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

# φ ⊦Lψ

General form of an argument in which L denotes a formal system
1. A formal system L consists of:
2. An alphabet containing symbols from which formulae are generated;
3. A well-defined syntax, specifying which formulae are wffs (or well-formed formulae);
4. A well-defined semantics, associating formulae with their respective meanings;
5. 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. x1, x2, etc)

a, b, c, …, w
Individual constants (lower-cased)

R, G, …
n-ary relations (upper-cased)

∼, ∧, ∨, →, ⟷
1. Truth-functional logical connectives:
2. ∼ denotes a negation: '∼p' denotes 'not-p'
3. ∧ denotes a conjunction: 'p ∧ q' denotes 'p and q'
4. ∨ denotes an inclusive disjunction: 'p ∨ q' denotes 'p or q'
5. → denotes a material conditional: 'p → q' denotes 'if p then q'
6. ⟷ denotes a material biconditional: 'p ⟷ q' denotes 'p iff q' (or 'if p then q and if q then p')

7. NOTE: An inclusive disjunction (denoted by '∨') is distinct from an exclusive disjunction (denoted by '⊻')
8. In an inclusive disjunction p ∨ q, at least one of the two disjuncts (p or q) would have to be true
9. In an exclusive disjunction p ⊻ q, at least and at most one of the two disjuncts (p or q) would have to be true
10. Suppose both p and q are true
11. ∴ p ∨ q (for 'p or q' in an inclusive sense) would be true
12. ∴ p ⊻ q (for 'p or q' in an exclusive sense) would be false
∀, ∃
1. Quantifiers in first-order predicate or quantificational logic:
2. '∀x' denotes 'for all x'
3. '∃x' denotes 'for some x' or 'there exists at least one x'

□, ◇
1. Modal operators in alethic modal logic:
2. '□p' denotes that 'p is necessarily true'
3. '◇p' denotes that 'p is possibly true'

O, P
1. Deontic operators in deontic logic:
2. 'Op' denotes that 'The action whose performance is described in p is obligatory'
3. 'Pp' denotes that 'The action whose performance is described in p is permissible'

B, K, C
1. Epistemic operators in epistemic logic:
2. 'Bap' denotes that 'Agent a believes that p'
3. 'Kap' denotes that 'Agent a knows that p'
4. 'CGp' denotes that 'A group of agents G has common knowledge that p'

F, G, X, U
1. Temporal operators in temporal logic:
2. 'Fφ' denotes that 'Finally, at some state on the path, the property φ will hold'
3. 'Gφ' denotes that 'Globally, along the entire path, the property φ will hold'
4. 'Xφ' denotes that 'At the next state of the path, the property φ will hold'
5. 'φUψ' denotes that 'For 2 properties φ and ψ, the 1st property φ holds at every state along the path until at some state the 2nd property ψ holds'