Rules of Inference for Proving p→(p→q)→(p→q)

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In summary, the conversation discusses a proof involving a table with blank spaces that need to be filled in with either a rule or axiom. The last stage (stage 7) is causing confusion as the speaker is unable to determine which rule or axiom was used. The conversation ends with the other person pointing out that stage 7 can be completed using modus ponens from stages 3 and 6.
  • #1
Yankel
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Hello to you all,

I am trying to prove the following:

\[\vdash \left ( p\rightarrow \left ( p\rightarrow q \right ) \right )\rightarrow \left ( p\rightarrow q \right )\]

I was given a table with the proof stages, and I had to fill the blanks. Sometimes the blanks were the rule or axiom used in this stage, and sometimes it was the result of using the rule/axiom. I filled all, but I can't figure out the last stage. More specifically, I can't figure out which rule / axiom was used in the last stage (stage 7). I have completed all other stages, and I think it's correct.

I am attaching as figures, the table of the proof, the 3 axioms I am allowed to use (I am using the L deductive system), and 4 statements which were proved already and can be used. In addition, the only inference rule is the modus ponens.

Thank you in advance for helping me complete the last stage.

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  • #2
I think 7 follows by MP from 3 and 6.
 
  • #3
I think you are correct, I didn't see it. Thank you ! (Yes)
 

1. What are rules of inference?

Rules of inference are logical principles that allow us to draw conclusions from premises. These rules are used in deductive reasoning, where the conclusion is necessarily true if the premises are true.

2. How many types of rules of inference are there?

There are three main types of rules of inference: introduction rules, elimination rules, and transformation rules. Introduction rules allow us to introduce new information or concepts into an argument, elimination rules allow us to remove or use existing information, and transformation rules allow us to manipulate the structure of an argument.

3. What is the difference between a valid and an invalid rule of inference?

A valid rule of inference is one that always produces a valid conclusion when applied to valid premises. In other words, if the premises are true, the conclusion must also be true. An invalid rule of inference, on the other hand, may produce a false conclusion even if the premises are true.

4. Can we use rules of inference in everyday life?

Yes, we use rules of inference in everyday life without even realizing it. Whenever we draw conclusions based on information or evidence, we are using some form of rule of inference. For example, if we see dark clouds in the sky, we may infer that it will rain soon.

5. How are rules of inference used in scientific research?

In scientific research, rules of inference are used to analyze data and draw conclusions based on evidence. Scientists use deductive reasoning to make logical connections between observations and theories. By using rules of inference, scientists can make valid conclusions and further advance their understanding of the natural world.

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