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Secret
Secret
5:21 PM
[The Cult of Infinity]
0. Begin by specifying an alphabet $\mathcal{A}$. This give us the symbols to be used. Being human beings, our alphabet is finite despite it can be left unspecified to account for the development of our society
1. Next define a formal language $\mathcal{L}$ which gave us the words assembled from the alphabet $\mathcal{A}$
2. Then define a formal grammar $\mathcal{G}$ which provide the rules in producing new strings and how they are concatenated, truncated etc.
3. After that, a logic $L$ is introduced to give us inference rules to reason with
Now to define various primitive notions in $L$
3a. Truth values: These are attributes assigned to propositions and predicates. There are 3 truth values: True (T), False (F), Null (O)
3b. Expression: A string or sentence constructed using $\mathcal{L}$ such that it is well formed (satisfy the syntax given by $\mathcal{L}$). They don't necessary have a truth value.
3c. Proposition P: An expression that has a fixed truth value. Can be though as a 0 argument predicate.
3d. Predicate P(S): An expression whose truth value depends on the argument(s)
3e. Procedure proc(S;T): It takes in an object in S and produce an object in T. The object can be any expression
A procedure can also take procedures as objects and return procedures
Note that S and T can be of any number of objects, thus one can get a relation which takes in one object and outputs multiple objects
3f. Algorithms: An expression consists of procedures applied in order to give some object(s) as a final output
3g. Proofs: These are algorithms that converts given objects into some target objects. IT can be as simple as a procedure
3h. The Boolean predicate bool(S): A predicate which gives the truth value of the object S
3i The inference rules were then introduced
3i.0. Membership ∈
(Note that in practice, bool is a proof of S that deteremines its truth value) 
∈ intro:
...
bool(Q(P))
True
--------
P ∈ Q

∈ elim:
...
P ∈ Q
--------
bool(Q(P))
True
Repeat
A
(Anything)
--------
A
∧ intro:
P ∈ Q
R ∈ Q
--------
(P ∧ R) ∈ Q


∧ elim:
(P ∧ R) ∈ Q
--------
P ∈ Q
R ∈ Q
¬ intro:
bool(Q(P))
False
--------
¬(P ∈ Q)

¬ elim:
¬(P ∈ Q)
--------
bool(Q(P))
False
$\lor$
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