Propositional and predicates logic

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This is my note of the video https://youtu.be/V49i_LM8B0E. It’s not a full guide of propositional and predicates logic.

Mathmatics as a language of Physics

• Physics “casts” concepts about the real world into rigorous mathematical forms.
• Mathematics is just a language for the Physics. Physics interprets the meaning of mathematical statements.
• But to describe Mathematics, we need logic (especially, standard logic).
• Psycologically, an equivalent but a different form will help us to understand an interpretation.

• Physical “spaces” are defined on a set. So the set theory would be a base.
• Axiomatic set theory is written in standard logic.
• To induce “contnuity”, we need topology on the set theory.
• One of interesting topology for Physics is topological manifolds.

Propositional Logkc

Quickly construct a propositional logic

A proposition $p$ is a variable that can take the values only “true ($t$)” or “false ($f$)”. It’s called law of excluded middle (principium tertii exclusi).

Propositional logic doesn’t decide, a proposition is true or false by itself. Propositional logic doesn’t convey meanings of propositions.

Logical operator

Unary operator: takes one proposition and makes a new proposition from it. There could be 4 unary operator as follows:

$p$$\neg p$$^{id}p$ (Identity operator)$^{\top}p$ (Tautology operator)$^{\bot}p$
$t$$f$$t$$t$$f$
$f$$t$$f$$t$$f$
• defined the logic rules by tables, not deduction rules (another way).
• some logix cant be expressed in tables.
• Defining by deduction rules is more abstract, but in standard logic, it’s OK.
• didn’t use a terminology “map”.
• A tautology is a proposition (statement) that is always true.

Binary operater: takes two propositions and makes a new proposition from it. There are 16 binary operater, and here are some of binary operators:

$p$$q$$p\land q$$p\lor q$$p\veebar p$ (Exclusive OR)$p\Rightarrow q$ (Ex falso quodlibet)$p\Leftrightarrow q$$p\uparrow q (NAND) t$$t$$t$$t$$f$$t$$t$$f$
$t$$f$$f$$t$$t$$f$$f$$t f$$t$$f$$t$$t$$t$$f$$t$

Completeness

The whole theme will diverge if I describe more details, so I’ll skip this time.

Sideway: Gödel’s first incompleteness theorem

Any consistent axiomatic system within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are propositions of the system which can neither be proved nor disproved.