User:Addemf/sandbox/Technical Reasoning/Logical Abstraction



In this lesson, we will look back over some of the arguments which we gave in the geometry lesson, and try to discern the logical principles at play. This will hopefully clarify the concepts of logic while also preparing us for future applications of logic to other subjects.

A Case Study
Consider the axiom from the previous lesson,

For any two points there is exactly one line through them.

Later on we will learn how to logically analyze an "exactly one" statement.

For now, however, let's focus on the simpler sentence,

For any two points, there is a line through them.

Abstraction
The goal of logic is to abstract away the specific content of any sentence.

Thereby, in logic, we study the principles of reasoning which should apply to every subject.

To abstract the "case study" sentence above, is to remove all of the content which is specific to geometry, while leaving the structure intact. We remove the word "point" and replace it with a variable, P, so that P could refer to a point or to anything else.

For every x and y, if $$P(x)$$ and $$P(y)$$, then there is a z which is $$L(z)$$, and $$T(z,x,y)$$.

What we have done to abstract the sentence is:

1. Explicitly talk about the three objects of the sentence, by giving them names, x, y, and z.

2. Replace the word "point" with the variable P; and replace "line" with L; and replace "through" with T.

3. Indicate which objects have which relationships, by writing the names of the objects after the variables.


 * For instance, by writing $$P(x)$$ we indicate that object x has property P. By writing $$T(z,x,y)$$ we capture the idea that, whatever relationship T is, it is somehow a relationship which holds between the objects z, and x, and y.

Specification
By replacing these words with variables, we are trying to eliminate any reference to geometry, or to any other specific topic. The symbol P could now mean anything.

For example, P could mean "terminal", and L could mean "wire", and T could mean "connects".

Whereas "abstraction" is the removal of details, what we are discussing here is taking an abstraction and supplying details. We will call this "specification".

The specification
 * $$[\![ P\to \text{terminal}, L\to \text{wire}, T\to \text{connects}]\!]$$

applied to the abstraction above, results in the sentence
 * "For every x and y, if x is a terminal and y is a terminal, then there is a z which is a wire, and z connects x to y."

In more natural English, this is the same as saying,
 * "For every two terminals, there is a wire which connects them."

Proper Named Objects
Let's see one more example, this time with a "proper named object". A proper name is like the name "Mike", which denotes one specific object. (Let's ignore for now the fact that many people may be named "Mike".) Another example might be "Mount Everest" which specifically names just that one mountain.

Take the mathematical example sentence,


 * "Zero is the least number."

In this example, the sentence specifically refers to one "named" object, which is the number zero.

The Least
Before we try to abstract this sentence, it will help if we first unpack some of what is meant by "least". We must have in mind the "less-than-or-equal-to" relationship.

To say that zero is the least number, is to say,


 * "Zero is a number. And zero is less-than-or-equal-to every number."

But it is even better to write the sentence as


 * "Zero is a number. And for every x, if x is a number then zero is less-than-or-equal-to x."

The Abstraction
This second version is much easier to abstract, and the abstraction is,


 * $$N(a)$$. And for every x, if $$N(x)$$ then $$L(a,x)$$.

N is the abstraction of "number". L is the abstraction of "less-than-or-equal-to". a is the abstraction of "zero".

Note that we tend to use letters a through t as abstraction symbols for properly named objects.

By contrast we use letters like x, y, and z for object variables. In the abstraction above, x is a variable because it could refer to anything. We will also use t through w for variables if needed, although we have a slight (and meaningless) preference for x, y, and z.

We typically use capital letters, like N and L, for property variables. Property variables state some property of the objects, whether they are properly named objects or variable objects.

Give the abstraction of each of the following sentences. If any of them have the same abstraction as "For every x and y, if $$P(x)$$ and $$P(y)$$ then there is a z which is $$L(z)$$ and $$T(z,x,y)$$," then indicate this.

 Part 1.

Every cat is a mammal.

''Part 2. ''

For every two numbers there is a number between them.

''Part 3. ''

The empty set is a subset of every set.

Part 4.

Every number is less than infinity.

Part 5.

For any two natural numbers, there is a rational number between them.

To summarize what abstraction of a sentence means, it is to take the specific things in the sentence and replace them with variables which could be anything.

The reverse of abstraction is specification. That is to say, if one takes some abstracted sentence and replaces its variables with specific references, we call this act "specification".

Consider the sentence, which we will call S, below.


 * S is the sentence "2 is even and positive."

Below is the abstraction of S.


 * A is the abstraction "t is $$E(t)$$ and $$P(t)$$."

The specification which takes us from A back to S can be expressed by a function $$\sigma$$ which maps each variable to a word. In particular $$\sigma(t) = 2$$ and $$ \sigma(E)$$ = "even" and $$\sigma(P) $$ = "positive".

Therefore applying this specification $$\sigma$$ to abstraction A results in sentence S.

Let us now define a different specification, $$\tau$$. Define this to be the function $$\tau(t)$$ = Biden and $$\tau(E)$$ = "president", and $$\tau(P)$$ = "Democrat".

What is the sentence which results from applying specification $$\tau$$ to abstraction A?

True or false: For any given sentence, it has just one abstraction.

True or false: For any given abstraction, it has just one specification.

Conjunction
It is helpful to symbolize our abstract sentences, because it will later allow us to inspect the meaning of each symbol.

We will have two classes of symbols for our symbolization: Quantifiers and propositional operators. The first four symbols are propositional operators and the last two are quantifiers.

Before directly attempting to symbolize our abstract sentence above, let us start with some smaller examples. To begin with, consider the (abstract) sentence "$$P(x)$$ and $$P(y)$$".

To symbolize this we would merely replace the conjunction "and" with the symbol $$\land$$. Therefore its symbolization is


 * $$P(x)\land P(y)$$

Simple enough, right?

We call such a sentence a conjunction, and we call the two clauses of the conjunction "conjuncts". Therefore $$P(x)$$ is the left conjunct and $$P(y)$$ is the right conjunct.

Disjunction
Consider the example sentence "6 is divisible by 3 or 4." Its abstraction is


 * "$$D(s,t)$$ or $$D(s,f)$$."

where s abstracts 6, t abstracts 3, f abstracts 4, and D abstracts "divisible by".

We use the symbol $$\lor$$ in place of the word "or" so that the symbolization of this is


 * $$D(s,t)\lor D(s,f)$$

Because of the use of "or" this sentence is a disjunction, and we call each clause a "disjunct".

Negation
Consider next the sentence "The cat is not on the mat," with abstraction


 * "Not $$O(c,m)$$"

where c abstracts the cat, m abstracts the mat, and O abstracts the "is on" relation.

We symbolize this by


 * $$\neg O(c,m)$$

Note that negation is a unary operator, unlike $$\land$$ and $$\lor$$ which are binary operators. Negation applies only to a single sentence at a time.

There is, as far as I know, no official word for "the clause under the negation" the way that there are words for conjuncts and disjuncts. However, if we would like a word, we might choose the Latin conjugation negationem. (Literally: the thing negated.)

Conditional
Consider the next sentence "If you park here between the hours of 9 a.m. to 5 p.m. your car will be towed." This is abstracted as


 * "$$P(y, n, f)$$ then $$T(c)$$"

where y abstracts you, n abstracts 9 a.m., f abstracts 5 p.m., c abstracts your car, P abstracts the "parks" relation, and T abstracts the "is towed" relation.

This is symbolized as


 * $$P(y,n,f)\to T(c)$$

It is worth appreciating how sometimes the alignment of English and symbolization is awkward. In English we often indicate the condition with the word "if" and then signal the consequence with "then".

However, in symbols we only have a single "infix" symbol. We understand that whatever is to the left, is the condition, and whatever is to the right, is the consequence.

Up to now, I've been describing the clauses of a conditional as "condition" and "consequence". However, the more technical terms which is used by logicians are "antecedent" and "consequent".

Therefore, from now on, we will use the more correct vocabulary. In an expression of the form "P \to Q", we will say that P is the antecedent and Q the consequent.

Universal Quantification
The last two symbols that we will study are the "quantifiers".

Consider the (false) sentence "Every number divisible by 2 is divisible by 4." Its abstraction is


 * "For every x, if $$D(x,t)$$ then $$D(x,f)$$."

where t abstracts 2, f abstracts 4, and D abstracts the "divides" relation.

Naturally "if (condition) then (consequence)" portion of this can be symbolized with $$\to$$. But what about the "for every x"?

Notice that the propositional operators above, $$\land,\lor,\neg,\to$$ all took some number of sentences, and used them to form a new sentence. But that is not what the "for all x" part of this sentence does.

Rather, "for all x" takes a so-called "open formula" and turns it into a sentence.

An open formula is something like x = x, which is technically not a sentence because it gives you no indicate of what x is.

But if we attach this to the universal quantifier, and say "For all x, we have x = x," this now becomes a sentence because we are told what x means. In particular, when we write this, x represents any arbitrary object. And this makes the sentence a claim about all objects in the universe.

For example, if the number 1 is something in our universe (and for the purposes of most mathematical conversations, it is) then "For all x, we have x = x" would entail that 1 = 1. If 1/2 is in the universe (and for most mathematical conversations, it is) then it would also entail 1/2 = 1/2. And so on.

Because quantification plays a significantly different role than propositional connectives, we give it different notation. The sentence "For all x, we have x = x," has symbolization


 * $$\forall x(x=x)$$

The symbol $$\forall $$ is the universal quantifier, read as "for all", or "for any", whichever the speaker prefers. It is always written with a variable, called the variable of its quantification.

After that we write an open formula, which is some sentence which uses x. In this example the open formula is "x = x".

To return to the earlier example, "For all x, if $$D(x,t)$$ then $$D(x,f)$$", this has symbolization


 * $$\forall x(D(x,t)\to D(x,f))$$

Notice the importance of the parentheses wrapping around the open formula $$D(x,t)\to D(x,f)$$. If we only wrote


 * $$\forall x D(x,t)\to D(x,f)$$

then it would be ambiguous whether the $$\forall x$$ is meant to apply only to $$D(x,t)$$ or the entire $$D(x,t)\to D(x,f)$$.

(The reader might be able to reasonably guess the intended reading. But it is never good to let your reader guess, even when they might be able to.  Math is hard enough when it's precise — let's not make it harder by being unnecessarily ambiguous.)

Existential Quantification
Consider the (true) sentence "There is an even prime number." This has abstraction


 * "There is an x such that $$E(x)$$ and $$P(x)$$."

We use the symbol $$\exists$$ for the phrase "There is" or "There exists". Therefore this sentence has symbolization


 * $$\exists x (E(x)\land P(x))$$

Existential quantification, like universal quantification, works at the level of objects. But universal quantification requires that whichever object you assign to the variable x, you always get a true sentence. The existential quantifier only requires that there is at least one such object.

In our example sentence, the number 2 is even and prime, which is what makes that sentence true.

Conclusion of the Case Study
Finally we may symbolize the sentence with which we started this lesson. Recall that the sentence was


 * "For every two points there is a line through them."

It had abstraction


 * For every x and y, if $$P(x)$$ and $$P(y)$$, then there is a z such that $$L(z)$$ and $$T(z,x,y)$$.

Notice that, this time, there are two "for all" variables. Therefore we need two quantifiers for the initial choice of x and y.

Moreover, there is an existential sentence "in the middle" of the conditional statement.

The rest of the symbolization should be readable at this point, and so we present it below.


 * $$\forall x\forall y\Big((P(x)\land P(y))\to \exists z(L(z)\land T(z,x,y))\Big)$$

You should spend a moment to look back at the original sentence, and this symbolized sentence, and see how they align.