Gold casket: The portrait is in here.

Silver casket: The portrait is in here.

Lead casket: At least two of the caskets have a false inscription.

Which casket should the suitor choose? Read more

]]>Portia has a gold casket and a silver casket and has placed a picture of herself in one of them. On the caskets she has written the following inscriptions:

Gold: The portrait is not here.

Silver: Exactly one of these inscriptions is true.

Portia explains to her suitor that each inscription may be true or false, but that she has placed her portrait in one of the caskets in a manner that is consistent with this truth or falsity of the inscriptions. If he can choose the casket with her portrait, she will marry him. The problem for the suitor is to use the inscriptions (although they may be either true or false) to determine which casket contains her portrait.

How can we solve this problem? Read more

]]>By studying this rule we can observe several facts about the infimum. First, since it is defined in terms of and we have

{Golden Rule}

{Symmetry of and }

{Golden Rule}

i.e. is symmetric.

**Exercise**: Show that is associative.

Furthermore, is idempotent:

{Golden Rule}

{identity of ; idempotence of }

With regard to last three properties, is just like . We might very well ask, is a fixed point of as it is of ? Let’s calculate:

{Golden Rule}

{fixed point of }

{identity of }

So is the identity of .

Next we have two very useful laws of absorption/introduction:

We shall prove the first, the second follows by interchanging and .

{Golden rule}

{idempotence}

{identity of }

What about distributivity properties?

{Golden rule}

{distributivity}

{idempotence, preparing for the Golden Rule}

{Golden Rule}

{ distributes over }

{absorption}

{ distributes over }

{associativity; symmetry}

{absorption}

So distributes over . What about ? The best we can do is

{Golden rule}

{distributivity}

{symmetry, preparing for Golden rule}

{Golden rule, twice}

As a consolation we do have

.

Finally we have

{pseudo-distributivity}

{idempotence; identity of }

]]>

has a

is read as

(Notice the greater space around in the first expression).

The first two properties of the supremum are familiar:Associativity:

Symmetry:

The next two are new:

**Idempotence:**

**Distributivity/Factoring:**

Armed with these rules, we can now prove our first property concerning the supremum:

*Proof*

Our next rule is

*Proof*

***

The first type we will explore is a very simple one – so simple it consists of only two values. The type is called **boolean **and its values are , pronounced “top” and , pronounced “bottom”.

Now we can talk about the value of :

“The value of is of type boolean.”

or, equivalently,

“The value of is either or .”

***

Equality between booleans is special, and so when we express equality between and , where and are boolean we write

pronounced “ equivales “.

What is so special about boolean equality – called **equivalence **– that we have special symbol for it? The answer gives us our first rule for calculating:

**Associativity:**

We give the rule a name to help us remember what it is. Here is how we would use the rule in calculations:

An important consequence of $\eqv$’s associativity is that when we have a series of expressions punctuated by $\eqv$ signs it does not matter where we put the brackets. Indeed, we need not write the brackets at all, as for instance in our next rule:\\

**Symmetry: **

We may parse this rule in several ways:

$ latex (X\equiv Y)\equiv (Y\equiv X)$

The last two expressions tell us that `equivaling’ with any boolean is . We say that is the *identity *of . Our next rule gives a name to the identity:

(2)

Up to this point we have simply postulated rules i.e. we have taken it as given that they hold. Our next rule, however, we will prove. How do we prove? When we postulate a rule we are saying that it is an expression which is equivalent to . So to show that an expression is a rule, we present a calculation showing that the said expression is also equivalent to .

**Reflexivity:**

*Proof*

{{(2), parsed as

{(2)}

Such is the equivalence.

**Challenge:** Does anyone know how to get the latex \quad and \begin{tabular} commands to work in WordPress?

***

* *Let us start with the simple notion of an **expression. **As its name suggests, an expression is used to express (or *denote*) a value. A value may be a sum of money or the weight of an object or the truth of an assertion.

There are two kinds of expressions: **constants **and **variables.**

A constant represents a particular value and that value cannot be changed. Examples of constants include , .

A variable can represent any value (though only one value at time). In addition the value of a variable can be replaced by another expression. We’ll see exactly how to do this later on. We use single italic letters for variables e.g. , .

All our expressions will denote at most one value, but a value may be denoted by more than one expression. For example the expressions and both denote the same value.

Now writing

“the expression denotes the same value as the expression ”

over and over would become quite tiresome. So we instead we just write

pronounced “ equals “.

Moreover, we may view X = Y not just as a statement of fact but as **an expression in its own right**. But what, then, is its value?

*Exercise *– Ask any mathematicians you know what they can tell you about the value of .