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36 Cards in this Set
- Front
- Back
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commutative laws
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p and q == q and p;
p or q == q or p; order changes |
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associative laws
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(p and q) and r == p and (q and r);
(p or q) or r == p or (q or r); requires all same operators |
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distributive laws
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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 |
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identity laws
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p and t == p; p or c == p;
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negation laws
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p or ~p == t;
p and ~p = c; |
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double negative law
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~(~p) == p
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idempotent laws
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p or p == p;
p and p == p; |
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universal bounds laws
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p or t == t;
p and c == c; |
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DeMorgan's laws
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~(p or q) == ~p and ~q;
~(p and q) == ~p or ~q; |
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absorption laws
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p or (p and q) == p;
p and (p or q) == p; |
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negations of t and c
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~t == c;
~c == t; |
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Define "Statement"
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A sentence that is either true or false, and therefore not both.
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Define "Statement Form"
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An expression of statement variables and logical connectives that becomes a statement when the statement variables are replaced with statements.
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Define "Tautology"
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A statement form that is true regardless of the statements subsistuted for the statement variables.
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Define "Contradiction"
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A statement form that is false regardless of the statements substituted for the statement variables.
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Define "Logical Equivalence"
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Two statement forms with the same truth tables.
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Define "Conditional Statement"
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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."
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Law of division into cases
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(p or q)->r == (p->r) and (q->r)
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Unnamed law for conditional statement
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p->q == ~p or q;
"If you don't turn in your homework you will fail." or "Turn in your homework or fail." |
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Define "Contrapositive of p->q" (can be used as a transformative law)
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Contrapositive of p->q is ~q->~p (see, it's reversed!)
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Define "Converse of p->q"
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Converse of p->q is q->p, and is a common logical error, NOT a law.
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Define "Inverse of p->q"
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Inverse of p->q is ~p->~q, and is a common logical error, NOT a law.
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Define "Biconditional Statement"
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"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).
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Define "Necessary" and "Sufficient" and combine
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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.
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Define "argument" and "argument form"
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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.
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Define "valid argument form"
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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.
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Define "modus ponens"
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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."
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Define "modus tollens"
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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."
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Define "rule of inference"
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A form of argument that is valid, e.g. modus ponens and modus tollens.
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rule of inference: generalization
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p; therefore p or q
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rule of inference: specialization
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p and q; therefore p
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rule of inference: elimination
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p or q; ~q; therefore p
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rule of inference: transitivity
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p->q; q->r; therefore p->r
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rule of inference: division into cases
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p or q; p->r; q->r; therefore r
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rule of inference: contradiction
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~p->c; therefore p
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rule of inference: conjunction
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p; q; therefore p and q
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