Indirect Proof
The method of indirect proof will be familiar to those of you who have studied this discipline (developed by Euclid).
What is geometry?
"Chicago is windy." Translated into predicate logic is...
What is Wc?
How this statement is represented in quantification: "All skyscrapers are tall."
What is: "(x) (Sx -> Tx)"?
In order to save space, logicians sometimes represent the conclusion on this final line.
What is the final premise line?
Words like “some,” or “there is a,” or “there is at least one” are represented with this quantifier.
What is the Existential Quantifier?
How this statement is represented in quantification: "No diamonds are rubies."
What is: "(x) (Dx -> ~Rx)"?
Indirect proof often referred to as "reductio ad absurdum" means this.
What is "reduced to absurdity"?
A quantificational symbol used to represent words such as “every,” “all,” “anything,” and “everything.”
What is the universal quantification, x in parentheses—(x)?
How this statement is represented in quantification: "Some men are paupers."
What is: (Ex) (Mx . Px)?
An indirect proof is constructed by assuming, as an additional premise, the negation of its conclusion, and then we derive this from the augmented set of premises.
What is an explicit contradiction?
The basic idea of quantification is to analyze propositions not into subject and predicate, as we’ve been doing with Aristotelian logic, but into these two things.
What is function and argument?
How this statement is represented in quantification: "Some vehicles are not motorcycles."
What is: (Ex) (Vx . ~ Mx)?
We emphasize an indirect proof by placing a line to the left of the numbers that were used and discharging the proof by rewriting the conclusion followed by the steps used and this abbreviation.
What is "I.P."?
This Boolean translation, "If anything is an S, then it is a P," began with this statement form.
What is an A Proposition, "All S are P"?
If you use Universal Instantiation (UI) first, you introduce a name, which cannot be used later by this instantiation.
What is Existential Instantiation (EI)? You must always use EI before using UI.