by Brian Bosse, Copyright May 01, 2010, all rights reserved. 784 views
There are three types of derivations that we will use in our propositional calculus: (1) Direct Derivation (DD), (2) Conditional Derivation (CD), and (3) Indirect Derivation (ID). The following is a proof in the propositional calculus using the direct derivation method.
Premise 1: If Socrates did not die of old age, then the Athenians condemned Socrates to death.
Premise 2: The Athenians did not condemn Socrates to death.
Show: Socrates did die of old age.
To begin, we need to translate this argument into symbolic sentences. Here is the translation…
P: Socrates did die of old age.
Q: The Athenians condemned Socrates to death.
Premise 1: ¬P → Q
Premise 2: ¬Q
Show: P
To begin our proof we begin with what it is we are trying to show. As such, our first line is…
1. Show P [DD]
At this point we will list our two premises…
2. ¬P → Q [Premise 1] 3. ¬Q [Premise 2]
We now consider what inference rules we can use, and see that MT can be applied to 2 and 3. If we let φ stand for ¬P and ψ stand for Q, then by MT we can conclude ¬ φ, which when we translate back is ¬¬P. This is our line 4.
4. ¬¬P [MT – 2 and 3]
We now are able to apply the rule of DN to line four and get…
5. P [DN – 4]
At this point we have derived P from our premises, and as such have shown P. To indicate this we put a line through ‘Show’ in line 1. This would be the full proof…
1.ShowP [DD] 2. ¬P → Q [Premise 1] 3. ¬Q [Premise 2] 4. ¬¬P [MT – 2 and 3] 5. P [DN – 4] Q.E.D.
This is considered a direct derivation because each line follows directly from previous lines (with the exception of the first line). Note: the brackets next to each line are not part of the propositional calculus. They simply are an aid to the reader to show which rule is being used to account for the symbolic sentence. Also, ‘Q.E.D.’ stands for the Latin phrase “quod erat demonstrandum,” which means, “which was to be demonstrated.”