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Propositional Logic Arguments (5A)
Young W. Lim10/11/16

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Propositional Logic (5A)Arguments 3 Young Won Lim10/11/16
Based on
Contemporary Artificial Intelligence, R.E. Neapolitan & X. Jiang
Logic and Its Applications,Burkey & Foxley

Arguments
An argument consists of a set of propositions :The premises propositionsThe conclusion proposition
List of premises followed by the conclusion
A1
A2
A
n
-------- B

Entail
The premises is said to entail the conclusion If in every model in which all the premises are true,
the conclusion is also true
List of premises followed by the conclusion
A1
A2
A
n
-------- B
wheneverall the premises are true
the conclusion must be truefor the entailment

A Model
A mode or possible world:
Every atomic proposition is assigned a value T or F
The set of all these assignments constitutes A model or a possible world
All possible worlds (assignments) are permissiable

Entailment Notation
Suppose we have an argumentwhose premises are A
1, A
2, , A
n
whose conclusion is B
Then
A1, A
2, , A
n B if and only if
A1 A
2 A
n B (logical implication)
logical implication: if A1 A
2 A
n B is tautology
The premises is said to entail the conclusion If in every model in which
all the premises are true, the conclusion is also true
(always true)

Entailment and Logical Implication
A
1, A
2, , A
n B
A1 A
2 A
n B
A1 A
2 A
n B is a tautology
(logical implication)
If all the premises are true, then the conclusion must be true
T T T TT T T FF X X X

Sound Argument and Fallacy
A sound argument
A1, A
2, , A
n B
A fallacy
A1, A
2, , A
n B
If the premises entails the conclusion
If the premises does not entail the conclusion

Entailment Examples
A, (A B) B
A, (A B) B
A (A B) B
A, B A
A, B A
A B A

Entailment Examples and Truth Tables
A B AB AB AT T T TT F F TF T F TF F F T
AB A
A B AB A(AB) A(AB)BT T T T TT F F F TF T T F TF F T F T
A(AB) B
any of the premises are false, still premises conclusion is true (FT and FF always T)
Tautology

Deduction System
Propositional logic
Given propositions (statements) : T or FDeductive inference of T or F of other propositions
Deductive Inference A process by which the truth of the conclusion is shown to necessarily follow from the truth of the premises

Deduction System
Deduction System : a set of inference rules
Inference rules are used to reason deductively
Sound Deduction System : if it derives only sound arguments
Each of the inference rules is sound
Complete Deduction System : It can drive every sound argument
Must contain deduction theorem rule

Inference Rules
Combination Rule A, B A B Simplification Rule A B AAddition Rule A A BModus Pones A, A B B Modus Tolens B, A B A Hypothetical Syllogism A B, B C A CDisjunctive Syllogism A B, A BRule of Cases A B, A B BEquivalence Elimination A B A B Equivalence Introduction A B, B A A BInconsistency Rule A, A B AND Commutivity Rule A B B AOR Commutivity Rule A B B ADeduction Theorem If A
1, A
2, , A
n,B C then A
1, A
2, , A
n, BC

Deduction Theorem
A1, A
2, , A
n, B C
A1, A
2, , A
n, BC
A1A
2A
nB C
A1A
2A
n BC
A,B C A BC
B C BC
A1, A
2, , A
n B if and only if
A1 A
2 A
n B
(A1 A
2 A
n B is a tautology)
If A is T, then BC is always T (for the tautology)Even if A,B is T, then BC is always TAnd if A,B is T, then B is TBy modus ponens in the RHS, A,B is T, then C is true

Deduction Theorem
,A B iff ABwhere : a set of formulas,
if the formula B is deducible from a set of assumptions, together with the assumption A, then the formula AB is deducible from alone.
Conversely, if we can deduce AB from , and if in addition we assume A, then B can be deduced.
http://planetmath.org/deductiontheorem
,A B
AB
AB
,A B

Deduction Theorem
The deduction theorem conforms with our intuitive understanding of how mathematical proofs work:
if we want to prove the statement A implies B, then by assuming A, if we can prove B, we have established A implies B.

Deduction Theorem
The converse statement of the deduction theorem turns out to be a trivial consequence of modus ponens:
if AB, then certainly ,A ABSince ,A A, we get, via modus ponens, ,A B as a result.
,A AB,A A,A B
A, A B B

Deduction Theorem
Deduction theorem is needed to derive arguments that has no premises
An argument without premises is simply a tautology
A Ano premises appear before the symbolan argument without premisesTautology if it is sound

Argument Example
1. Q2. P Q3. P R S4. R T5. U 6. U T7. P Q, P Q P8. R S P R S, P R S9. T U, U T T10. R T, R T R11. S R S, R S

Argument without premises
A assume AA A X X YA A A A A A A A A
discharge AA assume AA A X X YA A X Y Y XA A A A A A A A A
discharge AA A X Y, X Y Y
A A A , A A A A A A A

Argument without premises
A A assume AA A A A AA A A A A A
discharge AA A assume AA A A A AA A A A AA A A A A A
discharge AA A A A

To prove a sound argument
Prove using truth tablesWhether an argument is sound or fallacy
1. time complexity (2^n)2. not the way which humans do
Prove using inference rules To reason deductively

Double Turnstile
1. semantic consequence:a set of sentences on the lefta single sentence on the rightto denote that if every sentence on the left is true, the sentence on the right must be true, e.g. . This usage is closely related to the single-barred turnstile symbol which denotes syntactic consequence.
2. satisfaction: a model (or truth-structure) on the left a set of sentences on the rightto denote that the structure is a model for (or satisfies) the set of sentences, e.g. A .
3. a tautology: to say that the expression is a semantic consequence of the empty set.
https://en.wikipedia.org/wiki/Double_turnstile

Syntactic Consequences
A formula A is a syntactic consequence within some formal system FS of a set of formulas if there is a formal proof in FS of A from the set .
FS A Syntactic consequence does not depend on any interpretation of the formal system.
A formal proof or derivation is a finite sequence of sentences (called wwf),
each of which is an axiom, an assumption, or follows from the preceding sentences in th

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