outcomes ^__^

we have been working in our own group to come out with these outcomes.
these are the things we have been searching though out the web, books etc
again we don't own the copyrights, instead knowledge have the rights to be spread :)
enjoy the 'outcomes'


1. PROPOSITIONAL LOGIC

     
        Propositional Logic is concerned with propositions and their interrelationships. It also known as sentential logic and statement logic is the branch of logic that studies ways of joining or modifying entire propositions, statements or sentences. It were form more complicated propositions as well as the logical relationships and properties that are derived from these methods of combining or altering statements. 
well in simpler words, you can just imagine such combinations of the sudoku puzzles, the cubics, it is all the matter of logics and great numbers of different codes and ways produced of any events etc

Example and Observations:
  • "An argument is any group of propositions where one proposition is claimed to follow from the others, and where the others are treated as furnishing grounds or support for the truth of the one. An argument is not a mere collection of propositions, but a group with a particular, rather formal, structure.

    "The conclusion of an argument is the one proposition that is arrived at and affirmed on the basis of the other propositions of the argument.

    "The premises of an argument are the other propositions which are assumed or otherwise accepted as providing support or justification for accepting the one proposition which is the conclusion. Thus, in the three propositions that follow in the universal deductive categorical syllogism, the first two are premises and the third the conclusion:
All men are mortal.
Socrates is a man.
Socrates is mortal.
(Ruggero J. Aldisert, "Logic in Forensic Science." Forensic Science and Law, ed. by Cyril H. Wecht and John T. Rago. Taylor & Francis, 2006)

Example of Arguments
  • the software won’t compile or it produces a division by zero error.
      the software does not produce a division by zero error
      Therefore the software will not compile.
                 if its in general form it will defines appropriate propositional
                 variables.
                this example of argument is called 'disjunctive syllogism'.


Another disjunctive syllogism examples:
The cars are yellow or green.
The cars are not green.
Therefore the cars are yellow.
Let X be: the cars are yellow.
Let Y be: the cars are green.

Rewritten argument:
X or Y not Y
Therefore X
This argument is valid, its not that meaningful since both X and Y are not true.

Propositional variable
  • A variable which can neither be TRUE nor FALSE.
  • The atomic formulas of propositional logic.
  • Typically, the formulas built up recursively(repeatedly) from some propositional variable, some number of logical connectives and some logical quantifiers. 

Truth table
The general truth tables for each of the connectives informs the value of any possible statement for each of it.

Negation


Conjunction
(declare BOTH statements are true.)


Disjunction
(declare at least ONE statement is true.)


Material  Equivalence
(Asserts the statements always have the SAME truth value.)

Material  Implication
(Antecedent sufficient for consequent)
Assume that a material implication is TRUE except proven FALSE.


To determine how many lines a truth table will have use this formula:
Lines = 2n
n = the number of different letters (simple statements) in the statement.

Starting with the left most letter, divide the table in half.  
Assign T as the value of the left most letter for the first half of the table, 
and F as its value for the second  half of the table.
Move to the next new letter to the right, and cut the alternation between T & F in half.  
Repeat this process until you are alternating between T& F line by line for the right most letter.

Assuming you are playing puzzles! :)

Tautology:  A statement forms invokes as its become true. got T on every line beneath its main symbol on it's truth table
Self-Contradiction:  A statement in which the form of invoke that it be FALSE.  
It’s truth table has F on every line beneath its main symbol.
Contingent Statement:  A statement in which the truth value is depend on the particular combinations of values for its letters (simple statements). 
It’s truth table has T on some lines and F on some lines on the bottom of the main symbol.


#  Logically Equivalent Statements:  The 2 statements have the same equal truth value on every line beneath their main symbols.

# Logically Contradictory Statements:  The 2 statements have the opposite(different) truth values on every line beneath their main symbols.


Consistent Statements:
    1. The statements are neither equal nor opposed.
    2. The least is when one line statement's is true
Inconsistent Statements:
    1. The statements are neither equal nor opposed.
    2. there is not even a line should be true.
If there is no line on which all the premises are true and the conclusion false, then the argument is valid.

If there is even one line on which all the premises are true and the conclusion false, then the argument is invalid.




2. PROPOSITIONAL EQUIVALENCE

Definition:
1.Tautology : is a proposition that is always true.
Example:-p ˅ ¬p          




e.g:
P:the switch is on.
Q:the switch is on.
This is tautology.Both is true.

2.Contradiction :  is a proposition that’s is always false.
Example:-p Ʌ¬p 
e.g:
P:the switch is off.
Q:the switch is off.
This is contradiction.Both is false.

3.Contingency :is a proposition that’s is neither  a tautology nor a contradiction.
Example:-p ˅ q→¬r
e.g:
P:the switch is off.
Q:the switch is on.
This is contingency.Both is neither  tautology nor a contradiction.

LOGICAL EQUIVALENT
Definition-proposition r and s are logically equivalent if the statement  r↔s is a tautology.
Notation:If  r and s are logically equivalent, we write as

r ˂═> s





3. PREDICATES AND QUANTIFIERS

Predicates:

    • Is a method to determine statement  which is true or false value based on one or more variables. 
    • In a statement there are 2 things:

      >Subject.
      >Predicate



Propositional Function?
    • A statement of in term of p (x1,x2……,xn)
    - P(x1,x2,…,xn) propositional function value of p. P is a.k.a predicate.
      ** (x1,x2,…,xn) n is tuple

functions :
    • To determine either the statement is true or false
    • To prove of argument 1 or more which are given.
    • If the value is assigned, proposition can be concluded
Example:
    • A ( a,b,c,d), we denote it to“(a+1),(b+1),(c+1)= d+1”. Given that, A (2,3,4,11) and A(2,2,2,9).
      Which is the truth value? 


Quantifiers:

A  quantifier is a type of logical symbol that is used to make a quantification or assertion about the set of value which make the formula or formulas true

Simpler: quantifier is an expression that make the formula true

Types of Quantifiers:

(i) universal Quantifiers :


assert that p(x) is true for every values in particular domains

e.g of expression: 
for all” or “for every” or “each” or ”for any

Example in daily life:

e.g of universal quantifier in daily life:
*for every car that the factory want to sell, it must be painted.

*All driver must obey the laws to prevent themselves to get fine




(ii) Existential Quantifier:




e.g of existential quantifier in daily life:
*not everybody in your class who know how to talk in javanese
*at least 3 people in your class is in the dean list

*there is a couple of person in your class whose using macbook


(iii) uniqueness Quantifier



refrences:


http://en.wikipedia.org/wiki/Proposition
http://logic.stanford.edu/classes/cs157/2009/notes/chap02.html
etc






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