Propositional Logic– Arguments (5A) Logic (5A) Arguments 4 Young Won Lim 11/29/16 Arguments An...

Propositional Logic Arguments (5A)

Young W. Lim11/29/16

Copyright (c) 2016 Young W. Lim.

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Propositional Logic (5A)Arguments 3 Young Won Lim11/29/16

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

propositionspremises

conclusion


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 model or a 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

A B AB AB AT T T TT F F TF T F TF F F T

T T

T F

F T

F F

T TT F

T TT FF T

T TT FF TF F

Every atomic proposition : A, B

models


Interpretation

An interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system.

The study of interpretations is called formal semantics

Giving an interpretation is synonymous with constructing a model.

An interpretation is expressed in a metalanguage, which may itself be a formal language, and as such itself is a syntactic entity.

https://en.wikipedia.org/wiki/Syntax_(logic)#Syntactic_consequence_within_a_formal_system


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 T


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


Modal Accounts

Modal accounts of logical consequence are variations on the following basic idea:

A is true if and only ifit is necessary that if all of the elements of are true, then A is true.

Alternatively :

A is true if and only if it is impossible for all of the elements of to be true and A false.

Such accounts are called "modal" because they appeal to the modal notions of logical necessity and logical possibility. 'It is necessary that' is often expressed as a universal quantifier over possible worlds, so that the accounts above translate as:

A is true if and only if there is no possible world at which all of the elements of are true and A is false (untrue).

https://en.wikipedia.org/wiki/Syntax_(logic)#Syntactic_consequence_within_a_formal_system


Validity and Soundness (1)

An argument form is valid if and only if

whenever the premises are all true, then conclusion is true.

An argument is valid if its argument form is valid.

An argument is sound if and only if

it is valid and all its premises are true.

http://math.stackexchange.com/questions/281208/what-is-the-difference-between-a-sound-argument-and-a-valid-argument

premises : true conclusion : trueIf then

premises : true conclusion : falseAlways therefore

false truefalse false

true falseIf then never



A deductive argument is said to be valid if and only if

it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.

Otherwise, a deductive argument is said to be invalid.for the premises to be true and the conclusion is false.

A deductive argument is sound if and only if

it is both valid, and all of its premises are actually true.

Otherwise, a deductive argument is unsound.

http://www.iep.utm.edu/val-snd/

true falseIf then never

premises : true conclusion : falseAlways therefore




A B AB A(AB) A(AB)BT T T T TT F F F TF T T F TF F T F T

sound

valid


If premises : true then never conclusion : false

Always premises : true therefore conclusion : true



the author of a deductive argument always intends that the premises provide the sort of justification for the conclusion whereby if the premises are true, the conclusion is guaranteed to be true as well.

if the author's process of reasoning is a good one, if the premises actually do provide this sort of justification for the conclusion, then the argument is valid.

an argument is valid if the truth of the premises logically guarantees the truth of the conclusion.

it is impossible for the premises to be true and the conclusion to nevertheless be false:



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(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


A(AB) B

Deductive Inference

Entailment(logical implication)


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

A sound argument: If the premises entails the conclusion

A fallacy:If the premises does not entail the conclusion


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 A,A AB,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 P (1, 2)8. R S P R S, P R S (3, 7)9. T U, U T T (5, 6)10. R T, R T R (9, 4)11. S R S, R S (8, 10)


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


Logical Consequences

A S B

there is a derivation, in the proof-system S, from the premise A to the conclusion B. [If context fixes the relevant system S, we suppress the subscript.]

A L B

on every possible interpretation of the non-logical vocabulary of language L, if A comes out true, so does B. [If context fixes the relevant language L we suppress the subscript.]

A B

on the truth-functional interpretation, if the atomic wff A happens to be false and the atomic wff B happens to the false too, then AB evaluates as true. But we don't have A B (q isn't true on every valuation which makes p true).

http://math.stackexchange.com/questions/365569/whats-the-difference-between-syntactic-consequence-%E2%8A%A2-and-semantic-consequence-%E2%8A%A8

A B A BTrue True True True False FalseFalse True True False False True

Syntactic Consequences

Semantic Consequences

Material Implication

Logical Implication


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 which follows from the preceding sentences in the sequence by a rule of inference.

The last sentence in the sequence is a theorem of a formal system.


Sound argument Fallacy


Semantic Consequences

A formula A is a semantic consequence within some formal system FS of a set of statements

FS A

if and only if there is no model I in which all members of are true and A is false. the set of the interpretations that make all members of true is a subset of the set of the interpretations that make A true.



Summary

Syntactic consequence sentence is provable from the set of assumptions .

Semantic consequence sentence is true in all models of .

Soundness If [ ] then [ ].

Completeness If [ ] then [ ].

http://philosophy.stackexchange.com/questions/10785/semantic-vs-syntactic-consequence

The propositional logic has a proof system

(propositional calculus)Syntactic consequences

a semantics (truth-tables)Semantic consequences


Syntactic and Semantic Consequences (1)

Syntactic consequence sentence is provable from the set of assumptions .

Semantic consequence sentence is true in all models of .

http://philosophy.stackexchange.com/questions/10785/semantic-vs-syntactic-consequence


A

A B B


Syntactic and Semantic Consequences (2)

A, AB B Syntactic consequence if we take the assumptions A and AB as given, by modus ponens we can deduce B.

A, AB B Semantic consequence in any model for which it is the case that A is true and also AB is true, then, in that model, B is also true.

talks about the propostions themselves as syntactic objects, talks about what the propositions mean i.e. semantics.

https://www.quora.com/What-are-the-differences-between-semantic-consequence-and-syntactic-consequence-in-logic


Soundness and Completeness (1)

Sound Deduction System : if it derives only sound arguments

each of the inference rules is sound

Soundness. If [ ] then [ ].

Complete Deduction System : It can drive every sound argument

must contain deduction theorem rule

Completeness. If [ ] then [ ].

A sound argument: If the premises entails the conclusion

A fallacy:If the premises does not entail the conclusion


Soundness and Completeness (2)

Soundness is the property of only being able to prove "true" things.

Completeness is the property of being able to prove all true things.

So a given logical system is sound if and only if the inference rules of the system admit only valid formulas. Or another way, if we start with valid premises, the inference rules do not allow an invalid conclusion to be drawn.

A system is complete if and only if

all valid formula can be derived from the axioms and the inference rules. So there are no valid formula that we can't prove.

http://philosophy.stackexchange.com/questions/6992/the-difference-between-soundness-and-completeness


Invalid Argument Examples

P1 Grass is greenP2 Paris is the capital of FranceC A poodle is a dog

P1, P2 and C all true, but argument not deductively valid.

If you object that this doesnt count as an argument because there is no connection between the Ps or between the Ps and the C, try

P1 All atoms are tinyP2 The smallest particle of hydrogen gas is tinyC The smallest particle of hydrogen gas is an atom

All true, but not deductively valid. To see this, substitute oxygen for hydrogen (the smallest part of oxygen gas is a molecule not an atom, so C false) or pollen for hydrogen gas (the smallest particle of pollen is a grain, C false)

https://askaphilosopher.wordpress.com/2012/02/08/invalid-argument-with-true-premisses-and-true-conclusion/


Unsound Argument Examples

P1 Craig is a ScotP2 All Scots are drunksC Craig is a drunk

Here, P1 can be true, C follows from the Ps (validity), C can be true, but the argument is unsound because P2 is false. So, although the C is true we cant rely on the argument to establish it. It is an unsound argument.

P1, P2 C T, T |= T T, F |= T

https://askaphilosopher.wordpress.com/2012/02/08/invalid-argument-with-true-premisses-and-true-conclusion/



Either Elizabeth owns a Honda or she owns a Saturn. (True / False) Elizabeth does not own a Honda. Therefore, Elizabeth owns a Saturn.A valid argumenteven if one of the premises is actually false, that if they had been true the conclusion would have been true as well

All toasters are items made of gold. (False) All items made of gold are time-travel devices. Therefore, all toasters are time-travel devices.A valid and unsound argumenteven if one of the premises is actually false, that if they had been true the conclusion would have been true as well

No felons are eligible voters.(True) Some professional athletes are felons. (True) Therefore, some professional athletes are not eligible voters. (True)A valid and sound argument



Valid Argument Examples

All tigers are mammals. No mammals are creatures with scales. Therefore, no tigers are creatures with scales.A valid and sound argument [P1,P2 C], [P1, P2 C]

All spider monkeys are elephants. (False) No elephants are animals. (False) Therefore, no spider monkeys are animals. (False)A valid but unsound argument [P1,P2 C], [P1, P2 C]

These arguments share the same form: A valid arguments, Syntactic Consequences All A are B; No B are C; Therefore, No A are C.




All basketballs are round. The Earth is round. Therefore, the Earth is a basketball.An invalid and unsound argument [P1,P2 C], [P1, P2 C]

All popes reside at the Vatican. (True) John Paul II resides at the Vatican. (True) Therefore, John Paul II is a pope. (True)An invalid and unsound argument [P1,P2 C], [P1, P2 C]

These arguments also have the same form: an invalid arguments All A's are F; X is F; Therefore, X is an A.



Syntactic and Material Consequences

https://en.wikipedia.org/wiki/Logical_consequence


Logical Equivalences

, ,

, ,


References

[1] en.wikipedia.org[2] en.wiktionary.org[3] U. Endriss, Lecture Notes : Introduction to Prolog Programming[4] http://www.learnprolognow.org/ Learn Prolog Now![5] http://www.csupomona.edu/~jrfisher/www/prolog_tutorial[6] www.cse.unsw.edu.au/~billw/cs9414/notes/prolog/intro.html[7] www.cse.unsw.edu.au/~billw/dictionaries/prolog/negation.html[8] http://ilppp.cs.lth.se/, P. Nugues,` An Intro to Lang Processing with Perl and Prolog

http://www.learnprolognow.org/http://www.csupomona.edu/~jrfisher/http://www.cse.unsw.edu.au/~billw/cs9414/notes/prolog/intro.htmlhttp://www.cse.unsw.edu.au/~billw/dictionaries/prolog/negation.htmlhttp://ilppp.cs.lth.se/

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