## Further help with Reductio

## Example of a difficult derivation

6/1/09

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6/1/09

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2013

You may have derivations of your own that you wish to try. Just type, paste, or drag and drop, them into the panel, select your derivation, and click 'Start from selection'. [*Often copy-and-paste won't work directly from a Web Page; however, usually drag-and-drop will work!*]

You will need to use the correct logical symbols. Here they are

F ∴ F & G ∼ & ∨ ⊃ ≡ ∀ ∃ ∴ (or use the palette to produce them)

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You now have the tools to appraise arguments to the level of detail offered by predicate logic.

Let us run through how these might be used with a long and difficult example.

Consider the argument

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The semantics of relations proceeds in much the way one would expect-- the new item that has to be taken account of is the order of the terms (because, for example, Tab is not at all the same thing as Tba -- Arthur being taller than Beryl is not the same as Beryl being taller than Arthur).

Let us start with an Interpretation

Interpretation 1

Universe= {a,b}

F={a}

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11/30/06

[The core of this is from Leblanc and Wisdom [1972] p.117 and f.]

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2013

Thus far we have considered only 'monadic' predicates-- our atomic formulas consist of a predicate followed by only one term-- for example, Fx. But in English we regularly encounter dyadic predicates or relations. For example, 'Arthur is taller than Bert' cannot be symbolized with the tools we have used so far; what is needed is a relation to represent '...is taller than ...' Txy, say, and then the proposition would be symbolized Tab.

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2013

Bergmann[2004] The Logic Book Section 10.1.

Existential Elimination (often called 'Existential Instantiation') permits you to remove an existential quantifier from a formula which has an existential quantifier as its main connective. It is one of those rules which involves the adoption and dropping of an extra assumption (like ∼I,⊃I,∨E, and ≡I).

The circumstance that Existential Instantiation gets invoked looks like this.

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2013

Bergmann[2004] The Logic Book Section 10.1.

There is a rule for adding a Existential Quantifier, Existential Introduction (also commonly known as 'Existential Generalization'). This permits the step illustrated by the following proof fragments.

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2013

To learn the rule for adding a Universal Quantifier.

Bergmann[2004] The Logic Book Section 10.1.

There also is a rule for adding a Universal Quantifier (often called 'Universal Generalization'). This permits the step illustrated by the following proof fragments.

The rule is

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2013

To understand the concepts of scope, free, bound. To meet substitution and the rule for removing a Universal Quantifier.

Bergmann[2004] The Logic Book Section 10.1.