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  • 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

    φL ψ

    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

    The alphabet of a formal system
    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'