Introduction to Propositions and Predicates
Recap of Propositions:
For the purpose of propositional logic, propositions are (almost) atomic black-boxes: They have names and definitions, but their names are just atomic symbols in the logic syntax, and their definitions are outside the logic syntax (usually these are statements in a human language).
The set of all propositions, together with the special symbols ⊤ and ⊥, are the atomic symbols of the syntax of formulas. The connectives will be familiar from their roles as boolean operators: ¬, ∧ ,∨, →, etc. Thus, formulas are recursive trees with connectives as nodes and atomic symbols as leaves.
There are two truth values: 1 and 0. (These are not part of the syntax of formulas.) These can go by other names, commonly "true" and "false". The properties and isomorphisms of Boolean Algebra are all good fun, but for now we can simply take truth values ≜ {1, 0}.
The operator ⟦·⟧ maps formula to truth values; e.g. ⟦⊤⟧=1. (This example is actually part of the definition of ⟦·⟧; ⊤ and ⊥ are given meaning respectively as an atomic tautology and an atomic contradiction.) Although it's often left implicit, it technically has a subscript argument, either an assignment (a set of (proposition, truth value) pairs) or a pair (KB, σ) of a knowlege base (a list of statements of the same form as proposition definitions) and a storage of proposition definitions. In either case, ⟦·⟧₋ can be implemented in terms of lookups and if-then-else statements.
Predicates as Relations:
Consider the category B of truth values, and whatever category S of nouns/subjects.
A single proposition (e.g. p≜"Six is prime.") can be represented as a pair of form
(s∈S, R) where R is a relation over S and truth values
(e.g. (6, IsPrime)).
Rather than writing it as a tuple "(s, R)", we can instead write it as "R(s)", which is the normal notation.
Not any relation R⊆S⨯B will serve in the representation of a proposition! It must be a predicate*, a relation mapping every element of S to exactly one truth value. We use the notation F(s)≜(s,1)∈F, and we deliberately reuse the "¬" symbol to write ¬F(s)≜(s,0)∈F.
* Here we consider only unary predicates. Nullary predicates are the same as propositions. Higher-artiy predicates are also possible. Also, it's not clear if predicates really have to be total functions; if you know know please share!
In the context of a relational representation of predicates and propositions, we can introduce quantifiers.
Universal Quantification: ∀s∈S (P(s)) ≜ {(s,1) | s∈S} ⊆ P
Existential Quantification: ∃s∈S (P(s)) ≜ ∅ ≠ {(s,1) | s∈S} ∩ P
Typically the element-hood restriction on s is omitted, and S is assumed to be the domain of P, but quantification over subsets of the domain is fine. In particular, notice that ∀s∈∅ (P(s)) is a tautology and ∃s∈∅ (P(s)) is a contradiction, regardless of P.
Predicate Logic Syntax:
In the prior section we introduced the notions and notations of predicates and quantifiers; here we introduce a formal syntax for them. While in the prior section we gave predicates meaning as relations, and gave quantifiers meaning in terms of set-theory, here we will treat them as pure symbols. They'll get (possibly contextual) meanings again later when we define a semantics.
Predicates compose into formula just like propositions do. This is what that looks like in BNF:
| patterns | forms | notes |
|---|---|---|
| x | Variables (from some set of identifiers) | |
| P | Predicates (from some set of identifiers) | |
| F | ⊤ ⏐ ⊥ ⏐ P(x) ¬F ⏐ F∧F ⏐ F∨F ⏐ F→F ∀x (F) ⏐ ∃x (F) |
Atomic formula (could also include nullary predicates) Connectives borrowed from Propositional syntax Quantifiers |
Not inherent in the BNF, but fundamental to understanding predicate logic formula, are the ideas of free variables and bound variables. In the formula ∀x (F), we say "x is bound (by that specific quantifier node) in F"; usually we're only interested this if x actually appears in F. (Existential quantifiers bind variables the same way.) If x appears in some F and is not bound by any quantification node in F, then we say "x is free in F". Specifically, an appearence of x may be free; it is possible to have both free and bound appearances of a variable in a single syntax tree.
Re-write rules:
As a first step toward a semantics for predicate-logic syntax trees, we declare an equivalence relation (and therefor equivalence classes). When we discussed propositional logic, we considered the (reflexive, symmetric, transitive) relation a≡b ≜ ⟦a⟧=⟦b⟧ ("for all suitable assignments"). Equivalence here is similar, we just haven't discussed ⟦·⟧ yet.
Usually, we're interested in these equivalence classes because they define our re-write rules for formula: if F≡G then we can use G anywhere we were using F.
Example:
This example from class shows re-write steps from one representation of the required property of a predicate into its usual presentation:
| Justification | formula |
|---|---|
| Start ("P is a predicate") | ∀f ∀f' ∀b ( (f=f' ∧ (f,b)∈P) → ¬∃b' (b≠b' ∧ (f',b')∈P) ) |
| Distribute negation across existence | ∀f ∀f' ∀b ( (f=f' ∧ (f,b)∈P) → ∀b' ¬(b≠b' ∧ (f',b')∈P) ) |
| Distribute negation across conjunction | ∀f ∀f' ∀b ( (f=f' ∧ (f,b)∈P) → ∀b' (b=b' ∨ (f',b')∉P) ) |
| Move quantifier up (no name conflicts!) | ∀f ∀f' ∀b ∀b' ( (f=f' ∧ (f,b)∈P) → (b=b' ∨ (f',b')∉P) ) |
| Definition of implication | ∀f ∀f' ∀b ∀b' (¬(f=f' ∧ (f,b)∈P) ∨ (b=b' ∨ (f',b')∉P) ) |
| Distribute negation across conjunction | ∀f ∀f' ∀b ∀b' ( (f≠f' ∨ (f,b)∉P) ∨ (b=b' ∨ (f',b')∉P) ) |
| Associativity of disjunction | ∀f ∀f' ∀b ∀b' ( ((f,b)∉P ∨ (f',b')∉P) ∨ (f≠f' ∨ b=b') ) |
| Factor out negation from disjunction | ∀f ∀f' ∀b ∀b' (¬((f,b)∈P ∧ (f',b')∈P) ∨ (f≠f' ∨ b=b') ) |
| Definition of implication | ∀f ∀f' ∀b ∀b' ( ((f,b)∈P ∧ (f',b')∈P) → (f≠f' ∨ b=b') ) |
It's not clear that this example actually fits into the syntax as introduced. For the purpose of example, let's fix that. Our goal is to show the same sequence of re-writes, but at every step our formula should be a valid tree in our BNF.
First we'll extend our syntax to allow binary predicates. Here "=" is just an atomic symbol in our grammar/syntax, in the set of all predicates. We'll use it as a binary predicate; if we had semantics or a more advanced grammar we would enforce that. While we intuitively understand it to mean "equality", that idea is not encoded in the syntax. Similarly, I here use the un-italicized P to refer to a binary predicate "is a predicate", who's right-hand argument is a truth value, but neither the semantics of P nor the idea of a truth value is encoded in the syntax.
| patterns | forms | notes |
|---|---|---|
| x | f ⏐ f' ⏐ b ⏐ b' | Variables (that we'll use) |
| P | P ⏐ = | Predicates (that we'll use) |
| F | ⊤ ⏐ ⊥ ⏐ P(x) ⏐ x P x ¬F ⏐ F∧F ⏐ F∨F ⏐ F→F ∀x (F) ⏐ ∃x (F) |
Add binary predicates No change No change |
And then we can proceed:
| Justification | formula |
|---|---|
| Start | ∀f (∀f' (∀b ( (f=f' ∧ f P b) → ¬∃b' ( (¬b=b' ∧ f' P b') ) ) ) ) |
| Distribute negation across existence | ∀f (∀f' (∀b ( (f=f' ∧ f P b) → ∀b' (¬(¬b=b' ∧ f' P b') ) ) ) ) |
| Distribute negation across conjunction | ∀f (∀f' (∀b ( (f=f' ∧ f P b) → ∀b' ( (¬¬b=b' ∨ ¬f' P b') ) ) ) ) |
| Remove double negation | ∀f (∀f' (∀b ( (f=f' ∧ f P b) → ∀b' ( (b=b' ∨ ¬f' P b') ) ) ) ) |
| Move quantifier up | ∀f (∀f' (∀b (∀b' ( (f=f' ∧ f P b) → (b=b' ∨ ¬f' P b') ) ) ) ) |
| Definition of implication | ∀f (∀f' (∀b (∀b' (¬(f=f' ∧ f P b) ∨ (b=b' ∨ ¬f' P b') ) ) ) ) |
| Distribute negation across conjunction | ∀f (∀f' (∀b (∀b' ( (¬f=f' ∨ ¬f P b) ∨ (b=b' ∨ ¬f' P b') ) ) ) ) |
| Associativity of disjunction | ∀f (∀f' (∀b (∀b' ( (¬f P b ∨ ¬f' P b') ∨ (¬f=f' ∨ b=b') ) ) ) ) |
| Factor out negation from disjunction | ∀f (∀f' (∀b (∀b' (¬(f P b ∧ f' P b') ∨ (¬f=f' ∨ b=b') ) ) ) ) |
| Definition of implication | ∀f (∀f' (∀b (∀b' ( (f P b ∧ f' P b') → (¬f=f' ∨ b=b') ) ) ) ) |
| One more step for the fun of it | ∀f (∀f' (∀b (∀b' ( (f P b ∧ f' P b') → (f=f' → b=b') ) ) ) ) |
The double-implication isn't excellent, it might read more clearly if we took a different route from the sixth line:
| Justification | formula |
|---|---|
| Sixth line from above | ∀f (∀f' (∀b (∀b' ( (¬f=f' ∨ ¬f P b) ∨ (b=b' ∨ ¬f' P b') ) ) ) ) |
| Associativity of disjunction | ∀f (∀f' (∀b (∀b' ( ( (¬f P b ∨ ¬f' P b') ∨ ¬f=f' ) ∨ b=b' ) ) ) ) |
| Factor out negation from disjunction | ∀f (∀f' (∀b (∀b' ( (¬(f P b ∧ f' P b') ∨ ¬f=f' ) ∨ b=b' ) ) ) ) |
| Factor out negation from disjunction | ∀f (∀f' (∀b (∀b' (¬( (f P b ∧ f' P b') ∧ f=f' ) ∨ b=b' ) ) ) ) |
| Definition of implication | ∀f (∀f' (∀b (∀b' ( ( (f P b ∧ f' P b') ∧ f=f' ) → b=b' ) ) ) ) |
| Cull parentheses (not technically allowed) | ∀f ∀f' ∀b ∀b' ( ( f P b ∧ f' P b' ∧ f=f' ) → b=b' ) |
Returning to our intuitive understanding of the nature of predicates, we can see that this doesn't entirely capture the refinement of Relations that we want; P is not necessarily Total. We could express totality as
∀f (∃b (f P b)
but that (combined with the above property) would just limit us to functions; predicates must range over truth values. There are a few approaches we could take, probably the easiest of which would be to associate variables bT and bF with "boolean true" and "boolean false". Then we could just say
∀f (f P bT ∨ f P bF)
Which suffices to get us a law-of-excluded-middle in this weird logic-inside-of-logic we've built.