Collecting different rules for Natural Deduction

In summary, the conversation is about different sets of rules of inference and the use of the symbol "therefore" or "|-" to indicate deducibility. The conversation also mentions the book Principia Mathematica and the author's personal opinion on its use of Modus Ponens as the only rule of inference.
  • #1
honestrosewater
Gold Member
2,142
6
I'm only interested in the inference rules. I've read that people use different sets of rules, and I'm trying to find all of those different sets. Here's the ones I have so far (read "/" as a line break):
1. Modus Ponens (M.P.)
2. Modus Tollens (M.T.)
3. Hypothetical Syllogism (H.S.): p -> q / q -> r / ∴ p -> r.
4. Disjunctive Syllogism (D.S.): p V q / ~p / ∴ q.
5. Constructive Dilemma (C.D.): (p -> q) & (r -> s) / p V r / ∴ q V s.
6. Absorption (Abs.): p -> q / ∴ p -> (p & q).
7. Simplification (Simp.): p & q / ∴ p.
8. Conjunction (Conj.): p / q / ∴ p & q.
9. Addition (Add.): p / ∴ p V q.
(1-9) is a complete set. I've found a set differing from (1-9) only by replacing (6) with
6b. Destructive Dilemma (D.D.): (p -> q) & (r -> s) / ~q V ~s / ∴ ~p V ~r.
Does anyone know of any more?
 
Physics news on Phys.org
  • #2
honestrosewater said:
I'm only interested in the inference rules. I've read that people use different sets of rules, and I'm trying to find all of those different sets. Here's the ones I have so far (read "/" as a line break):
1. Modus Ponens (M.P.)
2. Modus Tollens (M.T.)
3. Hypothetical Syllogism (H.S.): p -> q / q -> r / ∴ p -> r.
4. Disjunctive Syllogism (D.S.): p V q / ~p / ∴ q.
5. Constructive Dilemma (C.D.): (p -> q) & (r -> s) / p V r / ∴ q V s.
6. Absorption (Abs.): p -> q / ∴ p -> (p & q).
7. Simplification (Simp.): p & q / ∴ p.
8. Conjunction (Conj.): p / q / ∴ p & q.
9. Addition (Add.): p / ∴ p V q.
(1-9) is a complete set. I've found a set differing from (1-9) only by replacing (6) with
6b. Destructive Dilemma (D.D.): (p -> q) & (r -> s) / ~q V ~s / ∴ ~p V ~r.
Does anyone know of any more?

imo, rules of inference are implicative tautologies in use.

What does the square mean here?

Russell and Whitehead claim they used Modus Ponens as the only rule of inference in Principia Mathematica.
 
  • #3
Owen Holden said:
What does the square mean here?
Sorry, it's supposed to be "therefore"- I guess I'll just use ".:". I'm trying to find a code that works for everyone. :frown:
Russell and Whitehead claim they used Modus Ponens as the only rule of inference in Principia Mathematica.
Well, that's great but a little too insane for my tastes. Have you seen PM? ;)
 
  • #4
honestrosewater said:
Sorry, it's supposed to be "therefore"- I guess I'll just use ".:". I'm trying to find a code that works for everyone. :frown:
Well, that's great but a little too insane for my tastes. Have you seen PM? ;)

The assertion sign '|-' is often used to say that such and such is deducible,
e.g. |- (p & (p -> q)) -> q, or |-(p) & |-(p -> q) -> |-(q), etc.

Yes, I have a copy of the paperback PM.
 

1. What is Natural Deduction?

Natural Deduction is a system of logic used in mathematics and philosophy to prove the validity of arguments. It is based on the principle of using a set of rules to make deductions from premises to reach a conclusion.

2. How does Natural Deduction differ from other logical systems?

Unlike other logical systems, Natural Deduction focuses on the structure of arguments rather than their truth values. This means that the validity of an argument can be determined without considering whether the premises are actually true or false.

3. What are the benefits of collecting different rules for Natural Deduction?

Collecting different rules for Natural Deduction allows for a more comprehensive and flexible approach to solving logical problems. It also allows for a better understanding of the underlying principles and patterns involved in logical reasoning.

4. How do you determine which rules to use in a Natural Deduction proof?

The rules used in a Natural Deduction proof depend on the structure of the argument and the type of logic being used. Generally, one starts with the basic rules, such as introduction and elimination, and then applies more complex rules as needed.

5. Can Natural Deduction be applied to real-world situations?

Yes, Natural Deduction can be applied to real-world situations, such as legal arguments or scientific hypotheses. It can help identify flaws in reasoning and clarify complex arguments. However, it should be noted that Natural Deduction is a simplified system and may not always accurately reflect the complexity of real-world situations.

Similar threads

I Hilbert's omega rule, induction, omega-consistency
    • Set Theory, Logic, Probability, Statistics
Replies
15
Views
2K
Rules of Inference problem
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Prove a proposition using natural deduction
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
MHB Discrete Math Exercise: Solving Logic Puzzle in Computer Science
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
Challenge Math Challenge - June 2021
    • Math Proof Training and Practice
2
Replies
61
Views
6K
Help understanding Suppes' formal conditional definition
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
Challenge Math Challenge - July 2023
    • Math Proof Training and Practice
2
Replies
69
Views
3K
I New Naming Rules for Exotic Hadrons at the LHC: LHCb Collaboration Paper (2022)
    • High Energy, Nuclear, Particle Physics
Replies
7
Views
1K
(Symbolic Logic) Proving P v P = P (Idempotency) using a direct proof
    • Engineering and Comp Sci Homework Help
Replies
2
Views
3K
Challenge Math Challenge - December 2021
    • Math Proof Training and Practice
3
Replies
93
Views
10K

Hot Threads

Recent Insights

Back
Top