Die Logik ist keine Lehre, sondern ein Spiegelbild der Welt.
Logik ist transzendental.
- Ludwig Wittgenstein's (1921, 6.13) Tractatus Logico-Philosophicus in the original German

Logic is not a body of doctrine, but a mirror-reflection of the world.
Logic is transcendental.
- Pears/McGuinness translation

Symbolic Notation

φ ⊦_{L}ψ

General form of an argument in which L denotes a formal system

A formal system L consists of:

An alphabet containing symbols from which formulae are generated;

A well-defined syntax, specifying which formulae are wffs (or well-formed formulae);

A well-defined semantics, associating formulae with their respective meanings;

A well-defined proof theory, making precise how reasoning in that system is to proceed by manipulating formulae in accordance with certain rules

x, y, z

Individual variables (lower-cased)
NOTE: You could add subscript if you have many distinct individual variables (e.g. x_{1}, x_{2}, etc)

a, b, c, …, w

Individual constants (lower-cased)

R, G, …

n-ary relations (upper-cased)

∼, ∧, ∨, →, ⟷

Truth-functional logical connectives:

∼ denotes a negation: '∼p' denotes 'not-p'

∧ denotes a conjunction: 'p ∧ q' denotes 'p and q'

∨ denotes an inclusive disjunction: 'p ∨ q' denotes 'p or q'

→ denotes a material conditional: 'p → q' denotes 'if p then q'

⟷ denotes a material biconditional: 'p ⟷ q' denotes 'p iff q' (or 'if p then q and if q then p')

NOTE: An inclusive disjunction (denoted by '∨') is distinct from an exclusive disjunction (denoted by '⊻')

In an inclusive disjunction p ∨ q, at least one of the two disjuncts (p or q) would have to be true

In an exclusive disjunction p ⊻ q, at least and at most one of the two disjuncts (p or q) would have to be true

Suppose both p and q are true

∴ p ∨ q (for 'p or q' in an inclusive sense) would be true

∴ p ⊻ q (for 'p or q' in an exclusive sense) would be false

∀, ∃

Quantifiers in first-order predicate or quantificational logic:

'∀x' denotes 'for all x'

'∃x' denotes 'for some x' or 'there exists at least one x'

□, ◇

Modal operators in alethic modal logic:

'□p' denotes that 'p is necessarily true'

'◇p' denotes that 'p is possibly true'

O, P

Deontic operators in deontic logic:

'Op' denotes that 'The action whose performance is described in p is obligatory'

'Pp' denotes that 'The action whose performance is described in p is permissible'

B, K, C

Epistemic operators in epistemic logic:

'B_{a}p' denotes that 'Agent a believes that p'

'K_{a}p' denotes that 'Agent a knows that p'

'C_{G}p' denotes that 'A group of agents G has common knowledge that p'

F, G, X, U

Temporal operators in temporal logic:

'Fφ' denotes that 'Finally, at some state on the path, the property φ will hold'

'Gφ' denotes that 'Globally, along the entire path, the property φ will hold'

'Xφ' denotes that 'At the next state of the path, the property φ will hold'

'φUψ' denotes that 'For 2 properties φ and ψ, the 1^{st} property φ holds at every state along the path until at some state the 2^{nd} property ψ holds'