The Logic Of Quantified Statements

predicate calculus

statement calculus(propositional calculus)

2.1 Introduction to Predicates and Quantified Statements I

Formal Versus Informal Language

It is important to be able to translate from formal to informal language when trying to make sense of mathematical concepts that are new to you. It is equally important to be able to translate from informal to formal language when thinking out a complicated problem.

Universal Conditional Statements

Implicit Quantification

2.2 Introduction to Predicates and Quantified Statements II

Negations of Quantified Statements

The negation of a universal statement("all are") is logically

equivalent to an existential statement("some are not")

Note that when we speak of logical equivalence for quantified statements, we mean that the statements

always have identical truth values no matter what predicates are substituted for the predicate variables

and no matter what sets are used for the domains of the predicate variables.

Another way to avoid error when taking negations of statements that are given in informal language is to ask yourself,

"What exactly would it mean for the given statement to be false?" What statement, if true, would be equivalent to saying that the given statement is false?"

Negation of Universal Conditional Statements

Vacuous Truth of universal Statements

Variants of Universal Conditional Statements

Necessary and Sufficient Conditions, Only If

2.3 Statements Containing Multiple Quantifiers

Exp:

There is a person supervising every detail of the production process.

Interpretation:

1) There is none single person who supervises all the details of the production process.

2) For any particular production detail, there is a person who supervises that detail, but there might be different supervisors for different details.

Considerer a statement of the form.

Note: because you do not have to specify the y until after the other person has specified the x, you are allowed to find a different value of y for each different x you are given.

Note that your x has to work for any y the person gives you: you are not allowed to change your x once you have specified it initially.

Ambiguous Language

 Once you interpreted the sentence at the beginning of this section in one way, it may have been hard for your to see that it could be understood in the other way. Perhaps you had difficulty even though the two possible meanings were explained, just as many people have difficulty seeing the second interpretation for the drawing even when they are told what to look for.

Although statements written informally may be open to multiple interpretations ,we cannot determine their truth or falsity without interpreting them one way or another.

There for, we have to use context to try to ascertain their meaning as best we can.

Negations of Multiply-Quantified statements

Order of Quantifiers

If one quantifier immediately follows another quantifier of the same type, then the order of the quantifiers does not affect the meaning.

Formal Logical Notation

The symbols for quantifiers, variables, predicates, and logical connectives make up what is known as the language of rest-order logic.

Prolog

2.4 Arguments with Quantified Statements

Universal Instantiation.

If some property is true of everything in a domain, then

it is true of any particular thing in the domain.

Universal Modus Ponens

Use of Universal Modus Ponens in a Proof

Mainly used in chapter 3 for logic deduction of proof.

Universal Modus Tollens

Proving Validity of Arguments with Quantified Statements

Give an example that the universal modus ponens is valid:

Using Diagrams to Test for validity

Creating Additional Forms of Argument

Remark on the Converse and Inverse Errors

1 All writers who understand human nature are clever.

 (for all writers if x understand human nature, then x is clever)

2 No one is a true poet unless he can stir the human heart.

(for All people x, x is not a true poet if x can't stir the human heart.)

(for all people x, if x can't stir the human heart, then x is not a true poet.)

(for all people x, if x is a true poet, then x can stir the human heart)

3. Shakespeare wrote Hamlet.

(for all people x, if x is Shakespeare then x wrote hamlet.)

4 No writer who does not understand human nature can stir the human heart.

(for all writer x, if x does not understand human nature then x can not stir the human heart)

(contrapostive:  for all writer x, if x can stir the human heart, then x understands human nature)

5 None but a true poet could have written hamlet.

(for all people x, if x it not a true poet then x haven't written hamlet.)

(for all people, if x have written hamlet, then x is a true poet)

1) if x is Shakespeare then x wrote hamlet.  by 3)

2) if x have written hamlet, then x is a true poet  by 5)

3) if x is a true poet, then x can stir the human heart by 2)

4) if x can stir the human heart, then x understands human nature. by 4)

5) if x understand human nature, then x is clever by 1)

there for. Shakespeare is clever.

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1. No bird except ostriches are at least 9 feat tall.

(for all birds x, if x is not ostriches, then x is less then 9 feat tall.)

(for all birds x, if x is at least 9 feat tall, then x is ostriches.)

2  There are no birds in this aviary that belong to anyone but me.

(for all birds x, if x is in this aviary then x is not belong to anyone (but me)

(for all birds x ,if x is in this aviary then x is belong to me)

3. No ostrich lives on mince pies.

(for all birds x if x is ostrich then x doesn't live on mince pies)

4. I have no birds less than 9 feet high.

(for all birds x if x is mine, then x is at least 9 feet high)

Reorder as:

1) If a bird is in this aviary then the bird is belong to me.  by 2)

2) if a bird is belong to me(it is mine) then it is at least 9 feet high. by 4)

3) if a bird is at least 9 feet high, then it is a ostriches.  by 1)

4) if a bird is ostriches then it doesn't live on mince pies.

there for. if a bird is in this aviary then it doesn't live on mince pies.

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Promise breakers are untrustworhy.
Wine drinkers are very communicative.
A man who keeps his promises is honest.
No teetotalers are pawnbrokers.
One can always trust a very communicative person.
conclusion:

1) Promise breakers are untrustworhy.

(for all people x, if x is a promise breaker, then x is untrustworthy)

(if x is trustworthy, then x is not a promise breaker)

2) Wine drinker are very communicative.

(for all people x, if x is a wine drinkers, then x is very communicative)

3) A man who keeps this promises is honest.

(for all people x, if x is a man who keeps this promises, then x is honest)

4) No teetotalers are pawnbrokers.

(for all people x, if x is a teetotaler, then x is not a pawnbroker)

(if x is not a wine drinker then x is not a pawnbroker).

5) One can always trust a very communicative person.

(for all people x, if  x is a very communicative person then we can trust x or x is trustworthy)

1) if x is a pawnbroker then x is a wine drinker. by 4)

2) if x is a wine drinker, then x is very communicative by 2)

3) if x is very communicative, then x is trustworthy by 5)

4) if x is trustworthy, then x is not a promise breaker.

5) if x is a not a promise breaker(keep promises), then x is honest.

therefore, the conclusion is that pawnbroker is honest.

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No boys under 12 are admitted to this school as boarders.
All the industrious boys have red hair.
None of the dayboys learn Greek.
None but those under 12 are idle.
conclusion:

1) No boy under 12 are admitted to this school as borders.

(for all boys x if x is under 12 then x is not admitted to this school as borders.)

2) All the industrious boys have red hair

(for all boys x if x is industrious then x has red hair.)

3) None of the dayboys learn greek.

(for all boys x if x is a dayboys then x doesn't learn greek.)

4) None but those under 12 are idle.

(for all boys if x is not under 12 then x is not idle)

(for all boys x if x is idle, then x is under 12).

Conclusion:

1) if x learns greek, then x is not a dayboys.

2) if x is not a dayboys;(it means that x is idle.