Reference Manual
Chapter Six:
Predicate Logic: Quantifiers and Categories

.pdf Version for Printing

This reference provides some of the basic points made in Chapter Six. In this chapter, we turn back to our formal language with names and predicates. But we add one small but terribly important element: the ability to talk about quantities. In English, this just means the words "all" and "some". This little wrinkle will take some time to develop. .

Contents
: Section 1: Predicate Logic Introduction; Section 2: Predicate logic and Venn diagrams; Section 3: Categorical Logic: Equivalences; Section 4: Categorical Logic: Syllogisms Section 5: PL for Categories

 

1. Predicate Logic Introduction

It is time to add quantifiers. Instead of just symbolize that Halpin is a faker (or whatever you might want to say) we will symbolize things like "everyone is a faker" or "someone is a faker".

For these we will need quantifiers. Jumping the gun a bit, we'll symbolize the word "every" (or equivalently "all", "any", etc.) with an upside down 'A': . And we'll use a backwards-E for "some" and it's synonyms: .

And soon we will want to get to the more useful "someone at O.U. is an employee and not a professor". Or, "everyone at OU is a professor or a student". (Of course, only the first of these two sentences is true.)

Note: You'll need The Logic Font to display our symbols. Use the '%' key for the backwards-E and the '^' for the upside down 'A'.

Symbolizing "All" and "Some"

But we start very simple. And with a very precise example. For purposes of this example will be discussing only sound arguments. We will ignore everything else. Again: for now the subject matter is the collection of all sound arguments. (We could talk instead about all cats, say. But we just need to restrict ourselves in some specific way for the example.)

So, t he subject matter for this example, the universe of discourse, is just the collection of all sound arguments.

Here's one, a disjunctive syllogism (DS) with only true premises:

(a) Halpin (whoever this guy is) is either a professor or a faker.
But (I'm telling the truth, really!) he is no faker.
So, Halpin is a professor

We call this one 'a'. Of course, there are lots of others. The collection of sound arguments is infinite in principle: you can make longer and longer true sentences that fit this form.

But we know that there is this one sound argument in existence and, of course many more out there to be found.

So, first we say:

Universe of Discourse = all sound arguments

W_: __ is valid.

a: the sound argument above in gray.

Now, it's easy to say of our one argument, a above, that it is valid:

Wa

But we need to say more. That a is representative in this way of any sound argument. We do this with variables: we use 'x' and later ('w', 'y', and 'z') as placeholders for anything in the universe of discourse. They are a little like the blank above in "__ is valid". In fact, hereafter when we say 'W' means "is valid" we will write this as 'Wx' means "x is valid". Let's make the change right now:

Universe of Discourse = all sound arguments

Wx: x is valid.

a: the sound argument above in gray.

Now, we say that some argument (in our universe of discourse) is valid with our backwards-E:

(%x)Wx

(read this "there is an x such that x is valid" or as "there is an x making 'Wx' true.")

So, '(%x)Wx' means that some x, some member of the universe of discourse, is a valid argument. There is a way to "fill the blank" making 'W_' true.

Of course, we already knew that some arguments are valid. Because argument a above is sound, it's an example of a valid argument.

More importantly, because we are only speaking about sound arguments, we can say that they all are valid. We symbolize that with the upside down 'A':

(^x)Wx

(read this as "every x is such that it is valid" or as "all x make 'Wx' true".)

So, '(^x)Wx' means that every x, every member of the universe of discourse, is a valid argument: Any way of filling in the blank of 'W_' is true (so long as we fill in with names for members of our universe of discourse: sound arguments).

Symbolizing "No" meaning "None"

We have two logically equivalent ways to think about our "no" statement: "No cats are reptiles". This statement can be understood to be it's not true that some cat is a reptile ('~(%x)Rx') or it can be equivalently rendered as all cats are non- reptiles ('(^x)~Rx'). (To say that all cats are non-reptiles is to say that every cat fails to be a reptile.)

We will see these two ways of expressing "no" again and again, so let's put it in a box:

QN: "no cats are reptiles" can be symbolized '~(%x)Rx' or, equivalently, '(^x)~Rx'.

We will return to the PL symbolizations later. For now keep in mind that "no", "some", and "all" have this complicated relation just called 'QN'.

 

Building Sentences

Some of these have more than on quantifier, something like '(%x)Wx & (%x)Dx'. This is built up as follows:

  1. Start with 'Wx' and 'Dx'. These are "open" sentences; they are like any English sentence with a blank: e.g., "___ is a valid argument".
  2. Build up from these by attaching a quantifier on the left of each to get: '(%x)Wx' and '(%x)Dx'. Quantifiers -- along with their parentheses and the variable 'x' -- work just like a tilde as far as the correct construction of a sentence goes. When an x-quantifier is added, it "binds" all x-variables in the expression to which it attaches.
  3. Finish by connecting the two sentences just formed with an ampersand. This gives '(%x)Wx & (%x)Dx' with main connective ampersand (the last connective used).
  4. In Genearal: Anything produced in the building process is called a "formula" of our language, PL. When a forumal has no "free variables", i.e., variables not bound by a quantifier, then that formula is a "sentence" of PL.

In general, you may build up a formula by starting with open atomic sentences like 'Wx' or 'Da' and applying connectives and quantifiers. But an occurrence of variable is not "bound" by a quantifier until, in the buiding process, a quantifer has been attached to a formula containing that occurrence. E.g., in 'Wx&Da', x is not bound by any quantifer. In 'Wx&(^x)Dx' only the second 'x' is bound. In (%x)(Wx&Dx)', the quantifier is applied to the whole formula 'Wx&Dx' in the building process: so both occurrences of 'x' are bound in 'Wx&Dx'. When all variable are bound in a formula, we say that formula is a sentence.

 

2. Predicate logic and Venn diagrams

Traditional Categorical Logic

We could move immediately to the full predicate logic at this point. But it is a bit easier to understand PL if we think in terms of the logic of subjects and predicates. So, we need to move away from our simplification to a single subject.

Within a normal universe of discourse, there are many subjects. If we are talking about cats, we may also be talking about birds, reptiles, humans, mice, carnivores, males, females, etc. Some of these other groups are mammals, some are not. Some of the groups overlap. We need to find a way to treat this more general case.

1. In traditional, categorical logic, we consider how groups are related to each other. Category terms are just words for groups of things, e.g., groups of animals. For examples from the categorical language, we might say,

All dogs are mammals.

This relates the one category, dogs, to the larger, mammals. We might also say:

Some mammals are carnivores.

or

No reptiles are mammals,

 

2. These three examples are of categorical statements. They relate categories. For example, "All dogs are mammals" takes the subject term "dogs" and relates it to the predicate term "mammals".

a) A categorical statement of this form...

All S are P

we call a "universal categorical statement". We'll call the form "universal".

b) A statement of this "existential form":

Some S are P

is an existential categorical statement.

c) A statement of the last or "negative form":

No S are P

is a negative categorical statement.

 

3. Semantics by Venn

Venn diagrams are the easy way to go to understand the meaning of categorical sentences. Think of everything in the universe (of discourse) as being contained within the box just below.

Everything that falls into the category S is located in a circle (the left one). Everything falling into category P is located in the other circle (the right one).

If something is both S and P, then it's located in the area of overlap of the two circles: the area in green.

Anything which is neither S nor P, is outside both circles and in the white area.

4. Example: Mammals and Carnivores

Let's let S be the collection of all mammals, and P be the collection of all land animals. Let's picture this. Human beings are both mammals and land animals. So, we all are in both S and P,; I'll give myself the name 'h' and you'll notice that I'm placed with the rest of you in the intersection of the two sets. Your job is to drag the names of animals below (pretending they are real!) and placing them where they belong in our universe.

h
s
m
n
c
t
u

The names are placed on the diagram (given the interpretation below)

So, the tiger ("Tony" or 't') is placed in the position of overlap: tigers are both land animals and mammals.

 

 

Universe of Discourse = Animals

c: Charlotte (the spider) h: Halpin (the professor), m: Moby Dick (the white whale), n: Nemo (the fish), s: Shamu (the killer whale), t: Tony (the Tiger), u: Ursula (the octopus)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5. Now, let's use these diagrams for understanding categorical statements.

We have three "official" categorical forms.

a) universal categorical form:

All S are P

 

The lines mean that the area inside S but outside P is empty. So, any S is P. Thus we have a diagram of our universal form.

 

 

 

b) existential categorical form:

Some S are P

 

The 'x' is an arbitrary stand-in for an object. The diagram simply says that there is something in the green area of overlap between S and P.

 

 

 

and c) negative categorical form:

No S are P

This diagram represents there being nothing in the overlap. So, there is nothing that is both S and P.

 

 

 

Symbolization into strict categorical form

Many English categorical forms don't exactly fit into our three forms. Much of the trouble surrounds the universal form. So, let's start with the easier two.

Existential Form can be manifested in English with phrases not including "some". For example, if one says any of...

There is a whale in Ohio.
There is at least one whale in Ohio.
Whales exist (or live) in Ohio.

one could be taken to mean

Some W are O

Where 'W' stands for the category of whales, 'O' for things in Ohio.

Trickier cases include:

A whale lives in Ohio.
Whales are in Ohio.

These clearly are also existential. But see below for similar case that are universal.

 

Think about this one:

Some dogs are not pets.

This doesn't fit our existential categorical form only because "not a pet" is not itself a category phrase. But we can easily turn it into one: "not a pet" goes to "non-pet". So, we can symbolize this one as

Some D are N

where 'N' stands for the collection of non-pets. (Warning: Traditional logic recognizes a forth form: "Some S are not P". Let's call this existential negation. It's easy to see it's meaning. Now, how would you diagram it?)

 

Negative Forms to come in different styles that are logically equivalent:

No S are P
Nothing is both S and P
None of S are P

but the biggest problem:

 

Universal Forms

 

For example, any of the following could be symbolized as a universal form statement:

1. Every whale is a mammal.
2. Each whale is a mammal.
3. Any whale is a mammal.

So, when your job is to symbolize is standard form, you will take such English and change them into

All W are M.

Where 'W' is interpreted as the predicate naming whales, 'M' for mammals.

Here are two more tougher ones:

4. Whales are mammals.
5. A whale is a mammal.
6. If a thing is a whale, then it's a mammal.

Usually these two, 4 and 5, would also be symbolized as of universal form, the same way as for 1-3.

All W are M.

If you say "whales are mammals" you are pretty clearly thinking about all whales.

When "a whale is a mammal" is used, this is normally about an arbitrary whale. So, 5 too is universal.

6 is similarly about any thing. Keep 6 in mind; it will help us symbolize categorical statements into English.

There are some that are pretty obvious, they just need subtle changes:

7. All losers complain.

7 is missing a category term for the predicate. But this is easy to fix. Let 'C' stand for "people who complain".

All L are C.

Here's another subtle change:

8. Tom always comes in late.

8 doesn't even look like a categorical statement at first. But it's about all times. "All times when Tom comes are times when he's late".

All C are L.

Notice that we've just changed what 'C' and 'L' mean. Context makes their interpretation clear.

 

A similar treatment is necessary for a spatial adverb:

9. Wherever one goes one finds competition.

This is "All places one goes are places one finds competition":

All G are F.

where 'G' stands for "places one goes" and 'F' for "places one finds competition".

And another similar one:

10. Whoever didn't pass should study harder.

This is "All people who did not pass should study harder".

All N are H.

 

The next few are trickier:

11. Only humans are rational.
12. None but humans are rational.

These say that nothing else is rational besides humans: All rational beings are human.

All R are H.

But they do not say that all humans are rational! So, "Only S are P"  and "None but S are P" amount to  "All P are S". Note the subject-predicate switch!

However, adding the word "The" makes all the difference!:

13. The only rational creatures are humans.

which is symbolized with no "switching":

All R are H.

In general, "The only S are P" is logically equivalent to "All S are P".

And, here's a last universal form symbolization:

14. One pays taxes unless one is lucky.

Unless means "if not" (you can prove this in SD). So, any person who is unlucky pays taxes.

All U are P.

Negations can be tricky too.

15. Not a single life should be wasted.

15 means that all life should be saved, not wasted.

All L are S.

Note that 15 is a negation of an existential: "Some life should be wasted". But 16 is different, it's the negation

16. Not all politicians are crooks.

This is "Some politicians are non-crooks":

Some P are N.

where now 'N' stands for "non-crook".

 

 

3. Categorical Logic: Equivalences

Let's begin with a mistaken inference. This will help us see what can go wrong in categorical logic before we say just what is right.

Illicit Conversion

YOU SAY:  All V are C.

but from this...

I CONCLUDE:   All C are V.

I've switched, "converted", the two categories!

As the subject matter becomes more complicated, it's easy to make this mistaken leap. But it is a mistake. Back in chapter one, we called this one the formal fallacy of illicit conversion. To make the point more obvious, it's best to have a specific example:

All bats are mammals, therefore all mammals are bats.

So, clearly we cannot convert in the universal form.

In Venn Diagrams, we can see this. "All bats are mammals" can be symbolized this way:

This diagram just means that there are no bats that are non-mammals. That should make sense for "All" and seems true!

 

 

 

But let's see what happens in the diagrams when we try to convert the English. Now we want to think about what the (obviously false) statement "All mammals are bats" comes to when we diagram it.

All mammals are bats:

This second diagram clearly is different from the one for "all bats are mammals". We know this second one is mistaken because it allows no room for humans, mice, and the rest of the non-mammals.

Bottom line: we can't just switch subject and predicate.

 

Conversion for Existential and Negative Categorical Statements

However, we can do the switch with our existential and negative forms. This is pretty easy to see by example.

1. Existential Form   If you say "Some mammals are flying creatures" then I can rightly infer from this that some flying creature are bats:

"Some M are F" is logically equivalent to "Some F are M".

In general:

"Some S are P" is logically equivalent to "Some P are S".

And we can see why by looking at the symmetrical diagram for "Some S are P":

 

2. Negative Form   We can say much the same thing about "No mammals are birds" and "No birds are mammals". Both are true and for the same reason: Both say that there is nothing in the overlap between mammals and birds.

In general:

"No S are P" is logically equivalent to "No P are S".

And we can see why by looking at the symmetrical diagram for "Some S are P":

 

 

Contraposition for Universal Categorical Statements

Conversion -- the switching of subject and predicate -- is always a valid inference for existential and negative categorical statements. But it's invalid for universal forms.

However, think about the following inference.

All bats are mammals. So, all non-mammals are non-bats.

This should sound right. And for good reason. But let's first get clear on "non". When we say "non-mammals" we mean everything that is not a mammal. This is sometimes called the "complement" of the category mammal. By any name, it's easy to see and diagram:

Now, with this understanding of complements, let's see just why "All bats are mammals" is logically equivalent to "all non-mammals are non-bats".

"All bats are mammals" means that all the bats have to be inside the green and so be mammals.

I.e., any non-mammal -- anything outside of the mammals circle -- cannot be a bat. So, this same diagram works for "all non-mammals are non-bats".

 

The general lesson here is this:

"All S are P" is logically equivalent to "All non-P are non-S".

 

Obversion

The operation of obversion can be defined on all categorical statements. But it's only really valuable as a means to transpose between universal and negative statements.

"All sound arguments are valid" means that no sound arguments are invalid. That's pretty much all there is to obversion. But let's make it clear.

First, our Venn Diagrams justify this equivalence.

When we first wrote up a Venn Diagram for the universal form, we put it this way:

This diagram specifies the meaning of "All S are P". The idea is that everything that is in P is also in S. I.e., there is nothing in S that is not also in P. This last just means:

       No S are non-P.

There is nothing in the overlap of S and non-P.

Second, a little thought shows that we've also justified the logical equivalence for this similar transposition:

"No S are P" is logically equivalent to "All S are non-P".

So, putting Which can be added to our first version

Obversion has two forms to remember:

"All S are P" is logically equivalent to "No S are non-P".

"No S are P" is logically equivalent to "All S are non-P".

And to reiterate, we also have

Conversion has two forms for equivalence:

"Some S are P" is logically equivalent to "Some P are S".

"No S are P" is logically equivalent to "No P are S".

Contraposition:

"All S are P" is logically equivalent to "All non-P are non-S".

 

4. Categorical Logic: Syllogisms

Traditional logic, going back to Aristotle and before, spent much time and energy working out an intricate systematization of the logic of categorical statements and "syllogisms", a.k.a. arguments. In this tutorial, we will address these syllogisms but use Venn Diagrams to circumvent the intricacies.

Let's begin with an example of a categorical syllogism:

No mammals are fish.
All whales are mammals.
So, no whales are fish.

1. Let's see how these arguments are defined. Each categorical syllogism consists of two premises, both of categorical form, and one categorical statement as conclusion.

The conclusion is of the form: "All/No/Some S are P" (choose one quantity word from the three types). Each premises contains one of the category terms, S or P, from the conclusion. In addition, there is a "middle term", M, in both premises. So, in the first argument above...

No mammals are fish.
All whales are mammals.
So, no whales are fish.

S  is "whales" and P is "fish" (see the conclusion) while the middle term, M, is mammals. Note that M is the one category term in both premises.

2. Because there are three terms in our syllogisms, our diagrams will be a little more complicated. We'll need three circles to represent the three groups.

Notice that the possibilities of overlap are increased. We could continue to talk about these in terms of their colors. Now there's an orange area that is inside the M and P circles but outside S. Right? This means that any objects in orange are in category M and P but they are non-S.

But it's better to have numbers:

Now, instead of "orange" we can just say 6.

The important thing is what it means to be in an area. Anything in area 3 is just P, it is neither M nor S.

Now let's see how we put these to work.

 

 

 

3. Here's that first categorical syllogism again:

No mammals are fish.
All whales are mammals.
So, no whales are fish.

Instead of writing out the whole category description, sometimes they get quite long!, we'll just do letters. So, let's rewrite this as

No M are F.
  All W are M.
So, no W are F.

Now we just do the Venn Diagram by diagramming what the premises mean together...the argument will be valid if the premises mean that the conclusion is true.

Here goes, let's first take our Venn Diagram with numbers and diagram the premises:

The first premise is "No M are F". To say what it means for this to be true, we just exclude everything in the area of overlap between M and F: areas 5 and 6.

Next we diagram the second premise right over the top of the first diagram. "All W are M" means that there is nothing in W outside of M: something can't be a whale without also being a mammal. OK? So, diagram this in the normal way, just strike-out the areas 1 and 2.

What do all these stripe mean? Well, the are of overlap of W and F is marked off. So, our premises together mean that there is nothing that is both W and F: No Whales are Fish. This is to say that the premises are enough. If they are true, then the conclusion "No W are F" must be. So, this argument is valid.

4. Here's the test of validity:

  1. Diagram both premises.
  2. Stop diagramming and see if premises together indicate that the conclusion is true.
  3. If they do make the conclusion true, then the argument is valid. Otherwise, it's invalid.

 

5. Now consider a second argument...

All whales are mammals.
No fish are whales.
So, no fish are mammals.

...is correctly diagrammed by the first option (to the left).

The first premise is diagrammed with the nearly horizontal stripes (like these: ) and the second premise is diagrammed with the nearly vertical stripes (the ones which look like this: ).

 

But, there is something wrong here. Or, I should say, something wrong with the argument above. It's not valid. Let's see why.

The point of the diagram is to spell out exactly what the two premises together mean. Our two premises tell us that there can be nothing in areas 4, 5, and 7. But this fact, the emptiness of the areas of 4, 5, and 7, is not enough to make the conclusion true.

No fish are mammals.

For this conclusion, we'd need to have 5 and 2 excluded. But 2 is not. The premises allow that some non-whales be fish that are mammals. So, the argument is not valid, the premises (alone) do not make the conclusion inescapable.

 

Existential Statements in the Diagram for a Syllogism

6. It's easy to diagram existential statements for a syllogism too. But there is one trick we'll see in a moment. To make life simpler, always diagram any non-existential statement before an existential one. We'll see why in a moment.

Here's a good argument to diagram:

Some fish are not carnivores.
All whales are carnivores.
So, some whales are not fish.

Step 1: Diagram the premises (leaving the existential premise to last).

So, first diagram the universal premise to get this:

 

 

Now, we'll diagram the existential premise. Because this is a negative existential, i.e., the predicate is "not a carnivore" which we read as "non-carnivore", we need to put a letter, 'x' inside the fish-circle but outside the carnivore-circle.

 

 

We diagram the existential statement last because now we can see that there is only one place to put the the 'x', viz. area 3:

Step 2: So, what conclusion can we draw from this about whales and fish?

We know that some fish (the 'x') is not a whale.

But the diagram does not tell us there is a whale that is a non-fish. Area 4 has no diagramming marks, so it tells us nothing about the existence or non-existence of whales there that would be non-fish.

Step 3: It follows that Our argument is not valid.

Some fish are not carnivores.
All whales are carnivores.
So, some whales are not fish.

 

To show the argument valid, the diagram would need to include the information that there is something in area 4 (there would need to be an 'x' marked there).

7. However, this same diagram is enough to show another argument is valid:

Some fish are not carnivores.
All whales are carnivores.
So, some fish are not whales.

 

8. There is one last wrinkle to consider. Begin diagramming this argument:

No M are P.
Some S are non-M.
So, some S are P.

Notice that it won't matter what 'S', 'M', and 'P' are about. This is formal logic. Their content does not matter!

Step 1: Because one premise, the first, is not existential. So, diagram it first:

 

 

The second premise require that we put something, named by 'x', inside S but outside P...

 

 

 

But when we try to place the 'x', there is some uncertainty:

 

 

 

Should we place the 'x' in area 1? Or in area 2? Nothing in the premises tell us where the object or objects should go. It's just that there are some S that are non-M. The best we can do is this:

 

 

 

 

We just put the 'x' right on the line...this indicates are uncertainty. Or, better, indicates the uncertainty in the premises.

 

 

 

Step 2: Because the conclusion of our argument...

No M are P.
Some S are non-M.
So, some S are P.

...is "some S are P" we now notice that the diagram is uncertain on its truth. The 'x' is not placed within area 2, so the diagram does not make the conclusion true.

Step 3: Hence this argument is invalid.

 

 

 

5. PL for Categories

Natural language provides many ways to quantify over objects. Often these involve very complicated constructions. Fortunately, the complications are often founded on our three categorical forms. And because obversion allows us to define a negative statement in terms of a universal one, we can take just two forms as basic. In this tutorial we focus on these two forms, the "existential". We will see how much of quantification in PL boils down to these two forms.

The Existential Form

The existential form is pretty easy to symbolize in PL.

An English sentence is of the form if it fits this mold:

Some S are P.

where 'S' (the subject) and 'P' (the predicate of the expression) range over English predicates. The predicates stand for categories (like students and freshmen).

1. Now, it's pretty easy to see how to symbolize simple sentences of this form. For instance,

Some whales are living in Ohio

means that there is some thing or things that are whales and that are living in Ohio. This may be symbolized as

(%y)(Wy&Oy)

(Read this as "there is a y such that y is W and y is O".)

assuming that 'Wx': "x is a whale", 'Ox': "x is living in Ohio".

2. For more complicate cases, we begin by rephrasing an English sentence into the "mold" of the appropriate existential English sentence:

(Step I) Some S are P.

Which can be seen in hybrid form as

(Step II) (%x)(x is an S  &  x is a P)

Such examples can be symbolized as

(Step III)       (%x)(Sx & Px)

 

3. Then, to take an example, to symbolize "Some happy whales live in Ohio" we first change the English to standard existential form letting S = "happy whale" and P = "creature living in Ohio".

(Step I)    Some happy whales are creatures living in Ohio.

This provides an easy transition to the hybrid symbolization:

(Step II)   (%x)( x is a happy whale & x is a creature living in Ohio )

How do we say "x is a happy whale" given the above symbolization key? The subject phrase here just comes to "x is happy and a whale" or 'Hx&Wx'. So, we just fill in to get the final answer:

(Step III) (%x)( (Hx&Wx) & Ox )

 

 

Universal Form

This form is a little less straightforward. Continue to think of the universe of discourse as all living creatures and begin with this easy example:

(*) All whales are mammals.

1. It's clear that these universal form categorical statements need to be symbolized with the universal quantifier because they are about "all" things in a some group. But this sentence is not about all things in our current universe of discourse of all creatures.

So, it's not to be symbolized as '(^x)Mx' (where 'Mx': "x is a mammal"): we don't want to say that all creatures are mammals. Nor can it be symbolized as '(^x)Mx&(^x)Wx' or as '(^x)(Mx&Wx)'. Both of these say that every creature is a mammal and a whale. That's not what's meant!

Instead, (*) asks us to restrict our attention to certain members of this group: the whales. And about these only, we say all are mammals. Now how do we symbolize this restriction?

Think about it this way: Of each creature in the entire animal kingdom, we say only that if it's a whale, then it's a mammal:

(^x)(Wx>Mx)

This is our symbolization for (*).

2. In general,

All S are P

means the same as

(^x)( x is an S > x is a P )

and so can be symbolized as:

(^x)(Sx>Px)

This gives us another three-step "mold" into which more complicated symbolizations will fit.

3. For instance, we might want to symbolize the universal form sentence:

All happy whales are mammals living outside Ohio.

Here S = "unhappy whales" and P = "mammals living outside Ohio". So, the hybrid form would be:

(^x)[ x is happy and a whale > x is a mammal and not living in Ohio ]

or, in pure PL:

(^x)[ (Hx&Wx) > (Mx&~Ox) ]

 

Categorical Logic

Categorical logic, then, including the negative existential (with which we have not been much concerned but which is a big part of traditional logic) can be represented with our quantifiers in this way. Also, these four forms are often given letter names as shown:

  Type English Form PL Form
A Universal form: All S are P (^x)(Sx>Px)
E Negative form: No S are P (^x)(Sx>~Px) or ~(%x)(Sx&Px)
I Existential form:

Some S are P

(%x)(Sx&Px)
O Negative Existential form: Some S are not-P (%x)(Sx&~Px)

 

Now, also notice that universal and O form sentences are "opposites": if one is true, then the other is false. The same relation of opposition holds between E and I forms. We call such pairs contradictories. This fact is represented in the following table:

The Modern "Square of Opposition"

(Pairs of sentences connected by diagonal lines are contradictory.)

Developed in ancient Greek times, categorical logic was the first systematic logic. "Aristotle" is the main name in this development. He also was the first systematizer of many of the sciences; quite an accomplishment! Aristotle's categorical logic differed a bit from the treatment just described. Our treatment of categorical logic is usually called "modern". There are many readily accessible descriptions of the difference between Aristotelian and modern categorical logic.