Lecture Notes 8: Monadic Predicate Logic Translations

The Language of Predicate Logic

The Limitations of Sentential Logic

In sentential logic, a letter represents an entire atomic statement.

It cannot represent common features between atomic statements.

Examples of common features in atomic statements:

1. Kanye West is an egotist.
   shares features in common with:
2. Kanye West is a musician.
   (Same subject, different predicate.)
3. Lindsay Lohan is an egotist.
   (Same predicate, different subject.)
4. Everyone is an egotist.
   (Same predicate, general subject.)
5. Someone is an egotist.
   (Same predicate, indefinite subject.)

Sentential logic treats these all as simple statements (K, M, L, E, S, etc.), and so loses any connection between them.

A valid argument not captured by sentential logic:

Kanye West is an egotist.
Therefore, someone is an egotist.

This argument is valid.

However the argument:
K /∴ S
is obviously invalid in sentential logic.

We now introduce a new style of symbolic logic, called predicate logic.

Subjects and Predicates

In predicate logic …

A subject is a specific individual person, place or thing. In predicate logic, lowercase letters (except x, y and z) represent subjects.

A predicate is something that can be said about one or more subjects. In predicate logic, uppercase letters are predicates.

Example subjects:

English	becomes
Amsterdam	a
Kevin	k
Britney Spears	b
the Pope	p
five (or 5)	f

Example predicates:

English	becomes
_______ is hungry	H
_______ plays tennis	T
_______ dawdles	D
_______ loves …	L
_______ is prime	P

Predicate Logic Statements

We put these together, capital letters first, to form statements.

Example Statements:

Kevin is hungry.

Translation: Hk

Britney Spears is hungry.

Translation: Hb

Britney Spears plays tennis.

Translation: Tb

5 is prime.

Translation: Pf

The pope loves Britney Spears.

Translation: Lpb

Monadic and Polyadic Predicates

Monadic predicates are those that are apply to just one thing on its own.

Polyadic or relational predicates are those that are applied to more than one thing.

With polyadic predicates, order usually matters.

Kanye loves Beyoncé.

Translation: Lkb

Beyoncé loves Kanye.

Translation: Lbk

In principle, predicates can apply to any number of things:

Using “B” to mean “_______ is between _______ and _______”:

Seven is between five and nine.

Translation: Bsfn

“And” with Polyadic Predicates

When considering Sentential Logic, we saw that sometimes when “and” occurs between two names, it is possible to translate with “&”.

When “and” means “&”:

Chandler and Monica are hungry.

This really means:

Chandler is hungry and Monica is hungry.

Translation: Hc & Hm

However, not always.

When “and” doesn’t mean “&”:

Chandler and Monica are engaged.

This does not mean the same as:

Chandler is engaged and Monica is engaged.

Instead, it means:

Chandler is engaged to Monica.

Translation: Ecm

(I.e., it uses a polyadic predicate.)

Molecular Statements

∨, →, ↔, & and ~ are used just as before:

Peter is a Republican only if he is ignorant.

Translation: Rp → Ip

Unless Peter plays tennis, then Alison loves him only if he does not dawdle.

Translation: ~Tp → (Lap → ~Dp)

If neither Ross nor Phoebe are hungry, then Monica will not cook.

Translation: ~(Hr ∨ Hp) → ~Cm  or (~Hr & ~Hp) → ~Cm

“~” can be used to translate prefixes like “non”, “un”, “in” and “im”.

Alison is unkind if and only if Peter is impolite.

Translation: ~Ka ↔ ~Pp

Quantifiers

General Statements and Variables

Not all statements are about specific individuals. Some are about all or some unspecified members of a group.

All dentists floss.
Some politicians are alcoholics.
Everyone loves Björk.

To capture these we use variables.

In algebra, variables stand for unspecified numbers. In predicate logic, variables can stand for anything.

Variables are signs that take the place of subjects, and can represent any possible subject relevant to the domain under discussion. In predicate logic, the lowercase letters “x”, “y” and “z” are used as variables.

They replace lowercase letters “a”, “b” for specific particular subjects in predicate logic statements.

If “Hk” means “Kevin is hungry”, then “Hx” means something like “it is hungry”, where it remains ambiguous who or what “it” is.

Statements containing variables are ambiguous as is.

We remove the ambiguity by prefixing a quantifier and the variable to which the quantifier applies to an ambiguous statement.

The Universal Quantifier, ∀, written with a variable means For every value of the variable …

The Existential Quantifier, ∃, written with a variable means For at least one value of the variable …

Simple quantified statements:

∃xHx

means

For at least one value of x, x is hungry.
I.e., Something (or someone) is hungry.

∀xBx

means

For every value of x, x is beautiful.
I.e., Everything (or everyone) is beautiful.

Quantified Variables and Their Values

“∀xBx” means that ALL of

Ba	Amherst is beautiful.
Bb	Britney is beautiful.
Bc	Chicago is beautiful.
Bd	Detroit is beautiful.
Be	Eminem is beautiful.
Bf	Five is beautiful.
Bg	Ganymede is beautiful.
⋮	(and so on for everything there is)

must be true.

If even one is false, “∀xBx” is false.

“∃xBx” requires only one or more of these to be true.

It’s false only if all of the above are false.

The Four Basic Forms of Quantified Statements

Universal Affirmative Statements

We don’t normally speak of about just everything, but every member of some group. We handle these by applying ∀ to a conditional.

Example universal affirmatives:

All frogs are green.
Every frog is green.
Any frog is green.
Each frog is green.

These all become: ∀x(Fx → Gx)

I.e., for every value of x, if x is a frog, x is green.

Plural and count noun phrases are translated with uppercase letters!

“∀x(Fx → Gx)” means that ALL of these are true:

Fa → Ga	If Amherst is a frog, then Amherst is green. (F, F)
Fb → Gb	If Britney is a frog, then Britney is green. (F, F)
Fc → Gc	If Chicago is a frog, then Chicago is green. (F, F)
⋮	⋮
Fk → Gk	If Kermit is a frog, Kermit is green. (T, T)
Fl → Gl	If Leaper is a frog, Leaper is green. (T, ?)
⋮	⋮
Fs → Gs	If Shrek is a frog, Shrek is green. (F, T)
⋮	⋮

The variable is not just restricted to frogs. But when the if-part is false, the conditional is true anyway.

Particular Affirmative Statements

Any statement about some (unspecified) member(s) of a group is translated with “∃” with a variable applied to a conjunction.

Example particular affirmatives:

Some politician is honest.
Some politicians are honest.
A few politicians are honest.
At least one politician is honest.

All become: ∃x(Px & Hx)

“Some” in English may suggest more than one, but we assume it doesn’t require it.

“∃”-statements can be interpreted as asserting existence: “There are honest politicians”, or “Honest politicians exist”.

Universal Negative Statements

A statement that something is true of no things in a certain group can be seen in one of two ways.

Example universal negatives:

No politician is ethical.
No politicians are ethical.

These can be paraphrased as

For all x, if x is a politician, then x is NOT ethical.

I.e., ∀x(Px → ~Ex)

Or as:

It is NOT the case that there is an x such that x is a politician and x is ethical.

I.e., ~∃x(Px & Ex)

These two formulations are logically equivalent.

Particular Negative Statements

These deny something of some members of a group.

Example particular negatives:

Some politician is not ethical.
Some politicians are not ethical.
Some politicians are unethical.

This can be paraphrased as:

There exists at least one x such that x is politician and x is not ethical.

I.e., ∃x(Px & ~Ex)

Or as: Not all politicians are ethical.
I.e., ~∀x(Px → Ex)

The Square of Opposition

Universal affirmatives (All Fs are Gs) and particular negatives (Some Fs are not Gs) are true opposites.

Particular affirmatives (Some Fs are Gs) and universal negatives (No Fs are Gs) are true opposites.

The Simple Square of Opposition

~∀xFx is not equivalent with ∀x~Fx, but it is equivalent with ∃x~Fx.

Similarly, ~∃xFx is equivalent not with ∃x~Fx, but with ∀x~Fx.

It’s important to watch where you place negations and parentheses.

Some Example Translations

Try them yourself.

Translations in Monadic Predicate Logic

Combined Predicates

A quantified statement can involve more than two predicates.

Often they should be joined with “&”. Consider:

Every rockstar is popular and sexy.

Translation: ∀x[Rx → (Px & Sx)]

These are often shortened to adjectives modifying nouns.

All popular rockstars are sexy.

Translation: ∀x[(Px & Rx) → Sx]

Some carrots are orange vegetables.

Translation: ∃x[Cx & (Ox & Vx)]

Complex predicates are put together in parentheses, and are not broken up, even when negations are involved.

No hamsters are endangered animals.

Translation: ∀x[Hx → ~(Ex & Ax)]  or ~∃x[Hx & (Ex & Ax)]

Some intelligent students are not pathetic geeks.

Translation: ∃x[(Ix & Sx) & ~(Px & Gx)]

Things can’t always be done this way.

A “computer programmer” isn’t someone who is a computer and is a programmer, but someone who programs computers.

A “basketball player” isn’t someone who is a basketball and a player, but someone who plays basketball, etc.

Disjunctive Combined Predicates

Sometimes disjunctions (“∨”-statements) are used to translate combined predicates.

Anything that’s plaid or striped is ugly.

Translation: ∀x[(Px ∨ Sx) → Ux]

All athletes are either quick or strong.

Translation: ∀x[Ax → (Qx ∨ Sx)]

Or with negations …

Some professors are not either quick or strong.

Translation: ∃x[Px & ~(Qx ∨ Sx)]

Hidden Disjunctive Combined Predicates

Some statements that use the word “and” in English can actually be translated with “∨”.

Consider:

All doctors and lawyers are rich.

This is NOT: ∀x[(Dx & Lx) → Rx]

That would mean “everything that is a doctor and a lawyer is rich”.

Instead make it: ∀x[(Dx ∨ Lx) → Rx]

This means “everything that is a doctor or a lawyer is rich.”

So where did the “and” come from in the English version?

It is equivalent with a conjunction of universal affirmatives.

An alternative: ∀x(Dx → Rx) & ∀x(Lx → Rx)

Only

What does it mean to say that only As are Bs?

On one way of thinking of it, it means that

No non-As are Bs.

On another, it is the converse of “all”: All Bs are As.

Only cats purr.

Translation: ∀x(~Cx → ~Px)  or ~∃x(Px & ~Cx)  or ∀x(Px → Cx)

But NOT: ∀x(Cx → Px) (Every cat purrs.)

Ambiguities Involving “Only”

“Only” creates ambiguities when complex predicates are involved.

Only happy cats purr.

This might be read as just about cats:

Among cats, only the happy ones purr.

Translation: ∀x[(Cx & Px) → Hx]  or ~∃x[(Cx & Px) & ~Hx]

 or ∀x[Cx → (~Hx → ~Px)]  or ∀x[Cx → (Px → Hx)]

Or it might be about everything:
The only things that purr at all are happy cats.

Translation: ∀x[Px → (Hx & Cx)]  or ∀x[~(Hx & Cx) → ~Px]

 or ~∃x[Px & ~(Hx & Cx)]

The Only

Note: “the only” does not mean the same as “only”.

Usually, “only A are B” means the same as “the only B are A”.

The only likable teachers are philosophers.

Translation: ∀x[(Lx & Tx) → Px]   or ∀x[~Px → ~(Lx & Tx)]

  or ~∃x[~Px & (Lx & Tx)]

“Only” is the converse of “all”. “The only” is the converse of “only”. Therefore, “all” and “the only” mean the same thing!

The

Normally, statements using “the” are about some specific thing, and may be translated with constant subject letters.

Other times “the”-statements are really about every member of a group.

Contrasting Cases of “The”:

The dog ran away.

Translation: Rd

The dolphin is a mammal.

Translation: ∀x(Dx → Mx)

Missing Quantifiers

Sometimes an quantifier word is simply left off in an English sentence.

These can mean different things in different contexts.

These can mean the same as “all”, as “some” or even “only” statements.

Frogs are amphibians.

Translation: ∀x(Fx → Ax) (Missing “all”.)

Students work at the library.

Translation: ∃x(Sx & Wxl) (Missing “some”.)

Employees are allowed in.

Translation: ∀x(Ax → Ex) (Missing “only”.)

All and Only

Rarely, “all and only” occurs as a single phrase.

This can be translated with “∀” and “↔”.

All and only bachelors are allowed in.

Translation: ∀x(Bx ↔ Ax)

Quantified Statements Joined Together

Entire quantified statements can be joined together with logical connectives like “and”, “or”, “if … then … ”, etc.

If nothing is permanent, then everything is illusory.

Translation: ~∃xPx → ∀xIx

No Republicans are environmentalists but some Democrats are environmentalists.

Translation: ~∃x(Rx & Ex) & ∃x(Dx & Ex)

If some students are confused, then all students are.

Translation: ∃x(Sx & Cx) → ∀x(Sx → Cx)

If nothing is permanent and nothing is good, then all religions are fictional.

Translation: (~∃xPx & ~∃xGx) → ∀x(Rx → Fx)

Unless the host cooks or some local restaurant delivers, no partygoer will be happy.

Translation: ~{Ch ∨ ∃x[(Lx & Rx) & Dx]} → ~∃x(Px & Hx)

Additional Example Translations

Try them yourself.