" and "Sufficient" and combine
r is a sufficient condition for s means r->s. r is a necessary condition for s means s->r == ~r->~s. r is a necessary and sufficient condtion for s is r<->s.
Define "argument" and "argument form"
An argument is a sequence of statements, and argument form is a sequence of statement forms. The final statement is the conclusion, and all predecessors are premises.
Define "valid argument form"
An argument form is valid if nomatter what statements are substituted for the statement variables, if all premises are true, then the conclusion is true.
Define "modus ponens"
p->q; p; therefore q. Ex: "If Socrates is a man, then Socrates is mortal; Socrates is a man; therefore Socrates is mortal. Means "method of affirming."
Define "modus tollens"
p->q; ~q; therefore ~p. Ex: "If Zeus is human, then Zeus is mortal; Zeus is not mortal; therefore Zeus is human. Means "method of denying."
Define "rule of inference"
A form of argument that is valid, e.g. modus ponens and modus tollens.
rule of inference: generalization
p; therefore p or q
rule of inference: specialization
p and q; therefore p
rule of inference: elimination
p or q; ~q; therefore p
rule of inference: transitivity
p->q; q->r; therefore p->r
rule of inference: division into cases
p or q; p->r; q->r; therefore r
rule of inference: contradiction
~p->c; therefore p
rule of inference: conjunction
p; q; therefore p and q