PHIL1005 Lecture Notes - Lecture 9: Object Language, Beagle, Albus Dumbledore
L9.1
Identity
- Some identity claims are trivial and obvious.
o‘Alex is Alex’
- Some identity claims are interesting and non-trivial
Expressing Identity
- Identity claims like ‘Hesperus is Phospherous’ are expressed by h=p
- This means ‘h and p are the same member of the domain.’
- Identity is symmetric so (h = p) (p = h)
- In this case the identity sign is flanked by proper names
Identity and Variables
- ( x Fx& y Gy) & (x = y) ‘There is at least one F and at least one G and they are the Ǝ Ǝ
same’ x (Fx& Gx)Ǝ‘Something is both F and G’
- x y(xRy) & (x=y) Ǝ Ǝ ‘There is at least one object that Rs itself’
-
-∃x xRx ‘There is at least one object that Rs itself.’
At least one
- x Fx there is a least one FƎ
- But this leaves the number of Fs open
- There could be one or a thousand Fs.
No more than one…
- ‘there is no more than one F’ or ‘there is at most one F’
-∀x (Fx ⊃ ∀y (Fy ⊃ x = y)
- If it is F then it is identical.
- This is compatible with there being no Fs, it can be true even if the following is true
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-~ x ƎFx
Exactly one…
- How do we put an upper bound on the number of objects in our statement?
- ‘There is exactly one F’ x (Fx&Ǝ∀y(Fy⊃x=y))
!x (aka E shriek)Ǝ
- Let’s introduce a piece of notation for uniqueness
- !x will mean ‘there is one and only one…’ or ‘there is exactly one…’, ‘there is a Ǝ
unique…’, ‘there is no more than one but at least one..’
- x (Fx&Ǝ∀y(Fy⊃x=y)) !x FxƎ
Identity between kinds
- What about ‘lighting is electrostatic discharge’?
- It is tempting to represent this as ‘l=e’.
- But what kind of designators are ‘I’ and ‘e’?
- They might be disguised predicates, ‘l=e’ might be equivalent to ‘∀x∀y(LxEx)’
- Or they could be more analoguous to proper names in that they pick out a set of
objects not necessarily chararcterised by a property
- Interesting topic for further study
Constraining the domain’s cardinality from within the object language
- With these tools, we can start constraining the domain from the object language
(without talking about the domain directly).
-∃x Fx
- Says that the domain is non-empty (has a non-zero cardinality)
- x Ǝ∀y x=y
- Says that there is exactly one object in the domain (it has a cardinality of exactly
one).
9.2 COUNTING
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Document Summary
Some identity claims are trivial and obvious: alex is alex". Identity claims like hesperus is phospherous" are expressed by h=p. This means h and p are the same member of the domain. ". Identity is symmetric so (h = p) (p = h) In this case the identity sign is flanked by proper names. Identity and variables ( x fx& y gy) & (x = y) there is at least one f and at least one g and they are the same" x (fx& gx) Something is both f and g" x y(xry) & (x=y) There is at least one object that rs itself". X xrx there is at least one object that rs itself. ". At least one x fx there is a least one f. But this leaves the number of fs open. There could be one or a thousand fs. There is no more than one f" or there is at most one f".