Predicates

Propositional logic sometimes isn't enough

• x > 5
• If the user is an admin, then let them edit
• When is this true? When is this false?

Propositional Functions

• We can express these as functions of a variable
• Consists of a subject and predicate
• P(x) = "x > 5"
• Q(x) = "If x is an admin, then they can edit"

Complex Functions

• Functions can have more than one variable
• P(x, y) = "x + y = 3"
• Q(x, y) = "User x is the author of post y"
• Functions of n values are n-place predicates or n-ary predicates

Preconditions/Postconditions

• Establish assumptions for your code
• Given the assumptions, your code does this
• Error handling, state change, etc.

// What are the preconditions and postconditions?

public void start() {
  if (!this.isStarted()) {
    this.started = true;
  } else {
    throw new Exception("Already started");
  }
}

Quantifiers

We can assign values

But some expressions have multiple valid values

P(x) = "x * x >= 0"

When is this true? When is this false?

Domain of Discourse

Set of values which we can use to quantify our function/expression. Must be stated in order to use a quantifier. Examples include "real numbers", "imaginary numbers", "users in the database", etc.

Domains can be restricted, too. For example "Given all real numbers x where x > 3" or "Given all users in the database where role = 'admin'"

Universal Quantification

For all values in a given domain, P(x) is true. Denoted as "∀". If there is an x in the domain for which P(x) is false, that value is a counterexample of "∀x P(x)".

Another way to think about it: Given (x1, x2, x3, ... xn), the universal quantification ∀x P(x) is the same as:

P(x1) ^ P(x2) ^ P(x3) ^ ... ^ P(xn)

Existential Quantification

For at least one value in a given domain, P(x) is true. Denoted as "∃". If the statement "∃x P(x)" is false for all x in the domain, then the statement is false.

Another way to think about it: Given (x1, x2, x3, ... xn), the existential quantification ∃x P(x) is the same as:

P(x1) v P(x2) v P(x3) v ... v P(xn)

Uniqueness Quantifier

• Special case of the existential quantifier
• There exists a single value that satisfies the predicate
• For all real numbers, ∃!x such that x + 1 = 0

Logical Equivalence

• Statements can be logically equivalent (given the same domain)
• Are "∃x (P(x) v Q(x))" and "∃x P(x) v ∃x Q(x)" logically equivalent?
• To prove this true, determine if statement 1 is true, then statement 2 is true and vice versa.
• To prove this false, find a counterexample

Negating Quantifiers

• We can negate quantifiers
• ¬∃x P(x) is equivalent to ∀x ¬P(x)
• ¬∀x P(x) is equivalent to ∃x ¬P(x)
• What is the negation of "For all students in this class, every student plays video games"?
• What is the negation of "For all students in this class, there is a student that is bored"?