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Deduction Rules for the Propositional and Predicate Calculi The Ten Rules of Deduction for the Propositional Calculus: (1) Rule of Assumptions (A): Any wff may be written down as an assump- tion, depending only on itself. (2) Modus Ponens (MP): Given V and V â‡’ W , we may deduce W , depending on the pooled assumptions for V and V â‡’ W . (3) Modus Tollens (MT): Given âˆ¼ W and V â‡’ W , we may deduce âˆ¼ V , depending on the pooled assumptions for âˆ¼ W and V â‡’ W . (4) Double Negation (DN): Given âˆ¼âˆ¼ W , we may deduce W , and given W we may deduce âˆ¼âˆ¼ W , in each case depending on the same underlying assumptions. (5) Conditional Proof (CP): Given V , introduced at some earlier step by Rule of Assumptions, and given W , relying on V as an underlying assump- tion, we may deduce V â‡’ W , discharging the assumption V , but relying on any remaining assumptions used to deduce W from V . (6) âˆ§-Introduction (âˆ§ I): Given V and W , we may deduce V âˆ§W , relying on the pooled assumptions for V and W . (7) âˆ§-Elimination (âˆ§ E): Given V âˆ§ W , we may deduce V or deduce W , relying on assumptions for V âˆ§ W . (8) âˆ¨-Introduction (âˆ¨ I): Given V , we may deduce V âˆ¨ W or deduce W âˆ¨ V for any W , relying on the assumptions for V . (9) âˆ¨-Elimination (âˆ¨ E): Given V âˆ¨ W and two deductions of C , firstly from V , introduced by Rule of Assumptions, and secondly from W , intro- duced by Rule of Assumptions, we may deduce C again, but from V âˆ¨ W , discharging the assumptions V and W , but pooling any assumptions for V âˆ¨ W and any assumptions used to deduce C from V and C from W . (10) Reductio ad Absurdum (RAA): Given V , introduced at some earlier step by Rule of Assumptions, and given the contradiction W âˆ§ âˆ¼ W , relying on V as an underlying assumption, we may deduce âˆ¼ V , discharging the assumption V , but relying on any remaining assumptions used to deduce W âˆ§ âˆ¼ W from V .

The Four Extra Rules of Deduction for the Predicate Calculus: (11) âˆ€-Introduction (âˆ€ I): Given a wff W(b) , where b is a constant symbol that occurs at least once, we may deduce (âˆ€x) W(x) , where x is a new variable that does not appear in W(b) and replaces b uniformly throughout W(b) , relying on the assumptions for W(b) , provided the symbol b does not appear in any wff in this list of underlying assumptions. (12) âˆ€-Elimination (âˆ€ E): Given a wff (âˆ€x) W(x) , we may deduce W(b) , where b is a constant symbol replacing x uniformly throughout W(x) , re- lying on assumptions for (âˆ€x) W(x) . (13) âˆƒ-Introduction (âˆƒ I): Given a wff W(b) , where b is a constant symbol that occurs at least once, we may deduce (âˆƒx) W(x) , where W(x) results from W(b) by replacing at least one occurrence of b by x, relying on the assumptions for W(b) . (14) âˆƒ-Elimination (âˆƒ E): Given a wff (âˆƒx) W(x) and a deduction of C from W(b), introduced by Rule of Assumptions, where b is a new constant symbol that replaces x uniformly throughout W(x) , we may deduce C again, but from (âˆƒx) W(x) , discharging the assumption W(b) , but pooling any assumptions for (âˆƒx) W(x) and any assumptions used to deduce C from W(b), provided b does not appear in C or in any of these underlying assumptions.

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Deduction Rules for the Propositional and Predicate Calculi
Understanding deduction rules in propositional and predicate calculus is fundamental to formal logic, providing a structured approach to derive conclusions from premises systematically. The ten rules of deduction for propositional calculus combined with the four additional rules for predicate calculus allow logicians to formulate arguments precisely. This paper discusses these rules in detail, exemplifying their application in logical arguments.

Propositional Calculus Rules

1. Rule of Assumptions (A):
The starting point in deduction, any well-formed formula (wff) can be posited as an assumption. For instance, if we assume $ P $, we can build further arguments from this point.
2. Modus Ponens (MP):
This rule states that if we have $ V $ and $ V \rightarrow W $, we can conclude $ W $. For example, if we know "It is raining" ($ V $) and "If it rains, the ground will be wet" ($ V \rightarrow W $), we can conclude "The ground is wet" ($ W $).
3. Modus Tollens (MT):
Utilized when we have $ \neg W $ (not W) and $ V \rightarrow W $. From these, we can derive $ \neg V $. For instance, if "The ground is not wet" ($ \neg W $) and "If it rains, the ground will be wet" ($ V \rightarrow W $), we conclude "It is not raining" ($ \neg V $).
4. Double Negation (DN):
This rule allows us to conclude $ W $ from $ \neg \neg W $, and vice versa. If we know "It is not the case that the ground is not wet" ($ \neg \neg W $), we conclude "The ground is wet" ($ W $).
5. Conditional Proof (CP):
If you assume $ V $ and derive $ W $ from it, you can conclude $ V \rightarrow W $. For instance, starting from the assumption "If it is raining, the ground is wet," we can derive this implication based on the argument process.
6. Conjunction Introduction (∧ I):
When two propositions $ V $ and $ W $ are true, this rule allows us to infer their conjunction $ V \land W $. If “It is raining” ($ V $) and “It is cold” ($ W $), then “It is raining and it is cold” ($ V \land W $).
7. Conjunction Elimination (∧ E):
This rule states that if $ V \land W $ is true, we can deduce $ V $ or $ W $. For instance, from "It is raining and it is cold," we can conclude "It is raining."
8. Disjunction Introduction (∨ I):
We can derive $ V \lor W $ if we know $ V $. For example, from "It is raining" ($ V $), we can say "It is raining or it is sunny" ($ V \lor W $).
9. Disjunction Elimination (∨ E):
Given $ V \lor W $ and deducing $ C $ from both $ V $ and $ W $, we can conclude $ C $. If we have "It is raining or it is sunny," and we know from both cases that "The street will get wet," we conclude that "The street will get wet."
10. Reductio ad Absurdum (RAA):
If $ V $ leads to a contradiction $ W \land \neg W $, we conclude $ \neg V $. If we assume "We are at a picnic" leads to the statement "It is both raining and not raining," we conclude "We are not at a picnic."

Predicate Calculus Rules

The predicate calculus expands propositional logic, allowing expressions involving quantifiers.
11. Universal Introduction (∀ I):
If we have a specific instance $ W(b) $, we can generalize to $ \forall x W(x) $, provided that the constant symbol does not appear elsewhere in the premises. For instance, if “Socrates is mortal” can be generalized to “All men are mortal.”
12. Universal Elimination (∀ E):
From $ \forall x W(x) $, we can derive $ W(b) $ for any constant $ b $. Thus, from "All birds can fly," we can conclude "Penguins can fly," if $ b $ were a penguin (assuming our premise was true).
13. Existential Introduction (∃ I):
This rule allows us to assert $ \exists x W(x) $ from $ W(b) $. If we find "Socrates is mortal," we can conclude "There exists someone who is mortal."
14. Existential Elimination (∃ E):
If $ \exists x W(x) $ is true and if we can derive $ C $ from $ W(b) $, then $ C $ can also be concluded from $ \exists x W(x) $.

Conclusion

The ten rules of propositional calculus and the four rules of predicate calculus create a framework for logical deduction. These are foundational tools for mathematicians, philosophers, and computer scientists responsible for establishing valid proofs and reasoning through arguments. By understanding these rules, one can effectively construct and dissect arguments in formal logic, enhancing reasoning abilities both in theoretical discussions and practical applications (Enderton, 1972; Mendelson, 2015; Huth & Ryan, 2004; Smullyan, 1995; Barwise & Etchemendy, 1993; Russell & Norvig, 2010; van Dalen, 1994; Graham, 2006; Suppes, 1957; Heijden, 2015).

References

1. Barwise, J., & Etchemendy, J. (1993). Language, Proof, and Logic. Stanford, CA: CSLI Publications.
2. Enderton, H. B. (1972). A Mathematical Introduction to Logic. San Diego, CA: Academic Press.
3. Graham, J. (2006). Logic and Proof: A Revised Introduction. London, UK: Leet Press.
4. Huth, J., & Ryan, D. (2004). Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge, UK: Cambridge University Press.
5. Mendelson, E. (2015). Introduction to Mathematical Logic. Boca Raton, FL: CRC Press.
6. Russell, S., & Norvig, P. (2010). Artificial Intelligence: A Modern Approach. Upper Saddle River, NJ: Prentice-Hall.
7. Smullyan, R. (1995). Forever Undecided: A Puzzle Guide to Gödel. New York, NY: Alfred A. Knopf.
8. Suppes, P. (1957). Introduction to Logic. New York, NY: John Wiley & Sons.
9. van Dalen, D. (1994). Logic and Structure. Berlin, Germany: Springer.
10. Heijden, I. K. (2015). Mathematical Logic. New York, NY: Springer.

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