CSE 311 Lecture 08: Inference Rules and Proofs for Predicate Logic

Topics

Propositional logic proofs

A brief review of Lecture 07.

A quick look at predicate logic proofs

Inference rules for quantifiers and a “hello” world example.

An in-depth look at predicate logic proofs

Understanding rules for quantifiers through more advanced examples.

Inference rules for propositional logic

Two rules per binary connective: to eliminate and introduce it.

Direct Proof Rule is special: not like the other rules.

Proving implications with the direct proof rule

Direct Proof Rule $\rule{A\implies B}{A\rightarrow B}$

The premise $A\implies B$ means “Given $A$, we can prove $B$.”

So the direct proof rule says that if we have such a proof, then we can conclude that $A\rightarrow B$ is true.

Why does the direct proof rule work?

Inference rules let us derive facts that are implied by the existing facts.

So, for every rule $\rule{P}{Q}$, $P \rightarrow Q$ is a tautology ($P \rightarrow Q \equiv \T$).

The proof $P\implies Q$ shows that $P \rightarrow Q$ is a tautology ($P \rightarrow Q\equiv \T$), since it just a series of implications that we know are tautologies.

So the Direct Proof Rule $\small{\rule{A\implies B}{A\rightarrow B}}$ says that we can add $A\rightarrow B$ to our set of facts, if we can show that $A\rightarrow B$ is a tautology.

One way to show that $A \rightarrow B\equiv \T$ is by writing a subproof, using all the facts we have inferred up to that point.

An example proof

Prove $((p\rightarrow q)\wedge(q\rightarrow r))\rightarrow(p\rightarrow r)$.

Which facts can be used in a subproof?

Which facts can be used in a subproof? A mnemonic

A general proof strategy

1. Look at the rules for introducing connectives to see how you would build up the formula you want to prove from pieces of what is given.

2. Use the rules for eliminating connectives to break down the given formulas so that you get the pieces need for 1

3. Write the proof beginning with what you figured out for 2 followed by 1.

Inference rules for quantifiers

Predicate logic proofs can use …

Predicate logic inference rules

Applied to whole formulas only.

Predicate logic equivalences

Even on subformulas.

Propositional logic inference rules

Applied to whole formulas only.

Propositional logic equivalences

Even on subformulas.

A “hello world” proof

Prove $(\forall x. P(x)) \rightarrow (\exists x. P(x))$.

Advanced proofs: considering domain semantics

So far, we have treated the predicate definitions as black boxes, and the domain of discourse as a set of objects with no additional properties.

In practice, we want to prove theorems for specific domains, and use the properties of those domains in our proofs.

For example, the set of integers is equipped with the operators $+,\cdot,=$.

We can use these operators in our predicates (below) and proofs (next):

A not so odd example

Prove that there is an even number: $\exists x. \text{Even}(x)$.

A prime example

Prove that there is an even prime number: $\exists x. \text{Even}(x) \wedge \text{Prime}(x)$.

We use a black-box definition of Prime because the proof won’t need to break it down further.

An equal example

Prove that $\forall y. \exists z. y = z$ follows from $\forall x. x = x$.

A square example

Prove that the square of every even number is even: $\forall x. \text{Even}(x) \rightarrow \text{Even}(x^2)$.

Why list dependencies? To avoid incorrect proofs.

Over the integer domain: $\forall x. \exists y. y\geq x$ is True but $\exists y. \forall x. y\geq x$ is False.