9.7: Identity: Symbolizations

Transcription

1 9.7: Identity: Symbolizations Among the many different relations, one stands out in logic: identity. Consider the following argument: George Orwell wrote George Orwell was identical to Eric Blair. Hence, Eric Blair wrote (o: Orwell; b: Blair; Wxy: x wrote y; Ixy: x is identical to y) This is obviously a valid argument, but we can t prove that it is given the rules we have so far: 1. Won 2. Iob Wbn Hence, we need to introduce identity explicitly into our system and treat it specially. We ll borrow the identity sign = from arithmetic to express the identity relation is identical to or, more succinctly, simply is. We will also use standard infix notion, that is, we will write o=b instead of =ob. Since identity statements are statements like any other, they can be used to form more complex statements. Sarah Jessica Parker is not identical with Clint Eastwood. (s: SJP; c: CE) Symbolized: ~s=c If Sarah is not Clint, then Clint is not the star of Sex and the City. (s: SJP; c: CE; t: Sex and the City; Sxy: x is the star of y) Symbolized: ~s=c ~Sct Note no parens are used when negating an identity statement.

2 Many statements can be symbolized with identity that cannot be symbolized without it! Salient examples follow: Only... English: Only Edison invented the phonograph. Logicese 1: Edison invented the phonograph, and for all x, if x invented the phonograph, then x is identical to Edison. Symbolized 1: Pe (x)(px x=e) Logicese 1I: For all x, x invented the phonograph if and only if x is e. Symbolized 1I: (x)(px x=e) The only... English: The only engineer to invent the telephone was Bell Logicese: Bell is an engineer and Bell invented the telephone and, for all x, if x is an engineer and x invented the telephone, then x is Bell. Symbolized: Eb Tb (x)((ex Tx) x=b) No... except [but other than]... English: No engineer except Bell invented the telephone. Logicese: Bell is an engineer and invented the telephone, and no engineer x other than Bell invented the telephone. Symbolized: Eb Tb ~( x)(ex ~x=b Tx) (Can also be symbolized as in The only....) All... except [but other than]... English: All philosophers other than Derrida study logic. Logicese: For all x, if x is a philosopher and x is not Derrida, then x studies logic. Symbolized: (x)((px ~x=d) Lx)

3 Superlatives English: Everest is the tallest mountain. Logicese: Everest is a mountain and, for all x, if x is a mountain and x is not Everest, then Everest is taller than x. Symbolized: Me (x)((mx ~x=e) Tex) The idea here is that, in a superlative, one says that a certain object in a certain class has more of a certain property than any other thing in the class. As seen above, other requires identity: the things other than x are the things not identical to it. At most n... English: There is at most one god. Logicese: For all x and for all y, if x is a god and y is a god, then x is identical to y. Symbolized: (x)(y)((gx Gy) x=y) English: There are at most two angels. Logicese: For all x, y, and z, if x, y, z are angels, then either x is y or y is z or x is z. Symbolized: (x)(y)(z)((ax Ay Az) (x=y v y=z v x=z)) English: At most two Democrats are conservative Logicese: For all x, y, and z, if x, y, z are all conservative Democrats, then either x is identical to y or y is identical to z or x is identical to z. Symbolized: (x)(y)(z)([(dx Cx) (Dy Cy) (Dz Cz)] (x=y v y=z v x=z)) Note that at most statements do not imply that there are any instances of the things of which there is said to be at most n. At least n... English: There is at least one god. Logicese: There exists an x such that x is a god. Symbolized: ( x)gx

4 English: There are at least two angels. Logicese: There exists an x and a y such that x and y are angels and x is not y. Symbolized: ( x)( y)(ax Ay ~x=y) English: At least two Democrats are conservative. Logicese: There exists an x and a y such that x and y are conservative Democrats and x is not y. Symbolized: ( x)( y)((dx Cx) (Dy Cy) ~x=y) Note that if we d omitted the ~x=y conjuncts in the latter two symbolizations they would only have expressed At least one statements, since it doesn t follow that x and y have to be distinct. Exactly n... English: There is exactly one god. Logicese: There exists an x such that x is a god and for all y, if y is a god, then x is y. Symbolized: ( x)(gx (y)(gy x=y)) English: There are exactly two angels. Logicese: There exists an x and a y such that x and y are angels and x is not y and for any angel z, either z is x or z is y. Symbolized: ( x)( y)[ax Ay ~x=y (z)(az (z=x v z=y))] Note that Exactly n... statements are equivalent to At least n and at most n statements. The (Definite Descriptions) The philosopher Bertrand Russell analyzed sentences of the form The A is B as essentially containing three assertions: 1. There is at least one A. 2. There is at most one A. 3. Every A is a B

5 Symbolized I: ( x)(ax) (x)(y)((ax Ay) x=y) (x)(ax Bx) We can also express Russell s idea in a single sentence: The A is B means Exactly one thing is A and that thing is B. Using the ideas above, we can express this succinctly in terms of quantifiers, variables, and identity: Symbolized II: ( x)[(ax (y)(ay x=y) Bx]