Beyond Standard Predicate Logic
Identity
What we have done so far has gone a long way toward allowing us
to symbolize expressions of natural language, but we aren’t done
yet. We need another logical symbol “=” which will express
identity in order to take care of many more expressions.
Brandon C. Look: Symbolic Logic II, Lecture 9
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Consider Tarski’s world. We want to say, “Only a is a cube”. We
can’t just write “Ca” because that just says “a is a cube”. Nor can
we write “Ca∧ ∼ ∃Cx” for that is a contradiction. What we need
to write is something that gets at the only — something along the
lines of “a is a cube and there exists nothing else which is a cube
and which is not identical to a.” Thus: Ca∧ ∼ ∃x(Cx∧ ∼ x = a).
Or, alternatively, Ca ∧ ∀x(Cx → x = a).
Brandon C. Look: Symbolic Logic II, Lecture 9
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Grammar and Semantics for the Identity Sign
We will simply add the following clause to our defintion of a wff:
I
If α and β are terms, then α = β is a wff.
And to our definition of truth-in-a-model, we will add the following
clause:
I
VM ,g (α = β) = 1 iff: [α]M ,g = [β]M ,g
So, for example, Hesperus = Phosphorus iff “Hesperus” and
“Phosphorus” refer to the same object.
Brandon C. Look: Symbolic Logic II, Lecture 9
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A translation guide:
Only a is F .
The only F that is G is a.
No F except a is G .
All F except a are G .
There is at most one F .
There are at least two F s.
There are exactly two F s.
Brandon C. Look: Symbolic Logic II, Lecture 9
Fa ∧ ∀x(Fx → x = a)
Fa ∧ Ga ∧ ∀x((Fx ∧ Gx) → x = a)
Fa ∧ Ga ∧ ∀x((Fx ∧ Gx) → x = a)
Fa∧ ∼ Ga ∧ ∀x((Fx ∧ x 6= a) → Gx)
∀x∀y ((Fx ∧ Fy ) → x = y )
∃x∃y (Fx ∧ Fy ∧ x 6= y )
∃x∃y (Fx ∧ Fy ∧ x 6= y ∧ ∀z(Fz → (z = x ∨ z = y )
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Beyond Standard Predicate Logic
Function Symbols
A singular term, such as “Socrates”, “Fido”, and “Lexington”,
refer to individual objects in a world. But there is another class of
terms that refer to individual objects but which are not among the
class of names from PL like a, b, c . . .: functions. For example,
“Socrates’ wife” refers to Xanthippe. And we will add a function
symbol to our vocabulary to allow for such references. Thus, f (a)
is the function that maps a to some referent; and where f is the
function for “is the wife of” and s is Socrates, we can write “f (s)”
and know that the referent will be Xanthippe. Moreover, we can
iterate these functions: if f is the father function, then f (f (a)) is
a’s paternal grandfather (and f (m(a)) could be a’s maternal
grandfather, for example).
Brandon C. Look: Symbolic Logic II, Lecture 9
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Grammar and Semantics for Functions
We will then add to our vocabulary the following:
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for each n > 0, n-place function symbols f , g , . . . with or
without subscripts
And to our definition of terms:
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if f is an n-place function symbol, and α1 . . . αn are terms,
then f (α1 . . . αn ) is a term
Brandon C. Look: Symbolic Logic II, Lecture 9
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And to the definition of a model:
I
If f is an n-place function symbol, then I (f ) is an n-place
(total) function defined on D.
And to the definition of denotation:
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[α]M ,g = I (f )([α1 ]M ,g . . . [αn ]M ,g ) if α is a complex term
f (α1 . . . αn )
Brandon C. Look: Symbolic Logic II, Lecture 9
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Beyond Standard Predicate Logic
Definite Descriptions
What about expressions that deal with definite descriptions —
“The so-and-so”? As Sider shows, one response is to introduce a
new symbol to stand for “the”. Thus, “the black cat” becomes
“ x(Bx ∧ Cx)”.
ι
ι
Brandon C. Look: Symbolic Logic II, Lecture 9
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