Predicate Logic and the language PL

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# Predicate Logic and the language PL - PowerPoint PPT Presentation

Predicate Logic and the language PL. In SL , the smallest unit is the simple declarative sentence. But many arguments (and other relationships between sentences) are actually based on sub-sentential units. Including: None of Mary’s friends supports Libertarians.

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Predicate Logic and the language PL
• In SL, the smallest unit is the simple declarative sentence.
• But many arguments (and other relationships between sentences) are actually based on sub-sentential units. Including:

None of Mary’s friends supports Libertarians.

Sarah supports Matlow and Matlow is a Libertarian.

So Sarah is no friend of Mary’s.

• Although we could symbolize this argument in SL, its logic would be lost.
Predicate Logic

Sub-sentential units in predicate logic

1. Singular terms:

• Names (The Washington Monument, Boston, Marie Curie, Harry Reid, Henry, Sherlock Holmes…)
• Definite descriptions (the Senate majority leader, the discoverer of radium, Michael’s only brother, the present king of France, the person Mary is now talking to…)
• Issues:
• “Non-designating” singular terms
• Singular terms and context
• Pronouns
Predicate Logic

Singular terms and pronouns:

• “If John voted for Hillary Clinton, then he’s no Libertarian.”
• “If John voted for Hillary Clinton, then John’s no Libertarian.”
• “This test is so easy that if anyone fails it, it’s his or her own fault.”
• We can’t use: “This test is so easy that if anyone fails it it’s John’s or Cynthia’s own fault”
Predicate Logic

Sub-sentential units continued

2. Predicates:

• Sentences can have more than one singular term, for example:

‘New York is between Philadelphia and Boston’

• Predicates of English are parts of English sentences that are obtained by removing one or more singular terms from an English sentence.
Predicate Logic

2. Predicates:

• (Or, a predicate is a string of words with one or more blanks in it such that when the blanks are filled in, a sentence results.)

‘New York is between Philadelphia and Boston’

_______ is between Philadelphia and Boston

New York is between _______ and Boston.

_______ is between ________ and Boston.

_______ is between Philadelphia and ______.

_______ is between _________ and _______.

Predicate Logic

2. Predicates:

• So there are one-place predicates such as

_______ is between Philadelphia and Boston

New York is between _______ and Boston.

• And there are many-place predicates.
• This is a two-place predicate:

_______ is between ________ and Boston.

This is a three-place predicate:

_______ is between _________ and _______.

Predicate Logic

2. Predicates:

• In general, where n is a positive integer, a predicate with n blanks is an n-place predicate.
• One way of generating a sentence from a predicate is filling the blanks with singular terms – any singular term may be put in any blank, and the same singular term can be put in more than one blank.
Predicate Logic
• Using variables:
• Instead of blanks, we use the lower case letters ‘w’, ‘x’, ‘y’ and ‘z’ (with numerical subscripts when necessary) to mark the blanks in predicates.
• So one predicate early discussed can be displayed as: x is between y and z.
• Another can be displayed as x is taller than y.
Predicate Logic
• So from the two-place predicate ‘x is taller than ‘y, and the singular terms ‘The Washington Monument’, ‘Mary’, ‘John’, and ‘the smallest prime number’, we can generate:
• Mary is taller than The Washington Monument.
• John is taller than Mary.
• Mary is taller than John.
• The Washington Monument is taller than John.
• The smallest prime number is taller than Mary.
• And so forth…
Predicate Logic
• We also retain the sentential connectives ‘and’, ‘or’, ‘if then’, ‘if and only if’, and ‘not’
• So given a stock of predicates, singular terms, and the sentential connectives, we can generate a wide variety of sentences of English.
• From the sentential connectives, the singular terms ‘Michael’, ‘Sue’ and ‘Rita’, and the predicates ‘x is easygoing’, ‘x likes y’, and ‘x is taller than y’, we can generate:
Predicate Logic
• From the sentential connectives, the singular terms ‘Michael’, ‘Sue’ and ‘Rita’, and the predicates ‘x is easygoing’, ‘x likes y’, and ‘x is taller than y’, we can generate:

Michael is easygoing.

Michael is easygoing but Sue isn’t easygoing.

Sue likes Rita and Rita likes Michael.

If Rita likes Michael, then Michael is taller than Sue and he is easygoing.

Either Rita or Sue is taller than Michael, but not both.

Predicate Logic
• Except when our domain is limited, what we can’t yet generate (but eventually will) are claims such as:
• Everyone is easygoing.
• No one is easygoing.
• Someone is easygoing.
• Someone is not easygoing.
• No one is taller than his or herself.
• Everyone likes him or herself.
• ‘every’, ‘some’, ‘all’, ‘each’, and ‘none’ are quantity terms and quantity terms are not singular terms.
The language PL
• Vocabulary:
• The sentential connectives ‘&’, ‘v’, ‘’, ‘’ and ‘~’.
• Individual constants (lowercase Roman letters ‘a’ through ‘v’, with or without subscripts) to symbolize singular terms that denote (names and definite descriptions)
• Predicates of PL: uppercase Roman letters ‘A’ through ‘Z’ with or without subscripts and followed by variables…
The language PL

Vocabulary:

Predicates of PL: uppercase Roman letters ‘A’ through ‘Z’ with or without subscripts and followed by one or more variables, n of the letters ‘w’, ‘x’, ‘y’ and ‘z’ after the predicate letter.

Fx is a one place predicate

Fxy is a two place predicate

Fxyz is a three place predicate..

The language PL

As with sentence letters in SL, we can use a predicate (say, ‘Lxy’) to symbolize, on different occasions, a variety of 2 place predicates of English, including

‘x loathes y’

‘x loves y’

‘x is larger than y’

‘x is less than y’…

The language PL

Vocabulary:

• Constants of PL: lower case Roman letters ‘a’ through ‘v’ are used to symbolize singular terms:

a is a constant

b is a constant…

• Sentential connectives and punctuation (parentheses and brackets)
The language PL

Symbolizing in PL

1. We begin with a symbolization key

a. Specify the universe of discourse (abbreviated ‘UD’) for the occasion.

Examples of UD’s:

the positive integers

the jellybeans in the jar on my desk

all persons

everything

people in Michael’s office

The language PL

Symbolizing in PL

1. We begin with a symbolization key

b. Specify symbols for the predicates

Ex: x is easygoing

Txy: x is taller than y

Lxy: x likes why

c. Specify symbols for constants (if there are any)

a: Anita

b: The Brooklyn Bridge

The language PL

A symbolization key:

UD: People in Michael’s office

Lxy: x likes y

Ex: x is easygoing

Txy: x is taller than y

m: Michael

r: Rita

s: Sue

UD: People in Michael’s office

Lxy: x likes y

Ex: x is easygoing

Txy: x is taller than y

m: Michael

r: Rita

s: Sue

‘Sue is easygoing’

Es

‘Sue is taller than Michael, and Michael is taller than Rita’

Tsm & Tmr

‘If Rita likes Sue, then Rita is taller than Sue

Lrs  Trs

‘If Michael is easygoing, Sue is not’

Em  ~Es

The language PL

We can symbolize some English sentences involving quantity terms with the resources we have so far if we have a UD that makes it possible. Given the

UD: People in Michael’s office

and the predicates and constants we have in the symbolization key – which include a constant for each of the people)

we can symbolize ‘Michael likes everyone’ as

(Lmm & Lmr) & Lms

The language PL

UD: People in Michael’s office

We can symbolize ‘Michael likes someone’ as

(Lmm v Lmr) v Lms

And ‘Michael likes no one’ as

(~Lmm & ~Lmr) & ~Lms

or

~[(Lmm v Lmr) v Lms]

And ‘Everyone is easygoing’ as

(Em & Er) & Es

Given the symbolization key shown or handed out, symbolize:

Alice was born in Boston, so she wasn’t born in Seattle.

Bonnie was born in Cleveland but she lives in Philadelphia.

If Bonnie is taller than Charles, and Charles is taller than Alice, then Bonnie is taller than Alice.

No one lives in Boston.

Everyone was born in Cleveland.

Create a symbolization key that has ‘Alex, Bruce, Cathy and Danielle’ as its UD

And the predicates:

x is attracted to y

x is intimidated by y

x is intelligent

x is shorter than y

x is sitting between y and z

And the singular terms:

Alex

Bruce

Cathy

Danielle

UD: Alex, Bruce, Cathy and Danielle

Axy: x is attracted to y

Ixy: x is intimidated by y

Cx: x is curious

Sxy: x is shorter than y

Bxyz: x is sitting between y and z

a: Alex

b: Bruce

c: Cathy

d: Danielle

Symbolize:

‘Cathy is attracted to Bruce, but she is intimidated by him’

(Acb) & (Icb)

‘Bruce is sitting between Alex and Danielle’

‘If Cathy is shorter than Alex, she is attracted to him’

Sca  Aca

‘No one is curious’

(~Ca & ~Cb) & (~Cc & ~Cd)

or

~[(Ca v Cb) v (Cc v Cd)]

Homework:

7.2E (all)

7.3E As much as you can of exercises 1-3.