Is This Logical Argument Valid?

• Euler2718
In summary, using the replacement rule ##p\to \neg q## being equivalent to ##\neg p\vee \neg q## and the rules of inference DNE (##\neg\neg q\equiv q##) and DS, we can prove that ##p \rightarrow \neg q , r \rightarrow q , r, \vdash \neg p ##.
Euler2718

Homework Statement

Determine whether the following is valid:

$$p \rightarrow \neg q , r \rightarrow q , r, \vdash \neg p$$

Homework Equations

Modus Ponens, disjunctive syllogism, double negation.

The Attempt at a Solution

I've boiled it down to

$$p \rightarrow \neg q , q, \vdash \neg p$$

However I do not understand how the book says to use disjunctive syllogism and double negation here. I've expanded the implication into its fundamental forum but I still can't see it.

These problems always depend on what rules of inference, axioms and replacement rules you've been given.
##p\to \neg q## is equivalent to ##\neg p\vee \neg q##. This may be specified as a replacement rule, or as the definition of one or the other of ##\to## or ##\vee##.

Using that replacement, are you able to do the problem using DNE (##\neg\neg q\equiv q##) and then DS?

Euler2718
andrewkirk said:
These problems always depend on what rules of inference, axioms and replacement rules you've been given.
##p\to \neg q## is equivalent to ##\neg p\vee \neg q##. This may be specified as a replacement rule, or as the definition of one or the other of ##\to## or ##\vee##.

Using that replacement, are you able to do the problem using DNE (##\neg\neg q\equiv q##) and then DS?

I've got it now, thank you. I wasn't seeing how ##neg neg q \equiv q ## would fit to DS, but now it's clear.

1. What is a "Rules of Inference problem" in scientific research?

A "Rules of Inference problem" is a type of logical problem that involves using a set of established rules in order to draw conclusions from given premises or facts. These rules are commonly used in scientific research to make logical deductions from experimental data or observations.

2. Why are "Rules of Inference problems" important in scientific research?

"Rules of Inference problems" are important in scientific research because they allow scientists to make logical and valid conclusions from their data. This helps to ensure that the results and findings of a study are accurate and can be used to make informed decisions and advancements in a particular field.

3. What are some common "Rules of Inference" used in scientific research?

Some common "Rules of Inference" used in scientific research include modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, and constructive dilemma. These rules help to establish relationships between different statements and make logical deductions from them.

4. How do scientists apply "Rules of Inference" in their research?

Scientists apply "Rules of Inference" in their research by using them to analyze and interpret their data. They start by identifying the premises or facts they have gathered, and then apply the relevant rules to draw conclusions from them. These conclusions are then used to support or reject a hypothesis or theory.

5. Can "Rules of Inference" be used in all areas of scientific research?

Yes, "Rules of Inference" can be used in all areas of scientific research. These rules are based on logic and reasoning, which are fundamental principles in the scientific method. They can be applied in various fields such as biology, chemistry, physics, and psychology to make logical deductions and draw conclusions from data.

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