How to think clearly: a preface to logic and reason

A few tips on how to think clearly and argue effectively:

  • Stick to the point – Do not drift from one issue to another.  Arguments for God’s existence: Without God there would be no purpose to life.  The issue at hand has drifted.
  • Avoid ambiguity – Ambiguity can lead to confusion.
  • Don’t argue in circles – “God exists because it has been proven by the Bible. The Bible is true because it is inspired by God’s word.” 
  • Don’t beg the question – I found this one on a Christian forum; this guy (or girl) is for real: “Our laws of science are based on our observations of the universe that God created, therefore God ‘wrote’ the laws of science.”  LOL.
  • Boil your argument down to the simplest manner; to its bare essentials.

pi = 3.1416

Ask yourself the question: What grounds have I for believing this?

  • Why do I believe this?
  • How do I know this?
  • Can I prove this?  If so, what is the evidence?

Most people believe things without any adequate grounds.  Yes, even in making that claim, I should be expected to provide some evidence.

How to accept expert authority?

  • Identify the expert or group.
  • Are they recognised in that area?
  • Check for bias.

Fact is simply fact regardless of perception and belief.  Facts perceived as sights, sounds, smells, etc. may be termed ‘perceived facts’.  Facts indirectly learnt by methods of inference may be termed ‘inferred facts’.

A true belief (truth) must correspond to all the facts which it covers.  Since facts are consistent with one another, it follows that true beliefs shall also be consistent with one another.

To check for reliability of an observation, also consider:

  • length of time between observation and it being recorded;
  • whether the “fact” observed is consistent with tested knowledge.*

*Extraordinary claims require extraordinary proof.

Generalisation is to make an inference like the following: x(in all observed cases) = T therefore x = T.  For example, Darwin once said that white cats with blue eyes are deaf — this statement is false.  Scientific generalisation differ from hasty generalisation:

  • Use of controlled experiments and conditions;
  • Use of variable conditions;
  • Use of instruments to measure and record, to eliminate or reduce human error;
  • Represent generalisation by mathematical formulae where posible;
  • Repeat and re-examine.

A theory is a logical explanation for observations and facts.  Theory is a statement of a fact or facts in relation to each other.  An alternative theory should at least equally explain all the facts.

  • theory should be consistent with all known facts;
  • theory should have explanatory and/or predictive power.

LOGIC is the study of inference.  It is concerned with assertions/propositions.  Every assertion/proposition is either TRUE or FALSE, providing that it is not ambiguous.

  • It is raining – ambiguous
  • It was raining in New York City at 12:00 PM (UTC-5) on 12/05/09. – unambiguous, is either true or false.

n < 3

Following ‘sentence’ is true when n = 2, n = 1, n = 0, when n is a non-negative integer.

 Three logical connectives: conjunction, disjunction and negation

Conjunction is the AND connective.

p ˄ q
p is true and q is true.

Disjunction is the OR connective.

p ˅ q
p is true or q is true or both is true.

Negation is the NEGATIVE connective.

not p; p is false.

Negation has precedence.  No precedence between Disjunction and Conjunction; brackets needed.

~p ˅ (q ˄ r)
not p or (q and r)

A tautology is a statement which is always true.

~p ˅ p
not p or p
always true. 

A self-contradiction is always false.

p ˄ not p
and not p
always false.

A statement which is neither a tautology or self-contradiction is a contingent statement.

A logical equivalence is when two statements are tantamount, for example:

P = ~(p ˅ q)
Q = ~p ˄ ~q
The two statements are logically equivalent, (producing the same truth table).

Some laws govern these logical equivalences: commutative, associative, distributive, de Morgan’s, idempotent, double negation; which I shall not go into detail.  Any truth table statement can be expressed as a disjunction of conjunctions of the variables or their negations.

Implications are shown by ═˃ (implies) sign.  Whenever we know that ═˃ q is true, and also that p is true, we can deduce that q holds.

p ═˃ q
is the same as (~p ˅ q)

Deductions are logical inferences from a general rule or principle.

1. “All fish are cold-blooded” (general principle)
2. “Whales are not cold-blooded” (connecting fact)
3. Therefore, “Whales are not fish.”

An example of incorrect deduction:

1. “All creationists are stupid.”
2. “Tim Cooley is not a creationist.”
3. Therefore, “Tim Cooley is not stupid.”

A better statement of (1) would be:
1. “All creationists (but not only creationists) are stupid.”

A different (2) works too:
2. “Tim Cooley is not stupid.”
3. Therefore, “Tim Cooley is not a creationist.”

 An example of a mathematical deduction:

Prove that the product of 2 odd numbers is odd.

1. Temporarily assume (hypothesise) that x and y are odd.
2. ie. x = 2n + 1; y = 2m + 1 where n and m are integers;
therefore xy = (2n + 1)(2m + 1).
= 4mn + 2n + 2m + 1
= 2(mn + n + m) + 1
3. Product is odd.
“x and y odd” ═˃ “(xy) is odd” Proved!

Lastly, an axiom is a statement which is so “self-evident” that it can be accepted to be true.  For example:

a + b = b + a


Tim Cooley is handsome.

Therefore, by scientific rationalism and mathematical logic, Tim Cooley is handsome.