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CH 8.1 (Predicate Logic translations)

Translating the old way (6.1):Ex.) All Fish are creatures that live underwater. F

Sharks are Fish S

Sharks are creatures that live underwater U

Theres a problem in that our translation doesnt show us a valid argument even though we know

this is a valid argument.

The Logic we learned in Ch 6 & 7 (Propositional logic) only allows us to represent relationships

BETWEEN simple statements & compound statements.

BUT it doesnt let us break inside the simple statement to show the relationship between thesubject and predicate of the statement. It also doesnt let us distinguish different quantities (All,

No, Some)

So we need a different kind of Logic that will enable us to do those 2 things (1. show the

relationship between subject & predicate & 2. be able to talk about groups of things).

I. Relationship between Subject & PredicateCapital letters designate the Predicate. The subject is designated by the lowercase letter coming

immediately after the Predicate letter.

I.e. Socrates is Mortal Ms

Kali is a puppy PkThis is a rock Rt

Xerxes was a king Ke (cannot use x)

You can create compound statements from these simple statements:

(Ms He), [Pk (Ms v Rt)] etc.

Only the first 23 letters of the alphabet can be used to designate specific subjects/names. These

are called constants. (x, y, z are reserved as variables).

Mx (x is mortal)

Px (x is a Puppy)(These are called Free variables because nothing is binding them). They have no truth value

(arent true or false) as Free Variables.

II. This brings us to the second thing well be able to do with this newlogic: Express the relationship between groups of things

(quantities).

So far weve just been looking at specific subjects (called constants). I.e. Socrates, Kali, This,

Xerxes

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We need a way to talk about characteristics a GROUP might have.We need a way to express statements like:

SOME things are puppiesSOME things ARE NOT men.

NO things are handsomeALL things are changing

We want to be able to talk about quantities.

To do this we pick the appropriate QUANTIFIER and place it IN FRONT of (to left of) thestatement function it applies to.

There are 2 types of quantifiers

Where M = is mortal:(x)Mx = Everything is Mortal, For any/all x, x is mortal.

( x)Mx = Something is mortal, There exists at least one x that is mortal.

The quantifiers BIND the variables (changing them from Free variables to Bound Variables

which do have a truth value now.)

When we add tildes we can express negative statements:

~(x)Mx = Its NOT the case that everything is mortal (some things are not mortal)

(x)~Mx = All things are NOT mortal (nothing is mortal)

~( x)Mx = Its NOT the case that some things are mortal (nothing is mortal)

( x)~Mx = Some things are NOT mortal

Relationships between groups:

To show the relationship between groups of things:

(Also see p.26 in Course Packet)

(x) = Universal QuantifierFor all x, Forany/every x Expresses ALL or NONE

( x) = Existential Quantifier There exists at leastone x that For some XEx resses SOME ARE or SOME ARE NOT

All S are P: (x)(Sx Px)No S are P: (x)(Sx ~Px)Some S are P: (x)(Sx Px)Some S are not P: (x)(Sx ~Px)

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To express Universal statements about groups we use the (x) and the horseshoe as the m.c of thestatement the quantifier binds to:

All S are P = (x)(Sx Px)For all x, if x is S then x is P

i.e. All Dogs are mammals = (x)(Dx Mx) (If its a dog, then its a mammal)

No S are P = (x)(Sx ~Px)For all x, if x is S then x is NOT P.

i.e. No Dogs are Fish = (x)(Dx ~Fx) (If its a dog, then its NOT a fish)

To express Existential statements about groups we use the ( x) and the dot as the m.c of thestatement the quantifier binds to:

Some S are P = ( x)(Sx Px)--There exists at least one x such that x is both S and P.

i.e. Some dogs bite. = ( x)(Dx Bx) (Some things are dogs, and they bite)

Some S are not P = ( x)(Sx ~Px)--There exists at least one x such that x is S but not P.

i.e. Some dogs dont bite = ( x)(Dx ~Bx) (Some things are dogs, but they dont bite)

Lets go back to our original argument about the sharks being creatures that live underwater and

symbolize it in the way weve just learned:

(x)(Fx Cx)(x)(Sx Fx)

(x)(Sx Cx) Now its an HS.

Some translation guidelines:

1. If a statement begins with the word ONLY , NONE BUT , NONE EXCEPT orNObut.. then use the (x) with the but reverse the subject and predicate.

i.e. ONLY egg-layers are chickens (x)(Cx Ex)

ONLY dogs are barkers: (x)(Bx Dx)

2. If the statement begins with the words THE ONLY, use the (x) with horseshoe, but DONOT switch Subject and Predicate.

3. The word AND doesnt always mean a doti.e. Dogs and cats are mammals: (x)[(Dx v Cx) Mx]General rule is to use a wedge to separate 2 nouns when they are both part of either the S or

P class.

Only dogs and cats make good pets: (x)[(Gx Px) (Dx v Cx)]

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4. When choosing your quantifier, consider the statements most likely meaning: Dogs are mammals: (x)(Dx Mx)

Dogs are barking: ( x)(Dx Bx)

Dogs are in college station: ( x)(Dx Cx)

Not every/ Not all = Some S are not P, or the negation of an All S are P.

(x)(Sx ~Px) = ~(x)(Sx Px)

None = Its not the case that something = ~( x)

Examples:

1. All dogs are happy mammals. (D,H,M)

(x)[Dx (Hx Mx)]For all x, if x is a dog, then x is a happy mammal.

2. A few chessmen were missing. (C,M)

( x)(Cx Mx)There exists some x such that x is a chessman and missing.

3. No lizards are firebreathers. (L, F)

(x)(Lx ~Fx)For all x, if x is a lizard, then x is NOT a firebreather.

4. Every dog and lizard is adored. (D,L,A)

(x)[(Dx v Lx) Ax]

For all x, if x is either a dog or lizard, then x is adored.

5. Only big lizards prefer radioactive food. (B,L,R,F)

(x)[(Rx Fx) (Bx Lx)]For all x, if x is radioactive food, then x is preferred by big lizards.

6. Hermit crabs are nocturnal and some of them live underwater. (H,N,L)

(x)(Hx Nx) (Ex)(Hx Lx)For all x, if x is a hermit crab, then x is nocturnal, AND some things are both hermit crabs and

live underwater.

7. If Jenny dances, then some parrots can dance. (D,P)

Dj ( x)(Px Dx)If Jenny dances, then some things are parrots and dance.