Open Statements, Quantifiers

An open sentence or predicate is almost like a proposition, except that it contains some unknown (non-propositional) variable in it. E.g. “\(x^2-2x+1=0\).” We don’t know its truth value unless we know what \(x\) is.
Symbolically, we denote an open sentence using the same notation as for functions. For example, \(P(x) ::= x^2-2x+1=0\).
Open sentences can be turned into propositions by quantifying the variables.
The existential quantifier \(\exists\) means at least one. In English, we can express it using these phrases
```
There is ...  
There are ...  
There exists ...
Some ...
For some ...
```

The universal quantifier \(\forall\) means all. Some ways to say it in English are

All ...
For all ...
Every ...
For every ...
Each ...
For each ...
Any ...
For any ...

Beware! People often switch the order of phrases in English. So you will often see a sentence like “\(40 = 2\cdot k\) for some interger \(k\)” instead of “for some integer \(k\), 40 = 2k.”

Predicate Calculus

The Predicate Calculus is called first-order logic while the propositional calculus is called zero-order logic.
In first-order logic, we always have some domain(s) of discouse in mind. E.g. the domains may be \(\mathbb{N}\) and \(\mathbb{Z}\). Then we can write a predicate like this \(\exists n\in\mathbb{N},\,\forall z\in\mathbb{Z}.\,z = n\).
Such an indication of membership right after the quantified variable is simply a short-hand notation. We can write without it. E.g. the previous predicate can be written in full like this \(\exists n.\,n\in\mathbb{N} \land (\forall z.\,z\in\mathbb{Z} \Rightarrow z = n)\).
When a predicate contains more than one quantifier, you need to pay attention to the order. Changing the order can result in a predicate with a different meaning.
A predicate may contain relations while propositional calculus cannot. E.g., if we define DIVIDE(d, n) ::= “d is a divisor of n,” then we can write a predicate like \(\forall n\,\exists d.\,\rm{DIVIDE}(d, n)\).

Question. What do the following English sentences mean?
- There is a female student in the MCS-228 class of the Fall 2021 semester at Gustavus Adolphus College.
- A rational number is not integral.
- Odd integers are multiples of 3.
- Some schiffs are fibs.
- All fibs are fobs.
- \(a + b = b + a\).
Exercise. Rewrite all ambiguous sentences above to disambiguate them.

A sentence of the form “\(\forall x\in X.\,\varphi(x)\)” is short for “\(\forall x.\,x\in X \Rightarrow \varphi(x)\).”
A sentence of the form “\(\exists x\in X.\,\varphi(x)\)” is short for “\(\exists x.\,x\in X \land \varphi(x)\).”

In propositional calculus we define validity of a proposition in terms of truth table.
In predicate calculus, we define validity in terms of the domain of discourse and the meaning of the predicate. A formula is valid iff it is true in all possible domains and all possible interpretation of the predicate.
An example valid formula is \(\exists x\,\forall y.\, P(x, y) \Rightarrow \forall y\,\exists x.\, P(x, y))\).
An example non-valid formula is \(\forall y\,\exists x.\, P(x, y) \Rightarrow \exists x\, \forall y.\, P(x, y))\).
A formula can be shown to be invalid by exhibiting a counter-model, i.e., a domain of discourse and an interpretation of the predicate that makes the formula false.

To negate a quantified prodicate, use the logical equivalence \[ \neg(\forall x.\,\varphi(x)) \equiv \exists x.\,\neg\varphi(x) \] or the logical equivalence \[ \neg(\exists x.\,\varphi(x)) \equiv \forall x.\,\neg\varphi(x) \]

Here are two common conventions in mathematics writing.
- Universal quantifiers are routinely omitted when possible.
- In definition, the word ‘if’ actually means ‘if and only if’.
Beware of the following.
- Depending on context, the word ‘one’ could mean ‘at least one’ or ‘exactly one’. Similarly for ‘two’, ‘there’, , etc.
- Oftentimes the word ‘a’ should be replaced by ‘any’.

A sentence of the form, “For every element of the nonempty set \(\ldots\) blah \(\ldots\)” is always true. E.g. “Every 10-feet-tall man is handsome” is true.
A sentence of the form, “There exists some element in the nonempty set such that \(\ldots\) blah \(\ldots\)” is always false. E.g. “There exists an 10-feet-tall mathematician” is false.
The term unique in mathematical writing means “exactly one.” We write the phrase “There exists a unique \(\ldots\)” by “\(\exists ! \ldots\).”
The term distinct in mathematical writing means ‘different’.
The term write in mathematical writing means ‘let’.
The phrase without loss of generality, or in short form, wlog, in mathematical writing has many meanings, as will be demonstrated throughout this course!