Me: I’m trying to come up with a model for a deductive system. It should contain four parts: axioms, a logic or rules of inference, a proof methodology, and an overall schema or system. What can you add to my thoughts?

ChatGPT: Certainly! Developing a deductive system involves careful consideration of axioms, rules of inference, proof methodology, and an overall system or schema. Here’s a breakdown of each component:

1. 1. Axioms:
      • Define the fundamental, self-evident truths or statements that serve as the starting point for the deductive system.
      • Axioms should be clear, consistent, and independent, forming the foundation upon which logical reasoning will be built.
      • Ensure that the axioms are relevant to the specific domain or problem you are addressing.
   2. Logic or Rules of Inference:
      • Establish a set of rules that dictate how conclusions can be drawn from the given axioms.
      • Common rules of inference include modus ponens, modus tollens, hypothetical syllogism, and more.
      • Choose rules that align with the nature of your deductive system and support valid logical reasoning.
   3. Proof Methodology:
      • Specify how proofs will be constructed within your deductive system.
      • Determine the structure and format of a valid proof, including the sequence of logical steps that lead from the axioms to the desired conclusion.
      • Consider the principle of mathematical induction, direct proof, proof by contradiction, or other proof techniques depending on the context.
   4. Overall Schema or System:
      • Develop a coherent and systematic framework that integrates the axioms, rules of inference, and proof methodology.
      • Clarify the relationships between different components and how they interact within the deductive system.
      • Consider whether your system aligns with classical logic, modal logic, propositional logic, or some other logical framework.

Here’s a more detailed breakdown:

• • Introduction:
    • Present the purpose and scope of your deductive system.
    • Define the terms and symbols used in the system.
  • Axioms:
    • List and explain each axiom.
    • Ensure clarity and completeness.
  • Rules of Inference:
    • Enumerate the rules of inference.
    • Provide examples to illustrate how the rules are applied.
  • Proof Methodology:
    • Specify the steps involved in constructing a proof.
    • Define the criteria for a valid proof.
  • Examples and Applications:
    • Demonstrate the application of your deductive system with specific examples or problem instances.
    • Show how the axioms and rules lead to valid conclusions.
  • Analysis:
    • Evaluate the strengths and limitations of your deductive system.
    • Discuss any assumptions or constraints.
  • Conclusion:
    • Summarize the key components of your deductive system.
    • Reflect on the effectiveness of the system in addressing the intended purpose.

Remember to iterate and refine your deductive system based on feedback and further analysis. It’s also important to be explicit and precise in your definitions and statements to ensure the soundness of your deductive reasoning.

Me: Thanks! I guess I was trying to decide if any model for a deductive system would have these four parts. What do you think?

ChatGPT: Yes, typically, deductive systems do encompass these four fundamental parts: axioms, rules of inference, proof methodology, and an overall schema or system. These components work together to create a coherent and systematic framework for logical reasoning. Let’s briefly revisit each part:

1. 1. Axioms: Axioms serve as the foundation of a deductive system, providing self-evident truths or statements from which logical reasoning begins.
   2. Rules of Inference: These rules dictate how conclusions can be drawn from the axioms, enabling the systematic derivation of new statements or theorems.
   3. Proof Methodology: The proof methodology outlines how to construct valid proofs within the deductive system, specifying the logical steps to move from the axioms to a desired conclusion.
   4. Overall Schema or System: This component ties everything together, providing a structured framework that defines the relationships and interactions between axioms, rules of inference, and proof methodology.

Together, these parts create a robust deductive system that allows for rigorous and valid logical reasoning. The specific details and nature of these components may vary depending on the type of logic or the problem domain you are addressing. However, having these four elements ensures a systematic and well-defined approach to deductive reasoning.

Further Reading:

https://en.wikipedia.org/wiki/Formal_system

https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)

https://www.sciencedirect.com/topics/computer-science/deductive-system

https://ncatlab.org/nlab/show/deductive+system

[*11.167]

Logic is not as absolute as we would like it to be. For example, linear logic breaks down normal logic into a realm of substructurality. There seems to be several ways to consider expanding classical two-valued logic to four values.

Let this post be a placeholder for considering expansion of classical two-valued logic to four values. For example, one might have:

• True
• False
• Both
• None

Further Reading:

https://en.wikipedia.org/wiki/Four-valued_logic

J. Michael Dunn / Two, Three, Four, Infinity: The Path to the Four-Valued Logic and Beyond
https://link.springer.com/chapter/10.1007/978-3-030-31136-0_6

Katalin Bimbo & J. Michael Dunn / Four-valued Logic
https://projecteuclid.org/euclid.ndjfl/1063372199

J. Ulisses Ferreira / A Four-Valued Logic
https://doi.org/10.5121%2Fcsit.2017.71206

[*12.6]

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