Tutorial 1: Identity, Semantics and Symbolization

Logical System
12/26/13
 

Skills to be acquired in this tutorial:

To start to learn about identities, the semantics of identities, and typical expressions in English that might use identities.

Why this is useful:

Reasoning with identity is vital for mathematics, philosophy, and many other areas.

The Tutorial:

In the real world, some people sometimes have aliases, that is, more than one name; so a young woman, 'Pascale' might also be known as (ie aka) 'Cal'. And several in the world at large might learn that Pascale and Cal were one and the same person (ie that Pascale=Cal). [In the real world, not only can people have aliases, but a single name can name more than one person (no doubt 'Jane Smith' names many women). We do not want that for our logic. We want every name (or alias) to name (or identify or pick out) exactly one thing; but we will allow one thing to have more than one name (or alias). So the relationship between names and things is many-to-one.]

In English, saying that two names actually name the same person (or thing) is not all that common, but it does happen and it is usually expressed by the word 'is'. A famous example is that of Samuel Clemens and the author Mark Twain. When Samuel Clemens started to write he took the pen name 'Mark Twain' (from the calls that were made from the deck of ships on the Mississippi as they were lowering a weighted measuring rope to get the depth of the river). We would express this identity with

Mark Twain is Samuel Clemens.

and then use that in arguments, like

Mark Twain wrote Huckleberry Finn.
Mark Twain is Samuel Clemens.
Therefore,
Samuel Clemens wrote Huckleberry Finn.

We wish to use something similar in our logic (to really increase its expressive power). We might make a start at symbolizing the previous argument by

Wm
?Mark Twain is Samuel Clemens.?
Therefore,
Ws

with m= Mark Twain, s= Samuel Clemens, Wx= x wrote Huckleberry Finn

It is the second premise that we need to focus on. It involves two terms, namely m and s, and what it says about them is that they denote the same thing.

We might symbolize this second premise by having a predicate, say Exy, which means x equals, or is identical with, y. This is a possibility. But identity permits some inferences (like substituting equals for equals), and it starts to be awkward to get these via a standard predicate.

What we will do instead is to have a special predicate '=' which we will write infix (ie between its arguments, which are terms). So we will have such expressions as

m=s

and this asserts the statement (true or false) that m is identical with s.

The word for 'names' in logic is 'terms' and thus far we have met two kinds of terms: constants (or proper names), and variables. For example, you are familiar with

Fa to mean, perhaps, 'Ann is friendly' (*Here, 'a' is a term, a constant*)

and

(∃x)Fx to mean, perhaps, 'There is an x such that x is friendly'. (*Here, 'x' is a term, a variable*)

We will now permit a new kind of atomic well formed formula, one that asserts the identity between two terms

<term1>=<term2>

as examples

a=a
a=b
b=x
x=y

and, from these new kinds of atomic formulas, many new compound formulas can be constructed, as examples

∼(a=a)
Pb⊃∼(a=a)
(∃x)(x=b)

The semantics of these new kinds of formulas proceeds pretty much as you would expect. a=b is going to be true in an Interpretation if a and b name the same thing; Pb⊃∼(a=a) is going to be true in an Interpretation if b lacks the property P (for ∼(a=a) is always going to be false); and so on.

(At present, in our drawings, we have entities in our Universe, for example {a,b,c}, and these are separate entities with 'a' naming one of them, 'b' a second. and 'c' the third. What we would like to have is for it be possible that a Universe has perhaps one entity {a} in it, but that this entity 'a' can have an alias or second name, say 'c'; then we would be able to write formulas like (a=c) which would be true or false as the case may be. The drawings will be adapted to permit this.)

These new kinds of well formed formulas can be used in arguments; if Ann and Beryl are aliases for the same person, and Ann is friendly it is valid to infer that Beryl is friendly. i.e.

a=b
Fa

Fb

is a valid inference. (And, eventually, there will be additional rules of inference that allow a derivation to be given of this inference).

While we do not often express plain identities between proper names in English. We do use plenty of constructions that require identity to be symbolized properly. Most prominent among these are statements which are, implicitly or explicitly, about numbers. For example, consider

Only Ann is happy.

now

Ha (with the conventions Hx= x is happy and a=Ann)

captures

Ann is happy

but it does not get the 'only'. To do this a symbolization like

Ha ∧ (∀x)(Hx⊃ x=a)

is needed and what this says is

Ann is happy and whatever x you may choose, if x is happy then x is identical to Ann.

ie Only Ann is happy.

Similarly for statements like

At least one person is happy is (∃x)(Px∧Hx)

Exactly one person is happy is (∃x)((Px∧Hx) ∧ (∀y)((Py∧Hy)⊃ y=x)) ie There is a happy person, x, and whatever y you may choose, if y is a happy person then y is identical to x.

At least two people are happy is (∃x)(∃y)((Px∧Hx)∧(Py∧Hy)∧∼(x=y)) ie There is a happy person, x, and there is a happy person, y, and x and y are different.

Exactly two people are happy is (∃x)(∃y)((Px∧Hx)∧(Py∧Hy)∧∼(x=y)∧(∀z)((Pz∧Hz)⊃( (z=x) ∨(z=y))) ie There is a happy person, x, and there is a happy person, y, not identical to x, and whatever z you may choose, if z is a happy person then either z is identical to x or z is identical to y.

and so on.

 

 


 

Exercise to accompany Identity Tutorial 1.

The Palette in the Interpretations Panel has a couple of extra items in it.

Palette

The one at the very bottom has a small letter, a constant, in it. This is for inserting aliases, choose it, click on the diagram, and you will be guided. What will be drawn is essentially an arrow, with its head inside an individual that is already in the Universe. Say the individual in the Universe is 'a', the arrow will have a label on it, say 'b'; and drawing this arrow into 'a' signifies that 'a' also has the alias 'b'. And the Interpretation shown in the Interpretation board will be changed to reflect the state of the diagram. [These aliases are what might be called zero-ary functors. They are like functions or functors and thus have name, say 'b', and they also have a value, say 'a', but they do not have or need an argument.]

The Interpretation for this exercise needs to show an Interpretation similar to this

 

If it does not, draw a similar one yourself.

Exercise 1(of 2)

Interpretations Widget

Exercise 2(of 2)

Interpretations Widget