This is the first installment in an occasional series (Read: Whenever I feel like it) on logic.
Why I’m Doing This
The first reason – or justification – is I like logic. I find logic interesting, fascinating, and – to be open about it – fun.
The second reason, is logic is widely misunderstood. Often people claim to be making a logical argument when the actuality is they ain’t. A logical argument (Applied or Formal) has strict requirements to which the argument must conform. If it does conform it is “logical.” If not, not.
Point 2B is something that most people either do not know or forget: a logical argument is not necessarily (and I’ll define that in a moment) True. The “Truth Value” of a logical argument is a separate issue. You can have a False Truth Value from a valid chain, or cascade, of logical relationships.
Definitions
Logic – from the ancient Greek logos meaning ‘rational words’ related to reasoning in the sciences, philosophy, and everyday life. In Formal Logic the abstract structure of the argument is the fundamental subject under examination or development. Applied logic adapts this machinery for use in a special subject where the matter – what you are talking about – is as important as the means.
Truth Value – A hot issue in “The Philosophy of Logic” I am going to elide (i.e. hand wave around) this for the simplest definition – the assignment of 0 (False) or 1 (True) to a logical proposition, statement, or argument. (Note: Yup, I’m asserting the Law of the Excluded Middle.)
Valid: correct deployment of Formal Logic.
Thesis: What is being argued for or examined.
Proposition: reason to accept or reject the Thesis.
Statement: a logical declaration capable of being analyzed.
Axiom: an unanalyzable declaration.
Argument, Complete: a chain, or cascade, consisting of Axioms, Thesis, Propositions, and Statements resulting in a Conclusion with the last 4 being assigned or capable of being assigned Truth Value(s).
Argument, Incomplete: an chain, or cascade, that does not include all the elements of a Complete Argument but either has or can have assigned Truth Value(s).
(More riveting prose after the break!)
Necessary: universally True or False.
Possible: existentially or sometimes True or False.
You like this stuff? Are you insane?
Yes and I hope not.
What I have done is declared, with some precision, exactly what I mean when I use a word. This is crucial to reasoning, overall, and necessary (always required, see?) for a logical argument. Partially this requirement stems from the development of logic but also from the fact that, in logic, I can do some pretty strange things. If I want to, tho’ why I would escapes me, I can declare:
1 != 0 AND 0 != 1; where {!=} means ‘definitional equal.’
And I’ve just made 1 = 0. In my argument you would see ‘1’ but know without shadow of doubt that I really mean ‘0’.
This is rather silly but there is a serious point to it.
Kurt Godel proved the logical consistency of the Axiom of Choice which states: if you have a set of sets then there exists a set consisting of one member from each of these sets.
And your immediate reaction is: Who Cares?
But the Axiom of Choice is absolutely vital in higher mathematics and several fields, Number Theory for one, would collapse without it. And, on the face of it, the Axiom of Choice seems reasonable but …
In 1963 another mathematical logican P. J. Cohen proved the negation – Thesis: Axiom of Choice is False – is also logically consistent and therefore valid. To put the strong case forward, if the Axiom of Choice is False then natural numbers cannot be assigned properties. Therefore, arithmetic has no relationship to reality.
Oops.
Fortunately, Godel also proved that statements exist in a formal system that cannot be proven or disproven – The Incompleteness Theorem – given a set of axioms. So if we want to “save” arithmetic (and why not? I mean, what the heck! It’s Saturday & we’re just goofing off anyway) we expand the set of axioms to include the Axiom of Choice and Life Is Good.
Wait a Cotton-Picking Minute! Didn’t You Just Palm That Card?
(Mix two metaphors and call me in the morning.)
NO!
Because it is clearly stated what is being done and the Axiom of Choice is an axiom which, by definition, is unanalyzable. We have the choice of asserting it, or not asserting it. And there is no rule against, but rather a rule approving, writing QED under any propositional chain leading to a valid conclusion inclusive of the assertion of the Axiom of Choice.
(QED – Quad erat demonstratum or, in English, ‘I done dood it – neener, neener.’)
What we have done is applied the Existential (“Sometimes”) quantifier to our conclusion rather than the Universal quantifier. Which, again, is perfectly acceptable.
What the #^&*! is an “Existential quantifier”?
Ah, you’ll have to wait until the ->next installment<- of “The Strange Loner Writes Again.”