Atomic%20Sentences

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Atomic Sentences
Language, Proof and Logic
Chapter 1
2
Overview
1.0
FOL is not really a particular language but
rather a family of languages. The members of this
family share the same grammar and certain
important (logical) vocabulary terms, known as
the connectives and quantifiers. Languages
of this family can differ, however, in the
specific (nonlogical) vocabulary used in their
most basic sentences, called atomic sentences.
Atomic sentences are formed by combining names
(also called individual constants) and
predicates. The differences between
different FOLs will thus be in what
particular names and predicates they contain.


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Individual constants
1.1
Individual constants are simply symbols that
are used to refer to some fixed individual
object. The individual constants of our blocks
language are the letters a through f plus
n1,n2,...
Conditions

• Every individual constant must name an actually
  existing object
• (unlike English, where Zeus or Santa Claus
  name non-existing things).

• No individual constant can name more than one
  object
• (unlike English, where there are many Toms and
  Bobs).

• An object can have more than one name, or no
  name at all
• (just as in English, where Morning Star and
  Venus name the same
• object, while most asteroids have no names at
  all).

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Predicate symbols
1.2
Predicate symbols are used to express some
property of objects or some relation between
objects, called its arguments. Predicate symbols
are also often called relation symbols.
Each predicate symbol has a fixed number of
arguments, called the arity of the predicate
(symbol). A 1-ary predicate is also said to be
unary, a 2-ary predicate is said to be binary,
and a 3-ary predicate is said to be ternary.
Some of the predicates of the blocks language
are 1-ary Cube, Small, Large 2-ary
Smaller, Larger, 3-ary Between
FOL assumes that every predicate is
interpreted as a determinate property, i.e., a
property for which, given any object, there is a
definite fact of the matter whether the object
has the property. E.g., while in English
small may have various and unclear degrees of
truth, in the blocks language every object is
just either Small or not.


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Atomic sentences
1.3
Atomic sentences look like
P(c1,...,cn) where P
is an n-ary predicate symbol and the ci are
names. An exception is the case with the
identity symbol, where we put the two required
names on either side of the predicate, as in

ab. This is called infix notation, since
appears between its two arguments. With the
other predicates we use prefix notation the
predicate precedes the arguments.
Each sentence makes a claim --- something that
is true or false which of these it is we call
its truth value. E.g., White(snow) expresses
a true claim while Black(snow) expresses a false
claim.
Do You try it on page 24


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General first-order languages
1.4

• FOLs differ in the names and predicates they
  contain.
• Sometimes you deal with a predefined language,
  as in Tarskis World.
• Sometimes you need to design a language of your
  own.
• Considerations when designing languages

• Universality
• Should be able to say everything you may
  potentially need to say
• Flexibility
• WifeOfBob(Jane) vs. Wife(Jane,Bob)
• Economy
• If you have Wife, no need for having Husband

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Function Symbols
1.5

• Function symbols allow us to form name-like terms
  from other name-like
• terms.
• In English, to function symbols correspond noun
  phrases
• Bobs mother, the square of 5, etc.
• Tarskis world has no function symbols.
• Function symbols, just like predicate symbols,
  have arities.
• Using function symbols, we can form arbitrarily
  complex terms
• father(mother(mother(bob)))
• You cannot do the same with predicates
• Father(tom,Mother(mary,bob)) is nonsense
• Do not confuse function symbols with predicates!

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The first-order language of set theory
1.6
No function symbols Two predicates and ?.
Both binary, both infix Let a be the name
of 2 b be the name of 2,4,6 c be the
name of x x is an integral root of x25
d be the name of x x is an elephant that can
fly True or false? a ? a a ? b b ?
b c ? b c d


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The first-order language of arithmetic
1.7
Individual constants 0,1 Predicate symbols
,lt (binary, infix) Function symbols ,?
(binary, infix) Terms 1. 0,1 are terms
2. if t an r are terms, then so are (tr) and
(t?r) 3. nothing is a term unless it can be
obtained by repeated application of (1)
and (2) Atomic sentences tr or tltr, where t,r
are terms. What are the following?
Express number 3

Express number 5 1. ((11)?(1(1?(10))))
Express
number 17 2. (11) (1?(10))
Express number
256 3. (11) lt (0?(11)) 4. (11) (1(0lt1))