# First Order Logic Resources

### From OpenWetWare

## Contents |

## Tutorials

- Rules and terminology by Wolfram Mathworld [FOL]
- Some [good note]s on how to start with FOL (very programming oriented)
- Peter Suber's [translating logic into English AND his [glossary] of set theory is immensely helpful for the uninitiated.

- [PD Magnus] forall x: an introduction to formal logic.

- Thorough intro, but very "textbookish" below

## Symbols in Logic

This chart can be found on the [wikipedia page]

## Other Logic Resources

## Other Tutorials

Symbol
| Name | Explanation | Examples | Unicode Value | HTML Entity | LaTeX symbol |
---|---|---|---|---|---|---|

Should be read as | ||||||

Category | ||||||

⇒
→ ⊃ | material implication | A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). ⊃ may mean the same as ⇒ (the symbol may also mean superset). | x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2).
| U+21D2 U+2192 U+2283 | ⇒ → ⊃ | \Rightarrow
\to \supset |

implies; if .. then | ||||||

propositional logic, Heyting algebra | ||||||

⇔
≡ ↔ | material equivalence | A ⇔ B means A is true if B is true and A is false if B is false.
| x + 5 = y +2 ⇔ x + 3 = y
| U+21D4 U+2261 U+2194 | ⇔ ≡ ↔ | \Leftrightarrow
\equiv \leftrightarrow |

if and only if; iff | ||||||

propositional logic | ||||||

¬
˜ ! | negation | The statement ¬A is true if and only if A is false.A slash placed through another operator is the same as "¬" placed in front. | ¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y)
| U+00AC U+02DC | ¬ ˜ ~ | \lnot
\sim |

not | ||||||

propositional logic | ||||||

∧
• & | logical conjunction | The statement A ∧ B is true if A and B are both true; else it is false.
| n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.
| U+2227 U+0026 | ∧ & | \land \&<ref>Although this character is available in LaTeX, the Mediawiki TeX system doesn't support this character.</ref> |

and | ||||||

propositional logic | ||||||

∨
+ | logical disjunction | The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
| n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number.
| U+2228 | ∨ | \lor |

or | ||||||

propositional logic | ||||||

exclusive disjunction | The statement A ⊕ B is true when either A or B, but not both, are true. A Template:Unicode B means the same.
| (¬A) ⊕ A is always true, A ⊕ A is always false.
| U+2295 U+22BB | ⊕ | \oplus | |

xor | ||||||

propositional logic, Boolean algebra | ||||||

⊤ T 1 | Tautology | The statement ⊤ is unconditionally true. | A ⇒ ⊤ is always true.
| U+22A4 | T | \top |

top | ||||||

propositional logic, Boolean algebra | ||||||

⊥ F 0 | Contradiction | The statement ⊥ is unconditionally false. | ⊥ ⇒ A is always true.
| U+22A5 | ⊥ F | \bot |

bottom | ||||||

propositional logic, Boolean algebra | ||||||

∀
| universal quantification | ∀ x: P(x) means P(x) is true for all x.
| ∀ n ∈ N: n^{2} ≥ n.
| U+2200 | ∀ | \forall |

for all; for any; for each | ||||||

predicate logic | ||||||

∃
| existential quantification | ∃ x: P(x) means there is at least one x such that P(x) is true.
| ∃ n ∈ N: n is even.
| U+2203 | ∃ | \exists |

there exists | ||||||

first-order logic | ||||||

∃!
| uniqueness quantification | ∃! x: P(x) means there is exactly one x such that P(x) is true.
| ∃! n ∈ N: n + 5 = 2n.
| U+2203 U+0021 | ∃ ! | \exists ! |

there exists exactly one | ||||||

first-order logic | ||||||

:=
≡ :⇔ | definition | x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).P :⇔ Q means P is defined to be logically equivalent to Q.
| cosh x := (1/2)(exp x + exp (−x))A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
| U+003A U+003D U+2261 U+003A U+229C | := : ≡ ⇔ | : = :=
\equiv \Leftrightarrow |

is defined as | ||||||

everywhere | ||||||

( )
| precedence grouping | Perform the operations inside the parentheses first. | (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. | U+0028 U+0029 | ( ) | ( ) |

everywhere | ||||||

inference | x Template:Unicode y means y is derived from x.
| A → B Template:Unicode ¬B → ¬A
| U+22A2 | \vdash | ||

infers or is derived from | ||||||

propositional logic, first-order logic |

## Logical Operator With Venn Explanations

[Wikipedia Page] is really best resource.

## Set Theory

*Failed to parse (syntax error): \{a \in \mathbf A }is used to denote that a is an element of a set A.

- is the set {0,1},
- is the set of all positive real numbers.

**Cardinal Number**

In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set.

**Function**

A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from to is an object such that every is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set of values at which a function is defined is called its domain, while the set of values that the function can produce is called its range. The term "map" is synonymous with function

## Basic Calc

**Integral** - An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive

**Derivative** - The derivative of a function represents an infinitesimal change in the function with respect to one of its variables.

The "simple" derivative of a function with respect to a variable is denoted either or

d f -- dx