Please enable JavaScript.
Coggle requires JavaScript to display documents.
Propositional Calculus (ARGUMENT (Valid Argument (valid argument if the…
Propositional Calculus
True or false statement
Propositional Variables
case letters starting from P onwards
COMBINATION OF PROPOSITIONS
Fundamental Connectors
Disjunction
ORing of two statements
p ∨ q
Negation
opposite of original statement
~ p
Conjunction
ANDing of two statements
p ∧ q
Derived Connectors
NOR
negation after ORing
p ↓ q
XOR
XORing of p and q is true if p is true or if q is true but not both and vice-versa
p ⊕ q
NAND
negation after ANDing
denoted by p ↑ q
Conditional
If p then q
p → q
Biconditional
if and only if
denoted as p ↔ q
⇔ was logically equivalent
∴ Therefore
∵ Because
PRINCIPLE OF DUALITY
A1 and A2 are said to be duals
replacing ∧ (AND) by ∨ (OR) and ∨ (OR) by ∧ (AND).
EQUIVALENCE OF PROPOSITIONS
same truth values
TAUTOLOGIES
if it is true under all circumstances
CONTRADICTION
statement that is always false
CONTINGENCY
true or false depending on the truth values of its variables
FUNCTIONALLY COMPLETE SETS OF CONNECTIVES
functionally complete if every formula can be expressed in terms of an equivalent formula containing the connectives from this set
ARGUMENT
group of propositions called premises
yields another proposition, called the conclusion
Conclusion
conclusion of an argument is the proposition that is asserted on the basis of other proposition of the argument.
Premises
propositions, which are assumed for accepting the conclusion, are called the premises of that argument
Valid Argument
valid argument if the conclusion is true whenever all the premises are true
Falacy Argument
falacy or an invalid argument if it is not a valid argument
EXISTENTIAL QUANTIFIER
The quantifier ∃ is called the existential
UNIVERSAL QUANTIFIER
The quantifier ∀ is called the universal quantifier
NEGATION OF QUANTIFIED PROPOSITIONS
De Morgan's law
~ 3 x p(x) ≅ ∀ x ~ p(x)
~ ∀ x p(x) ≅ ∃ x ~ p(x).
PROPOSITIONS WITH MULTIPLE QUANTIFIERS
The proposition which contains both universal and existential quantifiers, the order of these quantifiers can't be exchanged without altering the meaning of proposition
