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
Criterion for argumentative validity:
Assuming the truth of all the premises, the conclusion set cannot be false
(Alternatively) the conclusion set follows from the premises as a matter of logical necessity
Criterion for argumentative soundness:
CONDITION 1: The argument is valid
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
The criterion of argumentative validity relies on the truth values of propositions
A truth table represents the truth values of propositions
∴ A truth table test could in principle be constructed to illustrate the validity of an argument
CAVEATS:
CAVEAT 1: The number of rows increases exponentially as the number of propositional variables increases
Assuming classical bivalence and with n propositional variables, we get 2^{n} rows in our truth table
CAVEAT 2: Things also get more complicated if we admit more than 2 truth values
With a trivalent or 3-valued system of logic and n propositional variables, we get 3^{n} rows in our truth table
Where φ ⊦_{L}ψ denotes an argument:
STEP 1: Decompose all the constituents of the argument (viz. φ, ψ) into their propositional atoms
STEP 2: On the LHS side of the truth table, identify all possible combinations of input truth values for these propositional atoms
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)
STEP 4: Scan every row of the truth table
STEP 5: Identify the rows where the premises (i.e. members of φ) are all true
STEP 6: If the conclusion (i.e. members of ψ) is false in at least one of these rows, then the argument is invalid
Otherwise, the argument is valid
P1: p → q
P2: ∼q
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
It is only in INTERPRETATION 1 that all the argumentative premises (P1 and P2) are true
The output truth value of the conclusion (C) is true
∴ The argument is valid
We have just used the truth table test to illustrate the validity of this argument
This argument is a valid rule of inference: it is known as the modus tollens rule of inference