hausman

11/9/2007 10Software

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}

2/5/08

Reading

Hausman[2007] Logic and Philosophy Chapter 10

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

Symbolization using the Universal Quantifier (of non relational English)

10 Software

Reading

Hausman[2007] Logic and Philosophy Chapter 10

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.

2/5/08 10Software

Reading

Hausman[2007] Logic and Philosophy Chapter 9

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.

2/5/08 10Software

Reading

Hausman[2007] Logic and Philosophy Chapter 9

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

 

Tutorial 19 Simplified Universal Generalization.

2/5/08 10Software

Reading

Hausman[2007] Logic and Philosophy Chapter 9

The Tutorial

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

2/5/08 10Software

Reading

Hausman[2007] Logic and Philosophy Chapter 9

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.