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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

     

     

     

     

    Truth Table Test

    Argumentative criteria
    1. Criterion for argumentative validity:
    2. Assuming the truth of all the premises, the conclusion set cannot be false
    3. (Alternatively) the conclusion set follows from the premises as a matter of logical necessity

    5. Criterion for argumentative soundness:
    6. CONDITION 1: The argument is valid
    7. CONDITION 2: All the premises of the argument are true

    Truth table for p ⟷ q

    p q p ⟷ q
    0 0 1
    1 0 0
    0 1 0
    1 1 1


    1. The criterion of argumentative validity relies on the truth values of propositions
    2. A truth table represents the truth values of propositions
    3. ∴ A truth table test could in principle be constructed to illustrate the validity of an argument

    5. CAVEATS:
    6. CAVEAT 1: The number of rows increases exponentially as the number of propositional variables increases
    7. Assuming classical bivalence and with n propositional variables, we get 2n rows in our truth table

    9. CAVEAT 2: Things also get more complicated if we admit more than 2 truth values
    10. With a trivalent or 3-valued system of logic and n propositional variables, we get 3n rows in our truth table




    1. Where φ ⊦L ψ denotes an argument:
    2. STEP 1: Decompose all the constituents of the argument (viz. φ, ψ) into their propositional atoms
    3. STEP 2: On the LHS side of the truth table, identify all possible combinations of input truth values for these propositional atoms
    4. STEP 3: On the RHS of the truth table, compute all the associated output truth values of the argumentative constituents (i.e. members of φ and ψ)
    5. STEP 4: Scan every row of the truth table
    6. STEP 5: Identify the rows where the premises (i.e. members of φ) are all true
    7. STEP 6: If the conclusion (i.e. members of ψ) is false in at least one of these rows, then the argument is invalid
    8. Otherwise, the argument is valid

    1. P1: p → q
    2. P2: ∼q
    3. C: ∴ ∼p


    EXAMPLE of an argument


    Truth table test

    LHS (propositional atoms and all possible combinations of input truth values) RHS (associated output truth values of the argumentative premises and conclusion)
    p q p → q (or P1) ∼q (or P2) ∴ ∼p (or C)
    0 0 1 1 1
    1 0 0 1 0
    0 1 1 0 1
    1 1 1 0 0

    1. It is only in INTERPRETATION 1 that all the argumentative premises (P1 and P2) are true
    2. The output truth value of the conclusion (C) is true
    3. ∴ The argument is valid
    4. We have just used the truth table test to illustrate the validity of this argument
    5. This argument is a valid rule of inference: it is known as the modus tollens rule of inference
     

     

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