Use mathematical logic to prove this proposition

In summary, the given axioms prove the proposition $(p \to q) \to ((q \to r) \to (p \to r))$ using Modus Ponens as the rule of inference. The proposition can also be rewritten as $(A \to B) \to ((B \to C) \to (A \to C))$ and is proven by substituting the variables $p,q,r$ with $A,B,C$ and applying Modus Ponens repeatedly.
  • #1
solakis1
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0
Given the following axioms:
1) ##P\implies(Q\implies P)##
2) ##((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))## Where ##P,Q,R## are any formulas
3)##(\neg P\implies\neg Q)\implies (Q\implies P)## then prove:

##\{A\implies B,B\implies C\}|- A\implies C##
Without using the deduction theorem and as a rule of inference M.ponens
 
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  • #2
Your OP outlines the axioms of a Hilbert System. Go to the wiki page on Hilbert Systems and search "(HS2)" to see a proof of the following proposition from those axioms using Modus Ponens as rule of inference.
$$(p \to q) \to ((q \to r) \to (p \to r))$$
Relabel ##p,q,r## as ##A,B,C## to get
$$(A \to B) \to ((B \to C) \to (A \to C))$$
Then we have:
\begin{align}
&\vdash(A \to B) \to ((B \to C) \to (A \to C))\\
(A \to B), (B \to C)&\vdash(A \to B) \to ((B \to C) \to (A \to C))\\
(A \to B), (B \to C)&\vdash A\to B\quad\quad\textrm{[1st axiom]}\\
(A \to B), (B \to C)&\vdash(B \to C) \to (A \to C)
\quad\quad\textrm{[Modus Ponens on 3, 2]}\\
(A \to B), (B \to C)&\vdash B\to C \quad\quad\textrm{[2nd axiom]}\\
(A \to B), (B \to C)&\vdash A \to C
\quad\quad\textrm{[Modus Ponens on 5, 4]}
\end{align}
 
  • Informative
Likes berkeman

Related to Use mathematical logic to prove this proposition

1. What is mathematical logic?

Mathematical logic is a branch of mathematics that uses formal symbols and rules to study the principles of valid reasoning and proof. It provides a systematic way to analyze and evaluate arguments and statements in a precise and rigorous manner.

2. How is mathematical logic used to prove propositions?

Mathematical logic uses a set of axioms, definitions, and logical rules to construct a proof for a given proposition. This involves breaking down the proposition into smaller, more manageable statements and then using logical arguments to show that these statements are true.

3. What are the different types of mathematical logic?

There are several types of mathematical logic, including propositional logic, predicate logic, modal logic, and many others. Each type has its own set of rules and symbols for constructing proofs and analyzing arguments.

4. What is the importance of using mathematical logic in scientific research?

Mathematical logic is essential in scientific research as it allows scientists to make precise and accurate conclusions based on evidence. It helps to eliminate ambiguity and subjectivity in reasoning, making scientific arguments more rigorous and reliable.

5. Can mathematical logic be applied to real-world problems?

Yes, mathematical logic can be applied to real-world problems in various fields such as computer science, engineering, and philosophy. It provides a systematic and logical approach to problem-solving, making it a valuable tool in many practical applications.

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