Lecture Notes 9: Polyadic Predicate Logic Translations
Polyadic Predicates Revisited
Recall:
Polyadic predicates are those applied to more than one subject.
These are written with two or more names/variables, and the order usually matters.
Atomic Statements with polyadic predicates:
Angelina loves Brad.
Translation:
Lab
Brad loves Angelina.
Translation:
Lba
Normally we match the English order … except when passive voice is used (i.e., “ … is _______ed by … ”).
Brad is loved by Angelina.
Translation:
Lab
(same as “Angelina loves Brad”)
Quantifiers with Polyadic Predicates
Even with a single predicate, the possibilities are many.
| English | Translations |
|---|---|
| Everything loves Fergie. | ∀xLxf |
| Something loves Fergie. | ∃xLxf |
| Fergie loves everything. | ∀xLfx |
| Fergie loves something. | ∃xLfx |
| Nothing loves Fergie. | ~∃xLxf / ∀x~Lxf |
| Fergie loves nothing. | ~∃xLfx / ∀x~Lfx |
| Something doesn’t love Fergie. | ∃x~Lxf / ~∀xLxf |
| Fergie does not love something. | ∃x~Lfx / ~∀xLfx |
| Not everything loves Fergie. | ~∀xLxf |
| Everything loves itself. | ∀xLxx |
| etc. |
Reflexive Constructions
Reflexive constructions are those involving something having a relationship to itself (or herself, himself, themselves, etc.)
Reflexive constructions are translated by repeating the same subject letter or variable.
Fergie loves herself.
Translation:
Lff
(“Fergie loves Fergie.”)
Every singer who respects them/him/herself is happy.
Translation:
∀x[(Sx & Rxx) → Hx]
The “self-” prefix may be treated similarly.
No self-respecting musician admires Sanjaya.
Translation:
∀x[(Mx & Rxx) → ~Axs]
Some additional examples:
All men love Fergie.
Translation:
∀x(Mx → Lxf)
No beautiful women love Diddy.
Translation:
∀x[(Bx & Wx) → ~Lxd]
or
~∃x[(Bx & Wx) & Lxd]
All men love themselves.
Translation:
∀x(Mx → Lxx)
Only morons love Diddy.
Translation:
∀x(~Mx → ~Lxd)
or
∀x(Lxd → Mx)
or
~∃x(~Mx & Lxd)
Something Diddy loves loves itself.
Translation:
∃x(Ldx & Lxx)
Quantifier Movement
When the quantifier word occurs in the middle of the statement, try rewording it with the quantifier out in front.
Usually, this will result in a switch from active to passive voice
Diddy loves all guns.
This is the same as:
All guns are loved by Diddy.
Translation:
∀x(Gx → Ldx)
Angelina loves nothing that loves Brad.
This is the same as:
Nothing that loves Brad is loved by Angelina.
Translation:
∀x(Lxb → ~Lax)
or
~∃x(Lxb & Lax)
Multiple Quantification with Polyadic Predicates
Quantifiers can be applied to both terms of a polyadic predicate.
When multiple quantifiers are applied to the same part of the same statement, different variables (e.g., “x” and “y”) must be used.
The order of the quantifiers matters, and often points to subtle distinctions not often unambiguously marked in English.
Everything loves something.
This usually means:
∀x∃yLxy
I.e., for everything, there is something or other that it loves—
each of
∃yLay,
∃yLby,
∃yLcy,
… is true.
But it could mean instead:
∃y∀xLxy
I.e., there is some one thing in particular that everything loves—
one of
∀xLxa,
∀xLxb,
∀xLxc,
… is true.
Varieties of Multiple Quantification
The possibilities are endless! Here are just a few::
| Something loves something. | ∃x∃yLxy |
| Everything loves everything. | ∀x∀yLxy |
| Something loves everything.* | ∃x∀yLxy |
| Everything is loved by everything. | ∀x∀yLyx |
| Something is loved by something. | ∃x∃yLyx |
| Something is loved by everything.* | ∃x∀yLyx |
| Everything is loved by something.* | ∀x∃yLyx |
| Something loves nothing. | ∃x~∃yLxy / ∃x∀y~Lxy |
| Nothing loves everything. | ∀x~∀yLxy / ~∃x∀yLxy |
| Something does not love everything.* | ∃x~∀yLxy |
| Nothing loves anything. | ∀x∀y~Lxy / ~∃x∃yLxy |
Those marked with * are possibly ambiguous; I have given the most likely meaning.
More Complicated Forms of Multiple Quantification
These are perhaps best seen by example:
Every rockstar loves something.
Translation:
∀x(Rx → ∃yLxy)
Something is loved by every rockstar.
Translation:
∃x∀y(Ry → Lyx)
Every man loves every woman.
Translation:
∀x[Mx → ∀y(Wy → Lxy)]
Every beautiful woman is loved by some man (or other).
Translation:
∀x[(Bx & Wx) → ∃y(My & Lyx)]
Some men are not loved by every woman.
Translation:
∃x[Mx & ~∀y(Wy → Lyx)]
No beautiful women love every man.
Translation:
∀x[(Bx & Wx) → ~∀y(My → Lxy)]
or
~∃x[(Bx & Wx) & ∀y(My → Lxy)]
Restrictive Clauses
The words “that”, “which”, “who”, and “whom” are used to introduce restrictive clauses.
These operate much like complex predicates translated with “&”.
Example that/which/who/whom clauses:
All rockstars that dance are sexy.
= All dancing rockstars are sexy.
Translation:
∀x[(Rx & Dx) → Sx]
Some rockstars that love Diddy dance.
= Some Diddy-loving rockstars dance.
Translation:
∃x[(Rx & Lxd) & Dx]
When applied to “something” or “everything”, they take the place of the predicate they’d otherwise restrict.
Examples:
Everything that loves Fergie dances.
= Every Fergie-lover dances.
Translation:
∀x(Lxf → Dx)
Something that Diddy owns is not a gun.
= Some Diddy-owned-thing is not a gun.
Translation:
∃x(Odx & ~Gx)
Quantified Restrictive Clauses
These can get very complex.
In such cases, it is usually best to break the problem into steps.
Everything that loves all women loves Fergie.
Overall, this is an all-statement, which gets translated with “∀” along with “→”.
Mixing English and symbolic logic, we can do our first step as:
∀x[x loves all women → x loves Fergie]
Now “x loves all women” becomes: ∀y(Wy → Lxy)
And “x loves Fergie” becomes: Lxf.
Together we get: ∀x[∀y(Wy → Lxy) → Lxf].
More examples:
Some rockstars whom all women love dance.
STEP 1:
∃x{x is a rockstar whom all women love & x dances}
STEP 2:
∃x{[Rx & all women love x] & Dx}
STEP 3:
∃x{[Rx & ∀y(Wy → Lyx)] & Dx}
Some rockstars who own some guns are loved by no women.
STEP 1:
∃x{x is rockstar who owns some guns & x is loved by no women}
STEP 2:
∃x{[x is rockstar & some guns are owned by x] & no women love x}
STEP 3:
∃x{[Rx & some guns are owned by x] & ∀y(Wy → ~Lyx)}
STEP 4:
∃x{[Rx & ∃y(Gy & Oxy)] & ∀y(Wy → ~Lyx)}
Every jabberwock that only outgribes those things that every bandernatch outgribes gimbles no toves that gimble themselves.
STEP 1:
∀x{x is a jabberwock that only outgribes those things that every bandersnatch outgribes → x gimbles no toves that gimble themselves}
STEP 2:
∀x{[Jx & the only things that x outgribes are things that every bandersnatch outgribes] → no toves that gimble themselves are gimbled by x}
STEP 3:
∀x{[Jx & ∀y(x outgribes y → every bandernatch outgribes y)] → ∀y[y is a tove that gimbles itself → ~(y is gimbled by x)]}
STEP 4:
∀x{[Jx & ∀y(Oxy → ∀z(z is a bandersnatch → z outgribes y))] → ∀y[(Ty & Gyy) → ~Gxy]}
STEP 5:
∀x{[Jx & ∀y(Oxy → ∀z(Bz → Ozy))] → ∀y[(Ty & Gyy) → ~Gxy]}
Translations Review
Lower case letters (subjects/names) stand for specific individual persons, places or things.
Do not use them for plural or count noun phrases!
Upper case letters (predicates) stand for properties, traits and characteristics of specific things.
We write them before their subjects.
Monadic predicates stand for a property a thing has on its own.
Polyadic predicates stand for a relation between multiple things, and are followed by multiple subjects.
| Tk | Kevin is tall. |
| Lfa | France is larger than Amherst. |
| Pn | Neptune is a planet. |
| Lnf | Neptune is larger than France. |
| Df | France is democratic. |
| Age | Gabrielle admires (X)ena. |
Review: Connectives vs. Polyadic Predicates
“and”/“or” between names and predicates is sometimes translated with “&”/“∨” and sometimes with a polyadic predicate.
Think about whether it can be broken into two separate thoughts.
Examples:
France and Britain are democratic.
Translation:
Df & Db
Either Orion or Callisto is a constellation.
Translation:
Co ∨ Cc
Sanjukta is intelligent and talented.
Translation:
Is & Ts
Mercury and Pluto are similar.
Translation:
Smp
Callisto orbits either Jupiter or Neptune.
Translation:
Ocj ∨ Ocn
Warning: Logical connectives “&”, “∨”, etc. must always be flanked by things that can be true or false.
Never use them simply between two subjects or predicates. The first example cannot be written as D(f & b), etc.
Review: Quantifiers
Quantifiers in English are words like “all”, “some”, “only”, etc.
In predicate logic, we have two quantifiers.
∀x… — for all values of (variable) x …
∃x… — there is at least one value of (variable) x …
The simplest use of the quantifiers is to translate statements about “everything”, “something”, “nothing”, etc.
Everything is good.
Translation:
∀xGx
Something is infinite.
Translation:
∃xIx
Nothing is free.
Translation:
∀x~Fx
or
~∃xFx
Not everything is good.
Translation:
~∀xGx
or
∃x~Gx
Everything is either good or bad.
Translation:
∀x(Gx ∨ Bx)
Review: Categorical Judgments
More often we speak about all/some members of a group.
| All … are _______ | = ∀x( … x … → ___ x ___) |
| Some … are _______ | = ∃x( … x … & ___x___) |
| No … are _______ | = ∀x( … x … → ~___x___)
or ~∃x( … x … & ___x___) |
| Some … are not _______ | = ∃x( … x … & ~___x___) |
∀ almost always goes with →
∃ almost always goes with &
Be sure not to use the wrong combination!
All birds fly.
Translation:
∀x(Bx → Fx)
NOT: ∀x(Bx & Fx) (Everything is a bird that flies.)
Review: Combined Predicates
Adjectives modifying nouns are translated using an “&”.
Something similar happens with “that”, “who”, “whom” clauses, etc.
Some democratic countries are socialist.
Translation:
∃x[(Dx & Cx) & Sx]
Some countries are not socialist democracies.
Translation:
∃x[Cx & ~(Dx & Sx)]
No planets that rotate are inhabitable.
Translation:
∀x[(Px & Rx) → ~Ix]
Sometimes combined predicates require “∨”, surprisingly.
All trucks and motorcycles are prohibited.
Translation:
∀x[(Tx ∨ Mx) → Px]
Not: ∀x[(Tx & Mx) → Px]