**Mathematical Logic** is a logic that focuses on the formal study of logic within mathematics, major varients are model theory, proof theory, set theory, and recursion theory ^{[1]}.

## Variants[edit | edit source]

### Formal Logic[edit | edit source]

Formal logic is the mathematical study of logic, it represents sentences and objects as variables. It also has offspring such as modal logic, intiustianal logic, and ontological logic, and fuzzy logic

The three rules formal logic are:

- The rule of excluded middl: x is either true or false not both
- The rule of identity: x = x in all cases
- The rule of non-contradiction: x can't be y and not y in the same sense

#### Variables[edit | edit source]

we represent objects or nouns as variables, and predicates and capital letters surrounding the variables He is jumping becomes J(h)

The valency of predicates is how many variables are needed to slim down the possibilities to a minimal amount for example Gave needs a valency of 3. I gave him a apple: G(i,h,a)

Now J(h) G(i,h,a) propositions they can either be T or F, this will be important for truth tables later.

#### Connectives[edit | edit source]

In regular language, we connect ideas with conjunctions and the same goes for formal logic, the five basic conjunctions are as follows.

- ∧ : logical conjunctions/ AND|Both propositions are true|Apples AND oranges are fruits. F(a)∧F(o)
- ∨ : logical disjunction/or|one or both propositions can be true|(side note the middle is included). You can have apples or oranges or both. H(y,a)∨H(y,o)
- → : Implication/If then|if one proposition is true the other is true|If I bake pizza, it gets cooked.
- ↔ : Logical biconditional/if and only if|if one proposition is true the other is true and vice versa|It is raining outside if and only if it is a cloudy day.
- ¬ : not/|flips true to false and false to true| I am not happy.

#### Quantifiers[edit | edit source]

There are two quantifiers these refer to how much the set of x: S(x), a proposition applies to

- ∀(x) : universal quantifier/forall| all the set applies to the statement| for all even numbers are divided by two
- ∃(x) : existential quantifier/there exist| at least one the set applies to the statement| their exists

### Modal Logic[edit | edit source]

modal logic works on the basis of probablity,

□ necessity: It is nessacary| It is nessacarly true the 1+1=2 -> □ (1+1=2)

◇ possibility: means that it is possible that something may be true| It is possible today will rain -> ◇(T(r))

### Set Theory[edit | edit source]

**Set Theory** is a mathematical formalism used in foundations of mathematics and analytic philosophy.

Below are some of the most important theorems in set theory.

#### Cantor's Diagonal Argument[edit | edit source]

A set is called countably infinite if there is a bijection between and natural numbers. That is, you can label the elements of so that each positive integer is used exactly once as a label. Such a set is countable because you can count it (via the labeling just mentioned). Unlike a finite set, you never stop counting. But at least the elements can be put in correspondence with natural numbers.

On the other hand, not all infinite sets are countably infinite. In fact, there are infinitely many sizes of infinite sets.

Let be a set such that , that is, such that S is not a singleton.

Let be the set of all mappings from the natural numbers to :

Then is uncountably infinite.

### Symbolic Logic[edit | edit source]

## Further Information[edit | edit source]

### Wikipedia[edit | edit source]

## References[edit | edit source]

- ↑ Also known as computability theory