## Friday, February 1, 2013

### Logic

Proposition, or statement, is any declarative sentence which is either true (T) or false (F).

A paradox is not a statement.

Premise: a proposition supporting or helping to support a conclusion.

Argument: formally, a sequence of statements one of which is the conclusion and the remainder the premises

Fallacy: any of various types of erroneous reasoning that render arguments logically unsound.
-----------
・Induction ・Deduction

Induction: any form of reasoning in which the conclusion, though supported by the premises, does not follow from them necessarily.
Deduction: a process of reasoning in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true.
Inference: the process of deriving the strict logical consequences of assumed premises.
Logical consequence: X results of necessity from Y and Z if it would be impossible for X to be false when Y and Z are true.
----------
Syllogism: an argument the conclusion of which is supported by two premises, of which one (major premise)  contains the term (major term)  that is the predicate of the conclusion, and the other (minor premise)  contains the term (minor term)  that is the subject of the conclusion; common to both premises is a term (middle term)  that is excluded from the conclusion. A typical form is “All A is C; all B is A; therefore all B is C.”

Reductio ad absurdum or proof by contradiction demonstrates that a statement is true by showing that a false result follows from its denial.

Logic
1.A branch of philosophy and mathematics that deals with the formal principles, methods and criteria of validity of inference, reasoning and knowledge.
Logic is concerned with what is true and how we can know whether something is true. This involves the formalisation of logical arguments and proofs in terms of symbols representing propositions and logical connectives. The meanings of these logical connectives are expressed by a set of rules which are assumed to be self-evident.

Boolean algebra deals with the basic operations of truth values: AND, OR, NOT and combinations thereof.

Predicate logic extends this with existential quantifiers and universal quantifiers which introduce bound variables ranging over finite sets; the predicate itself takes on only the values true and false.

The rules of natural deduction describe how we may proceed from valid premises to valid conclusions, where the premises and conclusions are expressions in predicate logic.

Symbolic logic uses a meta-language concerned with truth, which may or may not have a corresponding expression in the world of objects called existance. In symbolic logic, arguments and proofs are made in terms of symbols representing propositions and logical connectives. The meanings of these begin with a set of rules or primitives which are assumed to be self-evident.

Deduction describes how we may proceed from valid premises to valid conclusions, where these are expressions in predicate logic.

The negation of p is the statement ~p, which we read "not p."
The conjunction of p and q is the statement p∧q, which we read "p and q."
The disjunction of p and q is the statement p∨q, which we read "p or q."

Tautology /tɔˈtɒlədʒi/ needless repetition of an idea, especially in words other than those of the immediate context, without imparting additional force or clearness, as in “widow woman.”

A compound statement is a tautology if its truth value is always T, regardless of the truth values of its variables.
It is a contradiction if its truth value is always F, regardless of the truth values of its variables.

~(~p)≡ p    Double Negative Law
p∧q ≡ q∧p Commutative Law for conjunction
p∨q ≡ q∨p Commutative Law for disjunction

Associative Law for conjunction
(p∧q)∧r ≡ p∧(q∧r)

Associative Law for disjunction
(p∨q)∨r ≡ p∨(q∨r)

DeMorgan's Laws
~(p∨q) ≡ (~p)∧(~q)
~(p∧q) ≡ (~p)∨(~q)

the Distributive Laws
p∧(q∨r) ≡ (p∧q)∨(p∧r)
p∨(q∧r) ≡ (p∨q)∧(p∨r)

Absorption Laws
p∧p ≡ p
p∨p ≡ p

The conditional p→q, which we read "if p, then q" or "p implies q,"
The only way that p→q can be false is if p is true and q is false

p→q ≡ (~p)∨q
p→q ≡ (~q)→(~p)

Biconditional
The biconditional p↔q, which we read "p if and only if q" or "p is equivalent to q,"
p↔q ≡ (p→q)∧(q→p)

p→q
p
----
q

p→q
~q
-----
~p

p∨q
~p
-----
q

p→q
q→r
-----
p→r

a proof is a way of convincing you that the conclusion follows from the premises, or that the conclusion must be true if the premises are. Formally, a proof is a list of statements, usually beginning with the premises, in which each statement that is not a premise must be true if the statements preceding it are true.

All men are mortal
For all x, if x is a man then x is mortal
∀x [ Px → Qx ]

∀x[Px→Qx] All men are mortal.
Ps     Socrates is a man.
------
Qs     Therefore, Socrates is mortal.

some politicians are honest
For some x, x is a politician and x is honest
∃x [ Px ∧ Hx ]
--------
Where is the definitive proof?

In discussions about soul, god, afterlife, etc, a lot of people ask for proof.

It is very difficult to find a definitive proof.

In logic there are 2 ways of thinking: induction and deduction.

Induction is extensively used in Physics.

Deduction is exensively used in Mathematics.

One example of induction:

Premise 1: John is mortal.

Premise 2: Mary is mortal.

Proposition: All humans are mortal.

Induction is a generalization. We see a certain characteristic in a sample, then we propose that such characteristic may also be present in the entire population.

In induction the premise is called EVIDENCE.

The proposition is CONFIRMED when more evidence is found.

But the evidence merely suggests that the proposition may be right.

The evidence is necessary but not enough to prove the proposition right.

One single counter-example can prove the proposition wrong.

But not matter how much evidence you have, it is impossible to prove a proposition right.

The only way to prove a proposition right would be to check the entire population. In most cases, the population is too large for such a check.

All scientific propositions come from induction.

It is impossible to prove a scientific proposition right.

The evidence merely suggests that the scientific proposition may be right.

One single counter-example can prove a scientific proposition wrong.

One example of deduction:

Premise 1: All humans are mortal.

Premise 2: Socrates is human.

Conclusion: Socrates is mortal.

In deduction the conclusion is necessarily right if the premises are right. The premises are enough to prove the conclusion right.

But there is no way to prove the premises.

Nobody knows if the premises are right or wrong.

The premises are assumed to be true.

The conclusion is right only if the premises are right.

In Mathematics the assumptions are called axioms or postulates.

Whether in deduction or induction, it is not possible to find a definitive proof.

Every proof is based on assumptions, which are unprovable by definition.