List of Logical Systems in First Order Logic:
Quantifiers:
Universial Quantifier:
Existential Quantifier:
Logical Connectives:
Conjuction
Disjuction
Implication =
Biconditional
Negation
Punctuation such as parentheses, brackets, etc.
Vari
Definition 1 (The Axioms of Identity). The axioms are as follow:
x=x
x=yy=x
(x = y P (x) P (x/y)
P (x) is any formula in which no yy occurs
Definition 2 (Modus Ponens). If and = are True in an interpretation
so is .
Every tautology is an axiom.
Given
MATH 406
Eli Sadoff
January 27, 2016
Interpretation is of a language
Model is of axioms in a language
Language of Preorder:
One binary relation
No functions
No constants
Two Interpretations
First: Domain N = cfw_0, 1, 2, . . .
Define: N = cfw_hx, yi N
MATH 406
Eli Sadoff
January 25, 2016
CTVA:
S : At cfw_T, F
Truth Function:
E : L cfw_T, F
We can then get S from this where we have
i ) = S(Ai )
S(A
)
S(P ) 6= S(P
(
Q) = T S(P ) = S(Q) = T
S(P
)=F
F S(P ) 6= S(Q)
S(P
Inductive definition of factor
MATH 406
Prof: Dr. McLarty
January 11, 2016
David Hilbert Greatest mathematician of the early 20th century. He is
responsible for:
Hilbert spaces
Integral equations
Algebra related to Galois Theory
Logic and foundations of mathematics
In 1900 he was i
MATH 406
Prof: Dr. McLarty
January 13, 2016
1
Sentential Logic (SL)
Atomic Sentences (to be interpreted as True or False)
A0 , A1 , A2 , . . . and if-then
Connectives:
not
and
or
then
equivalent
if and only if (iff)
2
A0
T
F
First Order Logic
A0
F
T