copi

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'. 

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

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

And the right syntax (the premises separated by commas and then a 'therefore' followed by the conclusion).

So, a selection needs to have a form similar to this

F⊃G,G⊃H∴F⊃H

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You now have to 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 Rule of Universal Instantiation UI

If a derivation contains a line of the form

n (<variable>)<scope> <any justification>

then a line of the form

<<scope>[<term>/<variable>]> 'n UI'

may be added to the derivation, provided that <variable> is free for <term> in the <scope> ie that the substitution [<term>/<variable>] is legitimate .

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The material explained here is conceptually at a higher level than the earlier tutorials. It also covers relatively rare cases.

Skills to be acquired:

To understand the concepts of scope, free, bound, and free for. To learn of the restrictions on the quantificational rules

Why this is useful:

The new predicate rules of inference using quantifiers have restrictions on them which are expressed in terms of these concepts. And, in turn, the restrictions on the rules allow them to be used properly.

The Tutorial

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

Symbolization using the Universal Quantifier (of non relational English)

‘Everything is lovely’ we understand as meaning ‘Whatever thing you may choose, it is lovely’ and this becomes

(x)Lx

The Tutorial

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.

The Tutorial

Existential Instantiation permits you to remove an existential quantifier from a formula which has an existential quantifier as its main connective.

The circumstance that Existential Instantiation gets invoked looks like this.

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The Tutorial

There is a rule for adding a Existential Quantifier. This permits the step illustrated by the following proof fragments.

5	Fa
6	(∃x)Fx	5 EG

 

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The Tutorial

There also is a rule for adding a Universal Quantifier. This permits the step illustrated by the following proof fragments.

5	Fx
6	(x)Fx	5 UG