First-Order Logic

Formation Rules and Translations Last Time

We wrapped up zeroth-order logic. We showed how to use the rule of conditional proof with a powerful proof strategy, which is sometimes called indirect proof or proof by contradiction. Today

We expand our system in order to handle talking about individuals and their properties and relations. The expanded system is called first- order logic or predicate logic. Motivation

When we first start looking around for an example of an obviously good argument, many of us think of something like the following:

The argument sketch is a Monty Python sketch. Every Monty Python sketch is funny.

Therefore, the argument sketch is funny. Motivation

But as we have seen, when we formalize arguments like this in zeroth-order logic, we miss something important.

The argument sketch is a Monty Python sketch. Every Monty Python sketch is funny.

Therefore, the argument sketch is funny. Motivation

But as we have seen, when we formalize arguments like this in zeroth-order logic, we miss something important.

P The argument sketch is a Monty Python sketch. Every Monty Python sketch is funny.

Therefore, the argument sketch is funny. Motivation

But as we have seen, when we formalize arguments like this in zeroth-order logic, we miss something important.

P The argument sketch is a Monty Python sketch. Q Every Monty Python sketch is funny.

Therefore, the argument sketch is funny. Motivation

But as we have seen, when we formalize arguments like this in zeroth-order logic, we miss something important.

P The argument sketch is a Monty Python sketch. Q Every Monty Python sketch is funny.

R Therefore, the argument sketch is funny. Motivation

But as we have seen, when we formalize arguments like this in zeroth-order logic, we miss something important.

P The argument sketch is a Monty Python sketch. Q Every Monty Python sketch is funny. ??? R Therefore, the argument sketch is funny. Motivation

Zeroth-order logic does not let us capture enough of the internal structure of sentences in order to show how sentences like the ones in the Monty Python argument are evidentially related.

First-order logic does. First Order Logic Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ... Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ... Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ...

Constants Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ...

Constants Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ...

Constants Variables Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ... Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ...

Predicates: G, H, L, ... Well-Formed Formulas

First-order logic (sometimes called predicate logic) adds three basic elements to our zeroth- order formal language:

Terms: a, b, c, … and x, y, z, ...

Predicates: G, H, L, ...

Quantifiers: ∀ and ∃ Well-Formed Formulas

We add two formation rules to the rules we already had for zeroth-order logic.

As before, the grammatically correct strings are called well-formed formulas. Well-Formed Formulas

New Formation Rules:

3. If Φ is a predicate and α1, α2, …, αn are terms, then Φα1α2 … αn is a well-formed formula. 4. Let ϕ be a well-formed formula, and let β be a variable. Then (∀β)ϕ and (∃β)ϕ are both well-formed formulas. Well-Formed Formulas

Positive Examples:

Ga (Ga → Hbc) (~Lcxy ᴧ Gb) (∀x)Gx (∃x)(∀y)Hxy ((∀x)Gx → P) Well-Formed Formulas

Positive Examples: Negative Examples:

Ga GaH (Ga → Hbc) aLx (~Lcxy ᴧ Gb) (∃ → a) (∀x)Gx ((∀x) ᴧ Gx) (∃x)(∀y)Hxy (∀B)Bc ((∀x)Gx → P) (∃P)(P → Ga) Translations

First-order logic lets us unpack sentences like:

Kermit is green. Kermit is a Muppet. Kermit plays the banjo. First-Order Logic

If P is a predicate symbol and a is a term, then Pa is a well-formed formula.

The basic idea is to think of a simple sentence as composed of a logical subject – a constant term, like “Kermit” – and a logical predicate – like “… is green.” First-Order Logic

Predicates may be interpreted in many different ways. A predicate …

… is a kind of truth function.

… assigns a property to an individual.

… indicates membership in a kind. First-Order Logic

Kermit is green.

Constant Predicate First-Order Logic

Kermit is green. TRUE

Constant Predicate First-Order Logic

Gonzo is green.

Constant Predicate First-Order Logic

Gonzo is green. FALSE

Constant Predicate Quantifiers

Add quantifiers and we can also represent sentences like: All Muppets are silly. Some Muppets wear hats. Quantifiers

Suppose (∀β)ϕ and (∃β)ϕ are well-formed formulas. Then we know that β is a variable term. Occurrences of β in ϕ are bound by the quantifiers.

The variable β is also bound in the quantifier expressions (∀β) and (∃β). Quantifiers

If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence. Quantifiers

If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence.

P (Q v R) Mxy Ga (∀x)Gx (∃x)(Mxy ᴧ Gy) Quantifiers

If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence.

P (Q v R) Mxy Which of these are sentences? Ga (∀x)Gx (∃x)(Mxy ᴧ Gy) Quantifiers

Previously, we had no need to distinguish between well-formed formulas and sentences because every well-formed formula stood for a sentence in ordinary language.

But a well-formed formula that contains a free variable does not stand for any sentence in ordinary language. Quantifiers

What do the quantifiers do though? Quantifiers let us talk about all of the constant terms without needing to actually list them all.

That is convenient, since there are infinitely many constant terms. Quantifiers

What do the quantifiers do though? Quantifiers let us talk about all of the constant terms without needing to actually list them all.

That is convenient, since there are infinitely many constant terms.

We would get tired trying to write them all down! Quantifiers

Quantified formulas tell us something about all of the constant terms.

The formula (∀β)ϕ says that for every constant c, if you substitute c for β everywhere β appears in ϕ, then ϕ will come out true. Quantifiers

Let G be the predicate “… is green.” Then the formula (∀x)Gx says that everything is green.

The formula (∀x)Gx says that for every constant c, if you substitute c for x in Gx, then Gx will come out true. Quantifiers

Let G be the predicate “… is green.” Then the formula (∀x)Gx says that everything is green.

In other words, (∀x)Gx says that for every constant c, Gc is true.

The universal quantifier works like an infinite conjunction. Quantifiers

Again, quantified formulas tell us something about all of the constant terms.

The formula (∃β)ϕ says that there is at least one constant c such that if you substitute c for β everywhere it appears in ϕ, then ϕ will be true. Quantifiers

Let H be the predicate “… is happy.” Then the formula (∃x)Hx says that something is happy.

The formula (∃x)Hx says that there is a constant c such that Hc true. Quantifiers

Let H be the predicate “… is happy.” Then the formula (∃x)Hx says that something is happy.

The formula (∃x)Hx says that there is a constant c such that Hc true.

The existential quantifier works like an infinite disjunction. Categoricals

We’ve seen that if G is the predicate “… is green,” then the formula (∀x)Gx says that everything is green.

What if we want to say that every green thing is colorful? Categoricals

We’ve seen that if G is the predicate “… is green,” then the formula (∀x)Gx says that everything is green.

What if we want to say that every green thing is colorful?

That would be a categorical sentence! Categoricals

There are four kinds of categorical sentence:

All Ss are Ps. All Muppets are silly.

Some Ss are Ps. Some Muppets wear hats.

No Ss are Ps. No Muppets wear hats.

Some Ss are not Ps. Some Muppets are not silly. Categoricals

There are four kinds of categorical sentence:

All Ss are Ps. All Muppets are silly.

Some Ss are Ps. Some Muppets wear hats.

No Ss are Ps. No Muppets wear hats.

Some Ss are not Ps. Some Muppets are not silly. Categoricals

There are four kinds of categorical sentence:

All Ss are Ps. All Muppets are silly.

Some Ss are Ps. Some Muppets wear hats.

No Ss are Ps. No Muppets wear hats.

Some Ss are not Ps. Some Muppets are not silly. Categoricals

Let’s translate the categorical sentence, “All Muppets are silly.”

Start with the predicates:

M = “… is a Muppet”

S = “… is silly” Categoricals

Suppose b stands for Beaker. Categoricals

Suppose b stands for Beaker. Categoricals

Suppose b stands for Beaker. Then … Categoricals

Suppose b stands for Beaker. Then …

Mb = “Beaker is a Muppet.”

Sb = “Beaker is silly.” Categoricals

Suppose b stands for Beaker. Then …

Mb = “Beaker is a Muppet.”

Sb = “Beaker is silly.”

(Mb ᴧ Sb) = “Beaker is a Muppet and Beaker is silly.” Categoricals

But we want to say that every Muppet is silly. Categoricals

But we want to say that every Muppet is silly. Categoricals

But we want to say that every Muppet is silly. Categoricals

But we want to say that every Muppet is silly. Categoricals

But we want to say that every Muppet is silly. Categoricals

But we want to say that every Muppet is silly.

How do we make sure to cover all of the Muppets? Categoricals

We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). Categoricals

We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb).

(Mh ᴧ Sh) Categoricals

We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb).

(Mf ᴧ Sf) (Mh ᴧ Sh) Categoricals

We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb).

(Mf ᴧ Sf) (Mh ᴧ Sh) (Mk ᴧ Sk) Categoricals

Could we use the formula (∀x)(Mx ᴧ Sx) to translate the categorical “All Muppets are silly”? Categoricals

Could we use the formula (∀x)(Mx ᴧ Sx) to translate the categorical “All Muppets are silly”?

NO Categoricals

Could we use the formula (∀x)(Mx ᴧ Sx) to translate the categorical “All Muppets are silly”?

NO Why not? Categoricals

The formula (∀x)(Mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. Categoricals

The formula (∀x)(Mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists.

But lots of things aren’t Muppets. Categoricals

The formula (∀x)(Mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists.

And some things aren’t silly either. Categoricals

The formula (∀x)(Mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists.

And some things aren’t silly either. Categoricals

So what do we mean when we say that all Muppets are silly? Categoricals

So what do we mean when we say that all Muppets are silly?

For any object you want to pick, if the object is a Muppet, then it is silly. Categoricals

All Muppets are silly.

(∀x)(Mx → Sx) Categoricals

All Muppets are silly.

(∀x)(Mx → Sx)

For any x you want Categoricals

All Muppets are silly.

(∀x)(Mx → Sx)

For any x you want x is a Muppet Categoricals

All Muppets are silly.

(∀x)(Mx → Sx)

For any x you want only if x is a Muppet Categoricals

All Muppets are silly.

(∀x)(Mx → Sx) x is silly For any x you want only if x is a Muppet Categoricals

All Muppets are silly.

(∀x)(Mx → Sx) x is silly For any x you want only if x is a Muppet Categoricals

All Muppets are silly.

(Ma → Sa) ᴧ One way to think of a (Mb → Sb) ᴧ universal quantifier is as a (Mc → Sc) ᴧ big conjunction. : Next Time

We will talk about the class of particular categorical sentences, and then we’ll talk about relations.