fundamental concept of math

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1A Class-Room Introduction to LogicMay 4, 2009Unit-X: Rules and Fallacies for Categorical SyllogismsFiled under: Logic,Rules for validity of syllogismDr Desh Raj Sirswal @ 8:09 amTags: Logic

Aristotle and other traditional logicians provided certain rules which determine the validly/invalidity of

syllogism. Here are some rules to check the validity of a syllogism.

Rule 1: Avoid Four TermsFallacy:Fallacy of four terms (A formal mistake in which a categorical syllogism contains more than three

terms.)

Example:

All men are rational animal.All chalks are white.

Therefore,.

Justification: A valid standard-from categorical syllogism must contain exactly three terms, each of which is

used in the same sense throughout the argument. If there is more terms than, it cannot be in standard-formsyllogism, we cannot call it syllogism.

Rule 2: The middle term must be distributed at least once.Fallacy: Undistributed middle( A formal mistake in which a categorical syllogism contains a middle term that isnot distribute in either premise.)

Example:

All sharks are fish.All salmon are fish.Therefore, All salmon are sharks

Justification: The middle term is what connects the major and the minor term. If the middle term is never

distributed, then the major and minor terms might be related to different parts of the M class, thus giving no

common ground to relate S and P.

Rule 3: If a term is distributed in the conclusion, then it must be distributed in a premise.

Fallacy:Illicit major(A formal mistake in which the major term of a syllogism is undistributed in the majorpremise, but is distributed in the conclusion.)

Illicit minor (A formal mistake in which the minor term of a syllogism is undistributed in the minor premise, butis distributed in the conclusion.)

Examples:

And: All horses are animals.Some dogs are not horses.Therefore, Some dogs are not

animals.All tigers are mammals.

All mammals are animals.

Therefore, All animals are tigers.
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2Justification: When a term is distributed in the conclusion, lets say that P is distributed, then that term is

saying something about every member of the P class. If that same term is NOT distributed in the major premise,then the major premise is saying something about only some members of the P class. Remember that the minor

premise says nothing about the P class. Therefore, the conclusion contains information that is not contained in

the premises, making the argument invalid.

Rule 4: No conclusion drawn from two negative premises.Fallacy:Exclusive premises (A formal mistake in which both premises of a syllogism are negative)

Example:

No fish are mammals.Some dogs are not fish.Therefore, Some dogs are notmammals.

Justification: If the premises are both negative, then the relationship between S and P is denied. The conclusion

cannot, therefore, say anything in a positive fashion. That information goes beyond what is contained in thepremises.

Rule 5: A negative premise requires a negative conclusion, and a negative conclusion requires a negative

premise. (Alternate rendering: Any syllogism having exactly one negative statement is invalid.)Fallacy:Drawing an affirmative conclusion from a negative premise, or drawing a negative conclusion from anaffirmative premise. (A formal mistake in which one premise of a syllogism is negative but the conclusion is

affirmative.)

Example:

All crows are birds.Some wolves are not crows.Therefore, Some wolves arebirds.

Justification: Two directions, here. Take a positive conclusion from one negative premise. The conclusion

states that the S class is either wholly or partially contained in the P class. The only way that this can happen isif the S class is either partially or fully contained in the M class (remember, the middle term relates the two) and

the M class fully contained in the P class. Negative statements cannot establish this relationship, so a valid

conclusion cannot follow.Take a negative conclusion. It asserts that the S class is separated in whole or in part from the P class. If both

premises are affirmative, no separation can be established, only connections. Thus, a negative conclusion cannot

follow from positive premises.

Note: These first four rules working together indicate that any syllogism with two particular premises is invalid.

Rule 6: If both premises are universal, the conclusion cannot be particular. And also there is noconclusion from two particular premises.Fallacy:Existential fallacy (As a formal fallacy, the mistake of inferring a particular conclusion from twouniversal premises.)

Example:

All mammals are animals.All tigers are mammals.Therefore, Some tigers are

animals.

Justification: On the Boolean model, Universal statements make no claims about existence while particular

ones do. Thus, if the syllogism has universal premises, they necessarily say nothing about existence. Yet if the

conclusion is particular, then it does say something about existence. In which case, the conclusion contains

more information than the premises do, thereby making it invalid.

Transformation rules

Propositional calculus

Predicate logic

Universal generalization

Universal instantiation
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Existential generalization Existential instantiation

3Rule of replacement

From Wikipedia, the free encyclopediaJump to: navigation, search

Transformation rules

Propositional calculus

Rules of inference

Modus ponens Modus tollens

Biconditional introduction Biconditional elimination Conjunction introduction

Simplification Disjunction introduction Disjunction elimination Disjunctive syllogism Hypothetical syllogism Constructive dilemma Destructive dilemma

AbsorptionRules of replacement

Associativity Commutativity Distributivity

Double negation

De Morgan's laws Transposition

Material implication Material equivalence

Exportation Tautology
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Predicate logic

Universal generalization

Universal instantiation Existential generalization Existential instantiation

v t e

4In logic, a rule of replacement[1][2][3]

is a transformation rule that may be applied to only a particular segmentof an expression. A logical system may be constructed so that it uses eitheraxioms, rules of inference, or both

as transformation rules forlogical expressions in the system. Whereas a rule of inference is always applied to a

whole logical expression, a rule of replacement may be applied to only a particular segment. Within the contextof a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in

propositional logic to manipulate propositions.

Common rules of replacement include de Morgan's laws, commutativity, associativity, distribution, double

negation, transposition, material implication, material equivalence, exportation, and tautology.

References

Rules of Replacement

Replacement

We complete our development of the proof procedure for the propositional calculus by making use of another

useful way of validly moving from step to step. Since two logically equivalent statements have the same truth-

value on every possible combination of truth-values for their component parts, no change in the truth-value ofany statement occurs when we replace one of them with the other. Thus, when constructing proofs of validity,

we can safely use a statement containing either one of a pair of logical equivalents as the premise for a step

whose conclusion is exactly the same, except that it contains the other one.

Although this would work for any pair of logically equivalent statement forms, remembering all of them would

be cumbersome. Instead, we will once again rely upon a short list of ten rules of replacement in our constructionof proofs, and we have already examined five of them:

D.N. p ~~p

DeM. ~(p q) (~p ~q)

~(p q) (~p ~q)

Impl. (pq) (~p q)

Equiv. [pq] [(pq) (qp)]

[pq] [(p q) (~p ~q)]
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Trans. (p q) (~q ~p)

We'll add just five more, making a total of ten tautologous biconditionals to be used as rules of replacement.

Commutation

The rule of replacement called Commutation (Comm.) shows that statements of

certain forms can simply be reversed.

In one form, this applies to all disjuctions:

(p q) (q p)

In its second form, Commutation establishes the same logical equivalence with respect to conjunctions:

(p q) (q p)

The truth-tables for these two varieties of commutation show that we can safely replace any disjunction or conjunction

with another in which the component elements of the original have been switched, since the truth values of the

commuted compound statements do not change under any of the possible conditions.

Association

Association (Assoc.) permits modification of the parenthetical grouping of certain statements.

p q (p q)(q p)

T T T T T

T F T T T

F T T T T

F F F T F

p q (p q) (q p)

T T T T T

T F F T F

F T F T F

F F F T F

p q r [p(qr)] [(pq)r]

T T T T T T

T T F T T T

T F T T T T
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Applied to disjunction, it has the form:

[p (qr)] [(pq) r]

This shows that the grouping of a string of disjuncts is irrelevant to the truth-value

of the compound statement form.

Applied to conjunction, it has the form:

[p (qr)] [(pq) r]

Used in tandem, the Commutative and Associative replacement rules make it possible to rearrange any series of

disjunctions or conjunctionsno matter how long and complicatedinto any new order and arrangement we wish to

have.

Distribution

T F F T T T

F T T T T T

F T F T T T

F F T T T T

F F F F T F

p q r [p(qr)] [(pq)r]

T T T T T T

T T F F T F

T F T F T F

T F F F T F

F T T F T F

F T F F T F

F F T F T F

F F F F T F

p q rp(qr) (pq)(pr)

T T T T T T


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The rule called Distribution (Dist.) exhibits the systematic features of

statements in which both disjunctions and conjunctions appear.

In one of its two forms, a conjunct is distributed over a disjunction:

[p (qr)][(pq) (pr)]

In the other form, a disjunct is distributed over a conjunction:

[p

(qr)][(pq) (p

r)]The truth-tables should make it clear that both forms of distribution are reliable rules of replacement.

Exportation

T T F T T T

T F T T T T

T F F F T F

F T T F T F

F T F F T F

F F T F T F

F F F F T F

p q rp(qr)(pq)(pr)

T T T T T T

T T F T T T

T F T T T T

T F F T T T

F T T T T T

F T F F T F

F F T F T F

F F F F T F

p q r (pq)rp(qr)

T T T T T T
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Exportation (Exp.) is a rule of replacement of the form:

[(pq)r)][p(qr)]

The truth-table at the right demonstrates that statements of these two forms are

logically equivalent.

Please take careful notice of the difference between Exportation as a rule of

replacement and the rule of inference called Absorption. Although they bear some

similarity of structure, the rules are distinct and can be used differently in the

construction of proofs.

Tautology

Finally, there are two forms of the rule called Tautology (Taut.):

the first involves disjunction,

p (p p)

and the second involves conjunction:

p (p p)

In each case, the rule permits replacement of any statement by (or with) another statement that is simply the

disjunction or conjunction of the original statement with itself. Although such reasoning is rare in ordinary life, it will

perform a significant formal role in our construction of proofs of validity.

Replacement in Proofs

Using the rules of replacement in the construction of proofs is a fairly straightforward procedure. Since the rulesare biconditionals, the replacement can work in either directionright side for left, or left side for right. What is

more, since the statement forms on either side are logically equivalent, they can be used to replace each other

wherever they occur, even as component parts of a line. (When applying the nine rules of inference, on the otherhand, we must always work with whole lines of a proof.) Consider the following argument:

T T F F T F

T F T T T T

T F F T T T

F T T T T T

F T F T T T

F F T T T T

F F F T T T

p(p

p)

T T T

F T F

p (p p)

T T T

F T F
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A (B ~C)

A D

~D C____________

D

As before, we begin by numbering each of the premises:

1. A (B ~C) premise

2. A D premise

3. ~D C premise

Next, notice that we can use our rules of replacement and inference to derive some part of the information conveyed by

the first premise:

1. A (B ~C) premise

2. A D premise

3. ~D C premise

4. (A B) (A ~C) 1 Dist.

5. (A ~C) (A B) 4 Comm.

6. A ~C 5 Simp.So long as each step is justified by reference to an earlier step (or steps) in the proof and to one of the nineteen rules, it

must be a valid derivation. Next, let's work with the third premise a bit:

1. A (B ~C) premise

2. A D premise

3. ~D C premise

4. (A B) (A ~C) 1 Dist.

5. (A ~C) (A B) 4 Comm.

6. A ~C 5 Simp.

7. ~C ~~D 3 Trans.

8. ~C D 7 D.N.

Again, each step is justified by application of one of the rules of replacement to all or part of a preceding line in theproof. Now conjoin the second premise with our eighth line, and we've set up a constructive dilemma:

1. A (B ~C) premise

2. A D premise

3. ~D C premise

4. (A B) (A ~C) 1 Dist.

5. (A ~C) (A B) 4 Comm.

6. A ~C 5 Simp.

7. ~C ~~D 3 Trans.

8. ~C D 7 D.N.

9. (A D) (~C D) 2, 8 Conj.

10. D D 9, 6 C.D.All that remains is to apply Tautology in order to reach our intended conclusion, so the entire proof will look like this:

1. A (B ~C) premise

2. A D premise

3. ~D C premise

4. (A B) (A ~C) 1 Dist.

5. (A ~C) (A B) 4 Comm.

6. A ~C 5 Simp.

7. ~C ~~D 3 Trans.

8. ~C D 7 D.N.

9. (A D) (~C D) 2, 8 Conj.


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5 10. D D 9, 6 C.D.11. D 10 Taut.

Truth of Statements, Validity of ReasoningPeter Suber, Philosophy Department, Earlham College

True Premises, False Conclusion

0. Valid Impossible: no valid argument can have true premises and a false conclusion.

1. InvalidCats are mammals.Dogs are mammals.Therefore, dogs are cats.

True Premises, True Conclusion

2. Valid

Cats are mammals.

Tigers are cats.

Therefore, tigers are mammals.

3. Invalid

Cats are mammals.

Tigers are mammals.

Therefore, tigers are cats.

False Premises, False Conclusion

4. ValidDogs are cats.Cats are birds.

Therefore, dogs are birds.

5. Invalid

Cats are birds.

Dogs are birds.Therefore, dogs are cats.

False Premises, True Conclusion

6. ValidCats are birds.Birds are mammals.

Therefore, cats are mammals.

7. Invalid

Cats are birds.

Tigers are birds.Therefore, tigers are cats.

The distinction between truth and validity is the fundamental distinction of formal logic. You cannot understand

how logicians see things until this distinction is clear and familiar.

The seven sample arguments above help us establish the following general principles of logic:

True premises do not guarantee validity.(Proved by cases #1 and #3 in the table above.) A true conclusion does not guarantee validity.

(Proved by cases #3 and #7.)

True premises and a true conclusion together do not guarantee validity. (Proved by case #3.)

Valid reasoning does not guarantee a true conclusion.(Proved by case #4.)

False premises do not guarantee invalidity.(Proved by cases #4 and #6.)

A false conclusion does not guarantee invalidity.
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(Proved by case #4.)

6 False premises and a false conclusion together do not guarantee invalidity.(Proved by case #4.)

Invalid reasoning does not guarantee a false conclusion.(Proved by cases #3 and #5.)

ValidityFrom Wikipedia, the free encyclopedia

Jump to: navigation, searchFor other uses, see Validity (disambiguation).

In logic, an argument is valid if and only if its conclusion is logically entailedby its premises and each step in

the argument is logical. A formula is valid if and only ifit is true under every interpretation, and an argumentform (or schema) is valid if and only if every argument of that logical form is valid.

Contents 1 Validity of arguments 2 Valid formula 3 Validity of statements 4 Validity and soundness

5 Satisfiability and validity 6 Preservation 7 n-Validity

o 7.1 -Validity 8 See also 9 References 10 External links

Validity of argumentsAn argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-

argument, or logical operation in the argument is valid. Under such conditions it would be self-contradictory toaffirm the premises and deny the conclusion. The corresponding conditional of a valid argument is a logical

truth and the negation of its corresponding conditional is a contradiction. The conclusion is a logicalconsequence of its premises.An argument that is not valid is said to be "invalid".

An example of a valid argument is given by the following well-known syllogism (also known as modus

ponens):

All men are mortal.Socrates is a man.

Therefore, Socrates is mortal.

What makes this a valid argument is not that it has true premises and a true conclusion, but the logical necessity

of the conclusion, given the two premises. The argument would be just as valid were the premises andconclusion false. The following argument is of the same logical formbut with false premises and a false

conclusion, and it is equally valid:All cups are green.Socrates is a cup.

Therefore, Socrates is green.

No matter how the universe might be constructed, it could never be the case that these arguments should turnout to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with

the following invalid one:

All men are mortal.Socrates is mortal.

Therefore, Socrates is a man.
http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case4http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case4http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case3http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case5https://en.wikipedia.org/wiki/Validity#mw-navigationhttps://en.wikipedia.org/wiki/Validity#p-searchhttps://en.wikipedia.org/wiki/Validity_%28disambiguation%29https://en.wikipedia.org/wiki/Logichttps://en.wikipedia.org/wiki/Argumenthttps://en.wikipedia.org/wiki/Entailmenthttps://en.wikipedia.org/wiki/Well-formed_formulahttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Interpretation_%28logic%29https://en.wikipedia.org/wiki/Logical_formhttps://en.wikipedia.org/wiki/Validity#Validity_of_argumentshttps://en.wikipedia.org/wiki/Validity#Valid_formulahttps://en.wikipedia.org/wiki/Validity#Validity_of_statementshttps://en.wikipedia.org/wiki/Validity#Validity_and_soundnesshttps://en.wikipedia.org/wiki/Validity#Satisfiability_and_validityhttps://en.wikipedia.org/wiki/Validity#Preservationhttps://en.wikipedia.org/wiki/Validity#n-Validityhttps://en.wikipedia.org/wiki/Validity#.CF.89-Validityhttps://en.wikipedia.org/wiki/Validity#.CF.89-Validityhttps://en.wikipedia.org/wiki/Validity#See_alsohttps://en.wikipedia.org/wiki/Validity#Referenceshttps://en.wikipedia.org/wiki/Validity#External_linkshttps://en.wikipedia.org/wiki/Argument_%28logic%29https://en.wikipedia.org/wiki/Entailmenthttps://en.wikipedia.org/wiki/Corresponding_conditionalhttps://en.wikipedia.org/wiki/Logical_truthhttps://en.wikipedia.org/wiki/Logical_truthhttps://en.wikipedia.org/wiki/Contradictionhttps://en.wikipedia.org/wiki/Logical_consequencehttps://en.wikipedia.org/wiki/Logical_consequencehttps://en.wikipedia.org/wiki/Syllogismhttps://en.wikipedia.org/wiki/Modus_ponenshttps://en.wikipedia.org/wiki/Modus_ponenshttps://en.wikipedia.org/wiki/Logical_formhttps://en.wikipedia.org/wiki/Logical_formhttps://en.wikipedia.org/wiki/Modus_ponenshttps://en.wikipedia.org/wiki/Modus_ponenshttps://en.wikipedia.org/wiki/Syllogismhttps://en.wikipedia.org/wiki/Logical_consequencehttps://en.wikipedia.org/wiki/Logical_consequencehttps://en.wikipedia.org/wiki/Contradictionhttps://en.wikipedia.org/wiki/Logical_truthhttps://en.wikipedia.org/wiki/Logical_truthhttps://en.wikipedia.org/wiki/Corresponding_conditionalhttps://en.wikipedia.org/wiki/Entailmenthttps://en.wikipedia.org/wiki/Argument_%28logic%29https://en.wikipedia.org/wiki/Validity#External_linkshttps://en.wikipedia.org/wiki/Validity#Referenceshttps://en.wikipedia.org/wiki/Validity#See_alsohttps://en.wikipedia.org/wiki/Validity#.CF.89-Validityhttps://en.wikipedia.org/wiki/Validity#n-Validityhttps://en.wikipedia.org/wiki/Validity#Preservationhttps://en.wikipedia.org/wiki/Validity#Satisfiability_and_validityhttps://en.wikipedia.org/wiki/Validity#Validity_and_soundnesshttps://en.wikipedia.org/wiki/Validity#Validity_of_statementshttps://en.wikipedia.org/wiki/Validity#Valid_formulahttps://en.wikipedia.org/wiki/Validity#Validity_of_argumentshttps://en.wikipedia.org/wiki/Logical_formhttps://en.wikipedia.org/wiki/Interpretation_%28logic%29https://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Well-formed_formulahttps://en.wikipedia.org/wiki/Entailmenthttps://en.wikipedia.org/wiki/Argumenthttps://en.wikipedia.org/wiki/Logichttps://en.wikipedia.org/wiki/Validity_%28disambiguation%29https://en.wikipedia.org/wiki/Validity#p-searchhttps://en.wikipedia.org/wiki/Validity#mw-navigationhttp://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case5http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case3http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case4http://legacy.earlham.edu/~peters/courses/log/tru-val.htm#case4


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7 In this case, the conclusion does not follow inescapably from the premises. All men are mortal, but not all

mortals are men. Every living creature is mortal; therefore, even though both premises are true and theconclusion happens to be true in this instance, the argument is invalid because it depends on an incorrect

operation of implication. Such fallacious arguments have much in common with what are known as howlers in

mathematics.

A standard view is that whether an argument is valid is a matter of the argument's logical form. Manytechniques are employed by logicians to represent an argument's logical form. A simple example, applied to two

of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men,

the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:All P are Q.

S is a P.

Therefore, S is a Q.Similarly, the third argument becomes:

All P are Q.

S is a Q.

Therefore, S is a P.An argument is formally valid if its form is one such that for each interpretation under which the premises are

all true, the conclusion is also true. As already seen, the interpretation given above (for the third argument) does

cause the second argument form to have true premises and false conclusion (if P is a not human creature), hence

demonstrating its invalidity.

Valid formulaMain article: Well-formed formula

A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of

the language. In propositional logic, they are tautologies.

Validity of statementsA statement can be called valid, i.e. logical truth, if it is true in all interpretations.

Validity and soundnessValidity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following

deduction is perfectly valid:

All animals live on Mars.All humans are animals.

Therefore, all humans live on Mars.

The problem with the argument is that it is notsound. In order for a deductive argument to be sound, the

deduction must be valid and all the premises true.

Satisfiability and validityMain article: Satisfiability and validity

Model theory analyzes formulae with respect to particular classes of interpretation in suitable mathematical

structures. On this reading, formula is valid if all such interpretations make it true. An inference is valid if all

interpretations that validate the premises validate the conclusion. This is known assemantic validity.[1]

PreservationIn truth-preserving validity, the interpretation under which all variables are assigned a truth value of 'true'

produces a truth value of 'true'.

In a false-preserving validity, the interpretation under which all variables are assigned a truth value of 'false'produces a truth value of 'false'.

[2]

Preservation

propertiesLogical connective sentences

True and false

preserving:Logical conjunction (AND, ) Logical disjunction (OR, )
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8 True preserving

only:

Tautology ( ) Biconditional (XNOR, ) Implication ( ) Converseimplication ( )

False preserving only:Contradiction ( ) Exclusive disjunction (XOR, ) Nonimplication ( )

Converse nonimplication ( )

Non-preserving:Proposition Negation ( ) Alternative denial (NAND, ) Joint denial

(NOR, )

n-ValidityA formula A of a first order language is n-valid iffit is true for every interpretation of that has a domain ofexactly n members.

-ValidityA formula of a first order language is-valid iffit is true for every interpretation of the language and it has a

domain with an infinite number of members.
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fundamental concept of math

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