Propositional Logic Proofs

This site is a placeholder for what is to become a reference for common logical relationships arising from the use of Howard Pospesel's propositional logic system.

The text that you see throughout this site are my proofs, mixed with my commentary on the details. I have several intentions:

To introduce interested parties to Howard Pospesel's logical system.
To produce a reference for students who are working through the book without a course.
To provide proof for all of the derived rules that the book engenders.

Notes on Logic

An argument has form and content.
An argument's form is what determines validity.
It's not valid to speak of an argument as being true or false.
Arguments are composed of statements which are the content of the argument.
The statements can be true or false, but not valid/invalid.
The only requirement for an argument to be valid is: It must be impossible for the argument to have a true premise and false conclusion.

1. Arguments in the form P & Q therefore P

This is a valid argument.
We can create all combinations of true or false premises and true or false conclusion, but not a true premise and false conclusion.

The following table contains all combinations of premises and conclusions with examples:

Combination	Example
True premise, true conclusion	Obama is a human and he is American Therefore Obama is American.
False premise, true conclusion	Donald Trump descends from an ape and he is a woman Therefore Donald Trump descends from an ape.
False premise, false conclusion	Hillary is a man and she is the devil, Therefore Hillary is a man
True premise, false conclusion	This isn't possible. It's straining to think of an example because it's impossible to. P & Q therefore P is a valid argument. And because of this, it's not possible for the argument to have a true premise and false conclusion.

2. Arguments in the form P & Q therefore P and R

Combination	Examples
True premise, true conclusion	The sky is blue and we breathe air Therefore the sky is blue and the moon is far away
False premise and true conclusion	Dogs are mammals and cats plot to kill us all Therefore dogs are mammals and we have cats.
False premise and false conclusion	All cats are black and the Pericles rode rockets Therefore all cats are black and a person can suffocate breathing pure oxygen.
True premise and false conclusion	Some cats are black and some Americans are human Therefore some cats are black and the moon is made of cheese.

Note: The argument P & Q therefore P & R is an invalid argument.

3. The ages of three sons

Description of the exercise goes here.

The ages of the kids are 9, 2, and 2. The way the author solves this is available near the end of page 145.

1. P & Q therefore P 2. P & Q therefore P & R 3. The ages of the three sons