proofs Flashcards
a demonstration that an argument is invalid/ valid
proof
true or false: a proof is an argument
false
what makes a good proof?
obvious and valid
how to prove an invalidity?
informal proof with counterexample
eliminates wide scope conjunction.
&Elim; 1
- cites one line
introduce a conjunction
&Intro;1,2
- cites TWO lines
eliminate 2 stacked negations
~Elim;1
- only applies to main connective
build a disjunction from a disjunct
vIntro;2
- disjuncts can be complex sentences
- can add as much as you want at a time from one disjunct
reiteration
Reit;1
- requires line number
- valid from circular reasoning
how to reason from a disjunction?
vElim;1, 2-3, 4-5
- proof by cases
- requires citation of disjunction and subproofs of both disjuncts
- requires the exact same sentence to appear 3 times: in each subproof AND in the main proof
how to reason to a negation?
reductio!!!
- ~Intro;2-7
- cites entire subproof
- make temporary assumption and show that a contradiction results
- each subproof ends with #
true or false: only a contradiction can entail a contradiction
true
what is the 5 step plan?
- pick a disjunct
- put it in a subproof
- build the disjunction
- intro the #
- intro the ~
always ends with ~Intro!!!
how do you prove a tautology (no premises) in bool?
reductio!
- never assume what you already know
2 more weird cases of validity
- logically false premise
- conclusion same as premise
true or false: if an argument is unsound, then it is invalid
false
true or false: a contingently false premise can entail a logically true conclusion
true
true or false: a logically true premise cannot entail a contingently true conclusion
true
place the sentence: some dog is a police dog
contingent truth
place the sentence: all police dogs are dogs
logical truth
place the sentence: rufus is a police dog or isnt a police dog
tautology
place the sentence: all dogs are police dogs
contingent falsity
place the sentence: some police dog isnt a dog
logical falsity
place the sentence: rufus is a dog and is not a dog
taut-falsity
what do you do when you have a negation around a conditional? ~(P->Q)
contradiction trick
reiteration trick
reductio
build P->Q
what is the reiteration trick?
~(P->Q) ==> P&~Q
- use REIT to build P->Q
how to reason from a universal quantifier?
universal instantiation
AIntro
how do you reason to an existential?
Existential generalization
EIntro
allows you to replace every instance of the variable with a name
AELim
name we declare to stand for an arbitrary member of the domain
arbitrary names
inferring something is true for all objects once we prove it is true for an arbitrary object
- starts w arbitrary assumption @n
universal generalization
how to reason from an existential claim?
existential instantiation
how do you use =Elim?
must cite 2 things: thing you are eliminating and the sentence you are doing the substitution into
- allows you to substitute one or more occurences of a name
G(p) premise
p = a premise
G(a) =Elim;1,2
true or false: a logically true conclusion follows from anything
true
what would your strategy be for a proof with a universal premise?
AElim
what would your strategy for a proof with a universal conclusion?
AIntro
what would your strategy be for a proof with an existential premise?
start EElim
what would your strategy be for a proof with an existential conclusion?
wait! look around for other ideas. you will use EIntro at some point
how would you prove ~ExP(x)?
reductio
how would you prove ~AxP(x)?
-start w reductio
-use 5 step plan to reductio assumption ~(~Pv~Q)
-assume ~Ex~P(x)
