derivations

11/27/11

Skills to be acquired in this tutorial:

a) Learning further sentential rules of inference. Carrying out simple sentential derivations using some of the Rules of Inference.

b) Learning elementary Tactics: how experts do derivations

Reading

Hausman[2007] Logic and Philosophy Chapter 4

Why this is useful:

Tactics will help you to do derivations.

The Tutorial:

a)

11/27/11

So that we can show certain arguments to be valid.

The focus of the course lies with the validity and invalidity of arguments. Now, invalidity can be established by counter-example (by producing an interpretation under which all the premises are true and the conclusion false, at the same time). But validity is a different matter. And the usual approach is to have rules of inference and to do derivations.

11/27/11

Skills to be acquired in this tutorial:

Learning further sentential rules of inference. Carrying out simple sentential derivations using some of the Rules of Inference.

Reading

Hausman[2007] Logic and Philosophy Chapter 4

The Tutorial:

Thus far we have encountered 3 sentential rules of inference: Simplification, Conjunction, and Addition. If we write these out as forms or patterns, these are the kinds of inferences they permit.

Simplication (Simp)

p.q
∴
p

11/27/11

Skills to be acquired in this tutorial:

Proving an argument to be valid by displaying a derivation. Carrying out simple sentential derivations using some of the Rules of Inference.

Reading

Hausman[2007] Logic and Philosophy Chapter 4

The Tutorial:

If you suspect that an symbolized argument might be valid, you should attempt to give a derivation of it.

A derivation is a proof of validity.

Formation Rules for System IV

10/8/09 Under construction

Terms, constants, variables, propositions, and predicates

Terms

<constant> ::= (['0'-'9']) |'∅'| 'U'|'{ }'
<subscript> ::=  ['₁'-'₉']
<functor> ::=  ['a'-'v'](< subscript >)*
<variable> ::=  ['w'-'z'](< subscript >)*
<term> ::=

Roll your own derivations

6/21/07 10 Software

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 ∼ ∧ ∨ ⊃ ≡ ∃ ∴

Roll your own derivations

6/21/07 10 Software

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)

6/20/07 10Software

Skill to be acquired:

To understand how the various restrictions on the quantificational rules work to exclude certain kinds of invalid inferences

Reading

Bergmann[2004] The Logic Book Section 10.1

The Tutorial

The Rule of Universal Elimination UE

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'

10 Software

Reading

Bergmann[2004] The Logic Book Section 10.1.

The Tutorial

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.