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Examples of Natural Deduction

Examples of Natural Deduction. D. Mott ETS, IBM UK. Using “Fitch”. All pets are happy, so what about “scruffy?”. Premises. Lines of proof, each applying a natural deduction rule to previous lines. Goal. In detail. Premises. All pets are happy. scruffy is a pet. Goal. scruffy is happy.

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Examples of Natural Deduction

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  1. Examples of Natural Deduction D. Mott ETS, IBM UK

  2. Using “Fitch”

  3. All pets are happy, so what about “scruffy?” Premises Lines of proof, each applying a natural deduction rule to previous lines Goal

  4. In detail Premises All pets are happy scruffy is a pet Goal scruffy is happy cites FORALL elimination -> elimination scruffy is happy QED

  5. But … • We can do this already in CE! • More interesting if we want to infer a new rule • Given “All pets are happy” • can we show that if something is unhappy then it is not a pet? • if not happy(X) then not pet(X) • Need to introduce the idea of a subproof first

  6. Subproof to show scruffy is not a pet All pets are happy Scruffy is not happy Is scruffy not a pet? SUPPOSE scruffy is a pet New subproof inside main proof We can show that scruffy is happy But we have proved an inconsistency Therefore the supposition is false, i.e. scruffy is not a pet

  7. Generalising the subproof SUPPOSE that is some thing, which we will call “c” that is not happy then “c” is not a pet But what “c” actually is is not relevant to the proof, so it works for ALL things So ANY thing that is not happy is not a pet, and we can create a generic rule

  8. In CE… • We could use negated information, eg: • it is false that the thing T is a pet • But in the logic problems I am using terms that include a negation: • cannot be wearing • So this new rule might be: • if the thing T is a nonpet then the thing T is an unhappy thing • This requires an additional step in the proof to perform the negation, based on the domain definition of “unhappy” as being the negation of “happy”, etc

  9. Could we build a similar system in CE? • In most cases there is a simple correspondance between the natural deduction rules and manipulations that could be made on the CE statements • But in the use of subproofs, it is necessary to manipulate the rationale graphs themselves • given a rationale link between a premise and its conclusion, we could construct a new rule • given a rationale link between a premise and an inconsistency, we could construct the negation of the premise.

  10. It might look like…

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