derivations

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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 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

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

 

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Skill to be acquired:

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

The Tutorial

The propositional rules of derivation carry over unchanged into Predicate Logic

Exercises to accompany Predicate Tutorial 5

Exercise 1 (of 1).

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You now have to tools to appraise propositional arguments.

Let us run through how these might be used with two examples.

Example 1.

Consider the argument

If no human action is free, then no one is responsible for what they do.
If no one is responsible for what they do, no one should be punished.
Therefore
If no human action is free, no one should be punished.

First it should be symbolized

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Skills to be acquired

Becoming familiar with common inference patterns and being able to use them via replacement rules. This helps with assessing ordinary everyday reasoning such as that found in the law, in newspapers, in advertisements, etc.

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Skills to be acquired:

Learning conditional proof.

The Tutorial:

There is an additional propositional rule of inference-- Conditional Proof-- which is slightly more difficult than the ones that we have met before. It is slightly more difficult in that it requires you to make new assumptions, and the correct new assumptions at that.