Logic Unit 3 Part 1:
Derivations with Negation and Conditional
2011
Niko Scharer
1
UNIT 3:
DERIVATIONS FOR SENTENTIAL LOGIC
NATURAL DEDUCTION
Part 1
3.1
What is a derivation?
A derivation is a proof or demonstration that shows how a sentence or sentences can be derived
(obtained by making valid inferences) from a set of sentences.
A derivation can be used to
demonstrate that an argument is valid; that a sentence is a tautology or that a set of sentences is
inconsistent.
In a derivation, you are proving that the conclusion logically follows from the premises.
We‟ll be using a natural
deduction system for firstorder logic that uses the symbolic language that we
have learned.
Our system is based on that presented by Kalish and Montague in their text,
Techniques
of Formal Reasoning
.
1
All of our derivation rules are truthpreserving, so that if we follow the rules
and the premises are true, we can only derive true conclusions.
Every sentence that is logically entailed by a set of sentences can be derived from that set of sentences
using our derivation system
–
the system is complete. Every one of the infinite number of valid
theorems and valid arguments (within the scope of firstorder sentential logic) can be proven.
Every argument arrived at through our sentential derivation system will be deductively valid
–
our
system is consistent.
Thus, true premises will always lead to a true conclusion.
1
Kalish, Donald, and Montague, Richard, 1964.
Logic: Techniques of Formal Reasoning
. Harcourt, Brace, and
Jovanovich.
Our system is based on Kalish
and Montague‟s natural deduction syst
em for firstorder logic, an elegant logical
system that treats negation and material conditional as primary.
It uses three types of derivation which we will
be learning in this unit:
direct derivation, indirect derivation and conditional derivation. Except for a single rule
of inference (
modus ponens)
no other logical symbol or rule of inference is necessary to express any sentence
(for any possible truthvalue assignment) or to complete a derivation in sentential logic.
However, the other
logical operators and rules of inference will make things a little easier and more intuitive!
2
This illustration is the property of Gerald Grow, Professor of Journalism, Florida A&M University.
http://www.longleaf.net/ggrow/CartoonPhil.html
Our natural deduction system should
be a little less painful than this!
©1996 Gerald Grow
2
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Logic Unit 3 Part 1:
Derivations with Negation and Conditional
2011
Niko Scharer
2
Three Types of Derivation for Sentential Logic:
Direct Derivation
Using the premises, you derive the sentence that you want to prove through the
application of the derivation rules.
Conditional Derivation
You can derive a conditional sentence by assuming the antecedent and deriving the
consequent from it using the derivation rules.
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 Winter '11
 Scharer
 Logic, Niko Scharer

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