[6.] Predicate Logic Symbolization

 

[6.1.] Introducing Predicate Logic.

 

So far we have been learning a type of symbolic logic called sentential logic (a.k.a. sentence calculus, or propositional calculus). This sort of logic is capable of representing the internal structure of compound sentences. For example,

 

A = Andy is an attorney.

B = Beth is a bass player.

 

The argument “A É B; A \B” is valid in part because of the internal structure of the compound sentence “A É B.”

 

But sentential logic cannot capture the validity of all arguments. Many arguments are valid, not because of connectives that relate shorter sentences to one another, but because of the internal structure of the atomic (non-compound) sentences that the arguments contain. For example:

 

All mammals are animals.

All cows are mammals.

Therefore, all cows are animals.

 

The best we can do to represent this argument in sentential logic is: “M; C; \ A”. In our system of symbolic sentential logic, the argument is invalid—there are no rules that allow us to derive “A” from “M” and “C.”

 

We need to expand our system of logic so that it can represent the validity of this sort of argument. We will do this by adding a new type of symbolization that will reveal the internal logical structure of atomic sentences. We will eventually add new rules to accommodate this new symbolization.

 

This expanded system of logic is called predicate logic.

 

 

[6.2.] “Individuals and Properties.” [Ch. 7:1]

 

Consider this very simple sentence, which predicates a single property of a single individual:

 

“Sam is diligent.”

 

To represent this in sentential logic, we would simply choose a capital letter to use as a sentence constant (probably either S or D).

 

In predicate logic, we can “break into” this atomic sentence and reveal its internal structure.

 

We will do this by using a lower-case letter (a through t only, for a reason to be revealed later) to stand for the individual who is the subject of the sentence (Sam), and we will use an upper-case letter to stand for the property that the sentence predicates of Sam (namely, the property dishonesty). The upper-case letter must precede the lower-case letter.

 

Sam is diligent.                        Ds

 

This is consistent with the examples your textbook gives (p.167):

 

Art is happy.                Ha

Betsy is happy.                        Hb

Art is friendly.             Fa

Betsy is friendly.         Fb

 

In each of these sentences, a property, or characteristic, or trait, is predicated of some individual, or entity, or thing.

 

Upper-case letters representing properties = property constants

Lower-case letters (a through t only) representing individuals = individual constants

 

This new way of symbolizing atomic sentences is being added to our existing system of sentential logic. All of the operators, rules, etc. of sentential logic will remain intact. So we will symbolize compound sentences using this new notation rather than sentence constants:

 

Art and Betsy are happy.                     Ha · Hb

Art is friendly, but Betsy isn’t.             Fa · ~Fb

If Art is happy, then so is Betsy.                      Ha É Hb

Either Art is happy, or Art is friendly

if and only if Betsy is.             Ha v (Fa º Fb)

 

Exercise 7-1 (pp.169-70)

·         do #2 and #4 in class

·         do all of these for next time; we’ll check some of the odds in class

 

 

[6.3.] “Quantifiers and Free Variables.” [Ch. 7:2]

 

[6.3.1.] Individual Variables.

 

Just as there is a distinction between sentence constants (A, B, C...) and sentence variables (p, q, r...) , there is also a distinction between individual constants and individual variables. Individual variables are lower-case letters (x, y, z, w, v, and u only) that serve as variables that may be replaced by individual constants.

 

We use individual variables when we want to make a general point without saying something about specific individuals.

 

For example, we can use the individual variable “x” to define a property constant:

 

Dx        =          “x is diligent”

Hx        =          “x is happy”

Lx        =          “x is loving”

etc.

 

 

 

[6.3.2.] Universal and Existential Quantifiers.

 

Recall that in sentential logic, we can create sentences out of sentence forms by replacing the sentence variables with sentence constants, e.g., by replacing p and q in “p º q” with constants, as in “A º B”.

 

In predicate logic, we can create sentences out of sentence forms by replacing individual variables with individual constants.

 

For example, “Tx” is a sentence form, and we can create a sentence from it by replacing “x” in “Tx” with an individual constant, as in “Ta” (which symbolizes the sentence “Albert is tall”). Other examples:

 

“Ha” and “Hb” are substitution instances of Hx.

 

“Fa” and “Fb” are substitution instances of Fx.

 

“Fa É Hb” and “Fc É Hd” are substitution instances of Fx É Hy.

 

But this is not the only way to create a sentence from a sentence form. We can also create sentences that predicate a property, not just of an individual, but of all individuals or some individuals.

 

We do this using quantifiers, symbols that indicate of how many individuals a property is predicated.

 

When you use a quantifier to create a sentence from a sentence form, you do not replace the individual variable.

 

Instead, you add the quantifier to the beginning of the sentence form:

 

(x)Tx   “For all x, x is T” [Everything is tall.]

 

The symbol (x) is the universal quantifier.

 

($x)Tx “For some x, x is T” or “There is an x such that x is T” [Something is tall.]

 

The symbol ($x) is the existential quantifier.

 

 

[6.3.3.] Parentheses and Scope.

 

Like sentential logic, predicate logic uses parentheses to indicate scope. This includes indicating the scope of quantifiers:

 

Everything is shiny and happy.	(x)(Sx · Hx)
Something is shiny and happy.	($x)(Sx · Hx)
If everything is shiny and happy, then everything is perfect.	(x)(Sx · Hx) É (x)Px

 

And just like parentheses are not necessary to indicate the scope of the tilde in

 

~A

 

when a quantifier includes in its scope a single atomic sentence, parentheses around that sentence are unnecessary (although you always need parentheses around the quantifier itself):

 

            (x)Sx

($x)Hx

 

 

[6.3.4.] Free and Bound Variables.

 

“($x)(Sx · Hx)” is a sentence, rather than a sentence form. It actually says something (namely, that something is shiny and happy) and is either true or false.

 

But “($x)Sx · Hx” is not a sentence; it is a sentence form. This is because it contains an individual variable that is not within the scope of a quantifier: the “x” in “Hx”.

 

An individual variable that is within the scope of a quantifier is a bound variable; it is bound to the quantifier the scope of which it is in.

 

An individual variable that is not within the scope of a quantifier is a free variable (a.k.a. an unbound variable); it is not bound by any quantifier.

 

Sentences contain only bound variables; if an expression contains a free variable, then it is a sentence form, not a sentence.

 

Examples:

 

(x)(Hx · Sx) É ($x)Px                         sentence (all variables are bound)

($x)Hx É (x)(Sx É Px)                                    sentence (all variables are bound)

Hx É Sx                                               sentence form (two free variables)

($x)Hx É Sb                                        sentence (one variable, one individual constant)

(x)Hx É [(x)Sx É Px]                          sentence form (one free variable)

 

 

Exercise 7-2 (p.172):

·         do all of these (there are only 6) -- make sure you do all three steps

 

 

[6.4.] “Universal Quantifiers.”

 

[6.4.1.] The Domain of Discourse.

 

To fully understand symbolization with quantifiers, we next need to consider the notion of a domain of discourse:

 

domain of discourse (df.): the set of items that a sentence or group of sentences deals with. It is the domain (territory, sphere) constituted by all the individual things a given discourse talks about. [a.k.a. universe of discourse]

 

We can choose to restrict the domain of discourse to a certain group of individuals (e.g., people, or physical objects, or numbers, or geometrical figures, or buildings on the UWG campus, or ...  etc.) For example, if we specify that the domain of discourse includes all and only human beings, then

 

(x)Px    =          For all x, x has a parent.

 

is true, since what it says is that every individual in the domain of discourse (i.e., every human being) has a parent.

 

But if we leave the domain of discourse unrestricted, so that it includes everything (not only all people, but all physical objects, all abstractions, all numbers, all times, all places...) then

 

(x)Px    =          For all x, x has a parent.

 

is false, since what it says is that every single thing (including your textbook, the dry-erase board in this room, the number seven, the concept of liberty, the orbit of Venus around the sun, etc.) has a parent.

 

So how you should symbolize an English-language sentence will depend on the domain of discourse that is being assumed.

 

 

[6.4.2.] Conditionals and Unrestricted Domains.

 

Suppose you want to symbolize “All humans are mortal” (this example is discussed in your textbook, pp.173-74).

 

If the domain of discourse is restricted to include all and only humans, then you can symbolize “All humans are mortal” as

 

(x)Mx                       [where “M” is a property constant representing mortality]

 

However, if the domain of discourse is unrestricted, then you should symbolize “All humans are mortal” this way:

 

(x)(Hx É Mx)   =         For all x, if x is human, then x is mortal; i.e. anything human is mortal.

 

And if the domain of discourse is unrestricted, you should not symbolize it this way:

 

(x)(Hx · Mx)   =          For all x, x is both human and mortal, i.e. everything there is, is a mortal human.

 

From this point forward, unless otherwise specified, when we symbolize sentences,

we will assume an unrestricted domain.

 

If a sentence says something about only a certain group of individuals, we can build the restriction into the sentence by symbolizing it as illustrated above. More examples:

 

(x)(Ex É Bx)             =             “Elephants are big.”

(x)(Px É Lx)             =             “Philosophers are lazy.”

(x)(Nx É Ax)            =             “Numbers are abstract objects.”

 

It is extremely common for a universal quantifier to range over a conditional. It is so common, in fact, that you should be suspicious of your translation if it contains a universal quantifier ranging over something else. For example,

 

(x)(Ex Ú Bx)             =             “Every individual thing is either an elephant or big.”

(x)(Px º Lx)              =             “All and only philosophers are lazy.”

(x)(Nx · Ax)             =             “Everything is a number and an abstract object.”

 

 

[6.4.3.] Negation and Universal Quantification.

 

It is important to understand how the tilde interacts with the universal quantifier...

 

(x)(Hx É Mx)           =             “All humans are mortal.”

~(x)(Hx É Mx)         =             “It is not the case that all humans are mortal.”

(x)(~Hx É Mx)         =             “All non-humans are mortal.”

(x)(Hx É ~Mx)         =             “All humans are non-mortal, i.e. no humans are mortal.”

 

An unusual case:

 

(x)~(Hx É Mx)         is equivalent to

(x)~(~Hx Ú Mx)        by way of Impl; and this is equivalent to

(x)(~~Hx · ~Mx)      by way of DeM; and this is equivalent to

(x)(Hx · ~Mx)          by way of DN; and this says that everything is an immortal human.

 

See long list of examples, pp.174-75.

 

Exercise 7-3, pp.175-76

·         do all for next time; we’ll check the odds

 

Exercise 7-4, pp.176

·         do all for next time; we’ll check the odds

 

 

[6.5.] “Existential Quantifiers.” (Ch.7:4)

 

($x), the existential quantifier, is used to indicate that something, i.e., at least one thing, has a given property:

 

($x)Hx             =                      Something is happy (i.e., at least one thing is happy.).

($x)Sx              =                      Something is shiny.

($x)Px              =                      Something is perfect.

 

As with universal quantifiers, existential quantifiers can be combined with tildes:

 

~($x)Hx            =                     It is not the case that something is happy (i.e., nothing is happy).

($x)~Hx            =                     Something is unhappy.

 

[Important -- these two sentences are not equivalent!]

 

You can use the existential quantifier to predicate more than one property at a time:

 

($x)(Hx · Sx)   =                      Something is happy and shiny.

Some happy thing is shiny.

Some shiny thing is happy.

 

($x)[(Hx · Sx) · Px]  =                        Some happy things are shiny and perfect.

Some things are happy, shiny and perfect.

 

The existential quantifier signifies any quantity greater than none and less than all:

 

($x)(Sx · Hx)   =                      A few shiny things are happy.

There are several shiny, happy things.

A number of shiny things are happy.

Many shiny things are happy.

Lots of shiny things are happy.

 

NOTE: Just as it is most common for a universal quantifier directly to bind a conditional, it is most common for an existential quantifier directly to bind a conjunction.

 

See examples on p.177.

 

Exercise 7-5, p.178

·         do all of these; we’ll go through the odds next time

 

 

Stopping point for Thursday March 1. For next time,

·         complete exercises 7-1, 7-2, 7-3, 7-4, and 7-5;

·         read Ch.7:5-7, 9 (pp.178-182, 186-188)