by Brian Bosse, Copyright April 20, 2010, all rights reserved. 458 views
The following are four inference rules we shall use with the propositional calculus. We will add more rules as we introduce the different sentential connectives. (It is assumed throughout that both φ and ψ represent symbolic sentences.)
1. Modus Ponens (MP): If one has both (φ → ψ) and φ, then one may conclude ψ.
Premise 1: If you believe, then you will be saved.
Premise 2: You believe.
If we let A stand for “you believe” and B stand for “you will be saved”, then we can symbolize the above premises as follows…
Premise 1: (A → B)
Premise 2: A
By the rule of MP we can conclude…
Conclusion: B (You believe.)
2. Modus Tollens (MT): If one has both (φ → ψ) and ¬ ψ, then one may conclude ¬φ.
Premise 1: If you believe, then you will be saved.
Premise 2: It is not the case you will be saved.
If we let A stand for “you believe” and B stand for “you will be saved”, then we can symbolize the above premises as follows…
Premise 1: (A → B)
Premise 2: ¬B
By the rule of MT we can conclude…
Conclusion: ¬A (It is not the case you believe.)
3. Double Negation (DN): If one has ¬¬φ, then one may conclude φ; or, if one has φ, then one may conclude ¬¬φ.
4. Repetition (R): If one has φ, then one may conclude φ.
We have simply asserted these four rules; although, we did provide some justification for MP and MT by referring to their use in the Bible. Hopefully, these rules are all intuitively obvious to the reader.