6/24/07 10 Software
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
6/20/07 10Software
To understand how the various restrictions on the quantificational rules work to exclude certain kinds of invalid inferences
Bergmann[2004] The Logic Book Section 10.1
If a derivation contains a line of the form
n (∀<variable>)<scope> <any justification>
then a line of the form
<<scope>[<constant>/<variable>]> 'n ∀E'
12/26/05
11/9/2007 10Software
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}
6/22/07
Bergmann[2004] The Logic Book Sections 7.7-7.8
[The core of this is from Leblanc and Wisdom [1972] p.117 and f.]
10 Software
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.
11/3/07
10 Software
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.
11/3/07
6/19/07 10Software
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.