SYMBOLIZATION TIPS

Predicate Logic: Relational Predicates

Predicate Logic: Multiple Quantifiers

Order of quantifiers. If an expression contains more than one quantifier, does it matter in what order they appear? If it does matter, which quantifier goes first and when?

When the quantifiers are of the same type, then their order does not matter. For example:

- (∀x)(∀y)(Axy → Bxy) : : (∀y)(∀x)(Axy → Bxy)
- (∃x)(∃y)(Axy · Bxy) : : (∃y)(∃x)(Axy · Bxy)
But when the quantifiers are of different types, then their order does matter. We will soon see the rule telling us which order to use. But first a note on what this rule corresponds to in English:

When quantifiers are of different types, their order matters. Follow this rule: when order matters, the first quantifier quantifies the subject of the sentence; the others quantify the objects of the verb.

For example, let our universe of discourse be human beings, and let Lxy mean x loves y.

- (∀x)(∃y)Lxy = Everyone loves someone (not necessarily the same person); everyone has a beloved.
- (∃y)(∀x)Lxy = Somebody (a particular person) is loved by everyone; everyone has the same beloved.
- (∀x)(∃y)Lyx = Everyone is the beloved of someone (not necessarily the same person); everyone is loved; everyone has a lover.
- (∃y)(∀x)Lyx = Somebody (a particular person) loves everyone; everyone is the beloved of the same person; everyone has the same lover.
- Note: The quantifiers governs only the variable in the variable slot captured by the quantifier. (∃y)(∀x)Lyx says exactly the same thing as (∃x)(∀y)Lxy
- Note: A quantifier can bind variables in multiple slots within a relational predicate. (∃x)Lxx says that someone is a self lover, that is, there is at least one narcissist among us.
For the sake of further analysis, take the second example above, (∃y)(∀x)Lxy: Somebody (a particular person) is loved by everyone; everyone has the same beloved. By putting the "y" quantifier first, we are making the individual in the "y" position, the beloved, not the lover, the subject of the sentence. We now have a choice. We can make the beloved the subject of the sentence, which requires the passive voice, "Someone is loved by everyone." Or we can shun the passive voice by reassigning the beloved to the object position, "Everyone loves the same person." So we can say that the "y" individual is only the subject of the sentence if we are willing to use the passive voice.

Here are 10 examples to help you distinguish the differences in meaning between similar looking quantified expressions. Let Rxy mean that x bears R to y.

1 Everything bears R to everything. (∀x)(∀y)Rxy 2 Everything is borne R by everything. (∀y)(∀x)Rxy 3 Something bears R to something. (∃x)(∃y)Rxy 4 Something is borne R by something. (∃y)(∃x)Rxy 5 Nothing bears R to anything. (∀x)(∀y)~Rxy 6 Nothing is borne R by anything. (∀y)(∀x)~Rxy 7 Everything bears R to something.

(Here "something" = "something or other".)(∀x)(∃y)Rxy 8 Something is borne R by everything.

(Here "something" = "something in particular".)(∃y)(∀x)Rxy 9 Everything is borne R by something.

(Here "something" = "something or other".)(∀y)(∃x)Rxy 10 Something bears R to everything.

(Here "something" = "something in particular".)(∃x)(∀y)Rxy Note that pairs 1-2, 3-4, and 5-6 are equivalent.

Here are some other members of the same family. What do they mean?

- (∃x)(∃y)~Rxy
- (∀x)(∃y)~Rxy
- (∃y)(∀x)~Rxy solutions
- (∀y)(∃x)~Rxy
- (∃x)(∀y)Rxy

Something. "Something" and similar words like "somebody," "sometime," and "somewhere" are ambiguous. In "Everything bears R to something", the word "something" is ambiguous. It could mean (1) something in particular, or (2) something or other. For later reference, let us say that the former is thedefiniteand the latter theindefinitesense of the word "something". If we read "Everything bears R to something" in its definite sense, it asserts that everything bears R to one discrete object; if we read it in the indefinite sense, it asserts that everything bears R to at least one thing, but it is not necessary that any two R bearers bear R , the same thing. These are clearly not equivalent and must be translated differently, but how?Order matters for quantifiers of different types, and that the first is regarded as quantifying the subject of the sentence. When an existential quantifier comes first, or applies to the subject of a sentence, it is to be understood in the definite sense, and when it does not come first, or applies to an object of the verb, then it is to be understood in the indefinite sense.

With this understanding, then our sentence in its definite sense would be translated, "(∃y)(∀x)Rxy". This says that something in particular is is borne R by everything, which is the same as saying that everything bears R to some particular thing. In its indefinite sense, it would be, "(∀x)(∃y)Rxy". This says that everything bears R to something or other, which is the same as saying that something or other is borne R by everything.

Rule:If the English is ambiguous, assume the indefinite sense of 'something'.

Another example to clarify the rule. "Everyone is offended by something". In its definite sense it means that somewhere there is a universally offensive object. We might be wise to find it and destroy it immediately. But in its indefinite sense it means that everyone is offended by something or other, not necessarily by the same thing, perhaps each by a different thing. What offends you might not offend me, and might even be precious to me. The policy of destroying or prohibiting what offends you might be one of the things that offends me. So no obvious policy for eliminating offense suggests itself. On the contrary, if my offended sensibilities count exactly as much as yours, and vice versa, then we might have to live with a little offense as a natural side-effect of differing from one another in our sensibilities. If we let Oxy mean that x offends y, then the sentence in its definite sense (the former), should be translated, "(∃x)(∀y)Oxy" (something in particular offends everyone). In its indefinite sense (the latter) it should be translated, "(∀y)(∃x)Oxy" (everyone is offended by something or other).

Although I've said "something in particular, some one thing" for simplicity and clarity, the definite sense of "something" could signify a plurality of particular things. So "(∃y)(∀x)Axy" could mean that some particular set of things attracts everything. Similarly, the indefinite sense of "something" could signify a plurality of things. So "(∀x)(∃x)Axy" could mean that everything attracts some plural number of things, but not necessarily the same plurality or set of things.

Spatial relations. The word "somewhere" can refer to places where a predicate is true of certain objects. "Somewhere a wild boar is enjoying sunlight" would be translated "(∃x)(∃y)(∃z)(Px · By · Sz · Eyzx)" ;there is a place x, a boar y, and some sunlight z such that that boar enjoys that sunlight at that place.

Numerical expressions. We can go beyond the crude quantities ofall,some, andnoneto express more precisely how many things have a certain property. We can express the natural numbers as adjectives ("three blind mice"), if not as nouns ("one, two, three").

Zero. We can already express zero: zero things are human, nothing is human, all things are not human: "(∀x)~Hx", or "~(∃x)Hx".

Introducing identity. To express natural numbers in predicate notation, we must introduce a predicate to express identity, that is, to do the work that the "=" unicode does in arithmetic. Let the two-place predicate, Ixy, mean that x is identical to y.Now let's abbreviate "Ixy" as "x = y". This kind of abbreviation is no different from introducing the "→" symbol so that "p → q" can abbreviate "~p v q".

Similarly, let "x ≠ y" abbreviate "~(x = y)" and "~Ixy".

Adjectival natural numbers. Exactly one thing is human: "(∃x)(∀y)[Hx · (Hy → (x = y))]": there is at least one thing, x, such that x is human, and if there is anything else, y, that is human, then it is the same as the first.Exactly two things are human: "(∃x)(∃y)(∀z)[[(Hx · Hy · x ≠ y · [Hz → ((z = x) v (z = y))]]": there is at least one thing, x, that is human, and another thing, y, that is human, and x is not the same as y, and if there is another thing, z, that is human, then it is either the same as x or the same as y. And so on for three, four, five...things.

Note that these formulas could have been expressed with our original identity predicate, Ixy, rather than with our abbreviation for it, x = y. We introduced the "=" unicode simply for convenience.

At least. At least one thing is human: "(∃x)Hx". This is the unadorned existential quantifier.At least two things are human: "(∃x)(∃y)[(Hx · Hy) · (x ≠ y)]". There is a human x, and a human y, and x is not the same thing as y.

At least three things are human: "(∃x)(∃y)(∃z) [(Hx · Hy · Hz) · (x ≠ y) · (y ≠ z) · (x ≠ z)]".

At most. At most one thing is human: "(∀x)(∀y)[Hx → (Hy → (x = y))]". For all x and for all y, if x is human, then if y is human too, then x is the same thing as y. By avoiding the existential quantifier in this expression, we are non-committal on the question whether there are any humans.At most two things are human: "(∀x)(∀y)(∀z) [[(Hx · Hy) · (x ≠ y)] → [Hz → [(z = x) v (z = y)]]]".

"Only [name]" and "All but [name]" expressions. Only Socrates is human: "Hs · (∀x)[(x ≠ s)→~Hx]". Socrates is human, and for all things x, if x is not Socrates, then x is not human.Only Socrates and Plato are human: "Hs · Hp · (∀x) [[(x ≠ s) · (x ≠ p)] → ~Hx]".

All but Socrates are human: "~Hs · (x)[(x ≠ s) → Hx]". Socrates is not human, and for all things x, if x is not Socrates, then x is human.

All but Socrates and Plato are human: "~Hs · ~Hp · (∀x)[[(x ≠: s) · (x ≠ p)] → Hx]".