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28 Cards in this Set
- Front
- Back
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argument form
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is a group of sentencee forms such that ALL its substituion instances are arguments.
p -> q p / :. q |
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Method of Proof means?
aka what? |
demonstrates the validity (nut not invalidity) of sentential arugments
aka Natural Deduction |
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Denying the Antecedent (invalid)
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p -> q
~p /:. ~q If for p we sub a False sentence and for q we sub a True sentence, result is Denying the Abntecedent that as True Premises and False Conclusion |
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Affirming the Consequent (invalid)
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p -> q
p /:. q |
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Modus Ponens (MP)
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p -> q
p /:. q |
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Modus Tollens (MT)
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p -> q
~p /:. ~q |
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Disjunctive Syllogism (DS)
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two forms:
1. p v q ~p /:. q 2. p v q ~q /:. p Given that at least on of two sentences is True and one of the sentences is definitely false, it follows that the other sentene MUST be True. |
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Hypothetical Syllogism (HS)
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p -> q
q -> r /:. p -> r |
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Simplification (Simp)
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two forms:
p . q /:. p and p . q /:. q 2 sentences are both True you can infer that one of the sentences is True. Rule used to take apart Conjunctions. |
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Conjunction (Conj)
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p
q /:. p . q Take any 2 lines of a proof and put them together tomake a conjunctions. Rule used to put Simplifications together |
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Addition (Add)
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p /:. p v q
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Constructive Dilemma (CD)
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p v q
p -> r q -> s /:. r v s requires you to cite 3 lines in the proof. You need a disjunction and 2 conditionals, the antecedednts of the 2 conditionals are the 2 disjuncts fo the disjuntion. |
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Implicational arument forms
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are one directional arugment forms.
A . B to A BUT NOT A to A . B |
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Principles of Strategy:
if you have a premise that is a conditional, see if you have another line that allos you to employ |
MP, MT or HS.
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Principles of Strategy:
If you have a premise that is a disjunction, look for ways to employ |
DS
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Principles of Strategy:
If you have a line that is a negation, see if you can use |
MT or DS
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Principles of Strategy:
Whenever the same sentence occurs on 2 differetn lines, look for ways to apply ___________ using those 2 lines. |
MP, MT, DS or HS
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Doulbe Negation (DN)
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p /:. ~~p or it can be ~~p /:. p
permits inferences from any substituion instance of p to the analogous sub instance of ~~p, and... from any sub instance of ~~p to the analogus sub instance of p. |
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Equivalence argument forms
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argument form that permit inferences in both directions.
because they permit inferences from given statements to statements with which they are logically equivalent. |
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DeMorgan's Theorem (DeM)
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~(p . q) :: ~p v ~q
~(p v q) :: ~p . ~q |
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Implicational argument forms MUST be used on ("parts of" or whole) lines only
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Whole
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Equivalence forms may be used on ("parts of" or whole) lines only.
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Parts of
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Replacement has to be _ throughout.
a) Uniform b) Not be Uniform |
Not be Uniform
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Substitution has to be _ throughout.
a) Uniform b) Not be Uniform |
Uniform
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A -> (B -> A) is not logiccally equivalent to p -> (q -> p) because the LATTER
a) statement form contains 2 variables and has no truth-value b) contains statement constants and has a definite truth-value. |
a) statement form contains 2 variables and has no truth-value
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A -> (B -> A) is not logiccally equivalent to p -> (q -> p) because the FORMER
a) statement form contains 2 variables and has no truth-value b) contains statement constants and has a definite truth-value. |
b) contains statement constants and has a definite truth-value.
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Commutation (Comm)
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p v q :: q v p
p . q :: q . p reversing the order of statements connected by "." or "v" Does NOT work for "->" |
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Association (Assoc)
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p v (q v r) :: (p v q) v r
p . (q . r) :: (p . q) . r THe movement of parentheses in either of the ways specified doesn't change the truth values of compound sentenctes in which they occur |
