Proposition
A proposition (statement) primarily a form of words which expresses what is either true or false. Here, the word "primarily" means that one does not exclude other verbal symbols, or even mere thoughts if they have a symbolic character. But, as Russell pointed out, the word "proposition" should be limited to what may, in some, be called "symbols," and further to such symbols as give expression to truth and falsehood. Therefore, "two and two are four" and "two and two are five" will be propositions, and so will "Socrates is a man" and "Socrates is not a man."
As an example,
(a + b)2 = a2 + 2ab + b2, whatever numbers a and b may be
is a proposition. On other hand, the bare formula alone
(a + b)2 = a2 + 2ab + b2
is not a proposition. The reason, Russell gives, is that the bare formula alone (such as the above) asserts nothing definite unless we are further told, or let to suppose, that a and b are to have all possible values, or are to have such-and-such values.
To sum up the above, we say
| A proposition is a declarative sentence which is true or false, but not both. |
Proposition Form
A propositional form is an expression made up of sentential
variables (proposition variables such as p, q, and r) and sentential operators
(logical connectives such as ~,
,
and →) that becomes a proposition when actual
propositions are substituted for the component proposition variables. The truth
table for a given propositional form displays the truth values that correspond
to the different combinations of truth values for the variables.
Atomic Proposition
The atomic proposition is the one that is not build up from
other propositions by quantifiers and sentential operators. A typical atomic
proposition consists of two objects (expressions), one on each side of
connective such as =,
, > etc.
As an example, the proposition
2 + 3 = 5
is an atomic proposition. It consists of two expressions, 2 + 3 and 5, on each side of a connective, =.
Another example, consider the following proposition:
a
{x : x 
b
x
c}
is an atomic proposition. Note that the connective symbol
expresses a relation between the objects on either side. The list of all the
atomic propositions to be seen in proposition, a
{x : x
b
x
c}, would consists of three propositions namely, x
b, x
c, and whole proposition itself. The important point here is that the
whole proposition must also be classed as atomic proposition.
As an example, consider the following proposition:
x ( x > 0 →
y (y < x y2 > 0))
The list of all atomic propositions consists of x > 0, y < x and y2 > 0.
Example: For proposition:
a (a = {x : x a})
The list of all atomic propositions contains x
a and a (a = {x : x a}).
Finally, consider the following huge proposition:
y (y {x
: x > 0} →
x (z {x : x < 0 }
z2 < y)
The list of all atomic proposition is x > 0, x < 0, z2
< y, y {x : x > 0} and z {x : x < 0 }.
Term
Any expression representing an object of a mathematical system
is called a term. For example, in arithmetic (mathematical system), 3 is a term
which enter into the structure of the term 2 + 3. Similarly, in a (language of)
set, b is a term in the term, x b
x c, which, in turn, is
a part of the term {x : x b
x c}.
As an example, consider the following proposition:
x (2 + y < 2 + y)
Terms representing following objects in the proposition: 2, y, 2 + y, 3, and 3 + y.
Another example, consider the following proposition:
{x : x a
x b} = {y : y a
y b}
Terms in this proposition are: x, a, b, {x : x a
x b}, y, and {y : y a
y b}.
And finally, the proposition, A = π r2, contains following terms: A, π, r, 2, r2, and π r2.
Constants and Variables
Mathematical propositions are not only characterized by the fact they are assert implications, but also by the fact they contain variables. The notion of the variable is one of the most difficult with which logic has to deal. You may say "what the hell?" and come up with the following example from Elementary Arithmetic:
1 + 1 = 2
At first sight it might appear that neither it contains variable nor it asserts implication. But it can be shown, as Russell has shown, the true meaning of this proposition is:
"If x is one and y is one, and x differs from y, then x and y are two."
Now it is easy to see that the proposition, 1 + 1 = 2, contains both variables and asserts an implication. If you cannot see this clearly - don't worry about it. In fact, this idea leads to the concept of universal quantifier and we shall discus this in great detail later. But the main point here is when dealing with mathematical proposition we always find ourselves using words such as "any" or "some"; and these words are the marks of a variable and an implication.
The difference between variable and a constant is fairly obscured by mathematical usage. For instance, it is not unusual to speak of parameters as in some sense constants, but we will completely reject this usage. Russell says,
"A constant is to be something absolutely definite, concerning which there is no ambiguity whatever."
In this regard, 1, 2, 3, pi, Socrates, are constants; so are man, and the human race, past, present and future, collectively. Furthermore, proposition, implication, class, etc. are constants. In arithmetic, we encounter such constants as 0, 1, +, and many others. Each of these terms has a well-determined meaning which remain unchanged throughout the course of the consideration.
As opposed to the constants, the variables do not possess any meaning by themselves. For example, consider the following question:
Does zero have such and such a property?
Since zero is a constant, the question can be answered in the affirmative or in the negative; the answer may be true or false, but at any rate it is meaningful. On the other hand, a question concerning x, for example the question:
Is x an integer?
cannot be answered meaningfully. Why we cannot answered it meaningfully? because x is not a constant or little more precisely, it does not denote one definite object.
Therefore, according to the above argument we say that a proposition, any proposition, some proposition, are not constants, for these phrases do not denote one definite object. From the above discussion, we see that what are called parameters are simply variables. As an example, consider the equation of straight line in the plane:
ax + bx + c = 0
Here we say that x and y are variables, while a, b, c, are constants.
But, as Russell pointed out, unless we are dealing with one absolutely particular line, say the line form a particular point in London to a particular point in Cambridge, so-called constants a, b, c are not definite numbers, but stand for any numbers, therefore, are also variables.
Here in these lecture notes, we restrict "constant" to mean a primitive symbol that represents a unique mathematical object. For example, a particular point in London or 3 etc. The uniqueness of that mathematical object may be postulated, or the uniqueness may be established by a theorem. Sometimes we temporarily assume the existence of a certain unique object to show that such an object does not exist. For example, in the classical proof that there is no rational number whose square is 2, we show that the assumption that there is such a rational number leads to a contradiction. How we establish the uniqueness that is another story. The main point here is the restriction of constant to unique object.
The point of this discussion is that any particular symbol can be a constant in one context and a variable in another, or it may be used to represent one unique object in one system and a different one in another. For example, 0 may be used as the name of the empty set in set theory, or the name of the identity element of an abelian group in algebra, or the name of the minimum element of a partially ordered set in lattice theory.
Let us take this point a little deeper. It is quite usual to view variables as restricted to certain classes. For example, in Arithmetic, variables are supposed to stand for numbers. But this only means that if variables stand for number, they satisfy some formula. It is easy to see that we have an implication here. In other words, we are saying that variables are numbers implies the formula.
Now this assertion tells us that in this proposition it is no longer necessary that variable should be numbers: the implication holds equally when variables are not numbers (that is the whole idea of p implies q.) For example, stating an above proposition a little more formally, we say the proposition
x and y are numbers implies (x + y) = x2 + 2xy + b2.
holds equally for x = Socrates and y = Pluto. The reason is that both hypothesis and conclusion will be false, but the implication will still be true.
Therefore, every proposition, when fully stated, the variables have an absolutely unrestricted field. That is, we my substitute anything for any one of variables without damaging the truth of a proposition.
We conclude our discussion of constants and variables by saying that the constants have restricted field consists of logical constants while variable have an absolutely unrestricted field.
Now that we know something about constants and variable, let us talk about proposition a little more. Let us start with a following proposition which is, in fact, a special theorem of arithmetic.
If 1 is a positive number and 1 < 2, then 1 is a positive number.
Obviously, this proposition is true since it contains constants only.
To generalize the above proposition we shall introduce the variables (say) "p" and "q", demanding that these variable symbols stands for whole proposition; variable of this kind are denoted as propositional variables (or sentential variables). In this manner, we arrive at the propositonal function:
if p and q, then p
So, lets conclude this section by saying that our symbolism starts with agreement to use the small letters a, b, c, ..., p, q, r, s, ..., sometimes with subscripts, to stand for propositions in general. We shall call these letters propositional variables (or sentential variables), when so used. Expressions involving such symbols will be called propositional functions.
A propositional form is an expression made up of propositional variables such as p, q, and r; and propositional operators (logical connectives such as ~, ∧, ∨ and →) that becomes a proposition when actual propositions are substituted for the component proposition variables. The truth table for a given propositional form displays the truth values that correspond to the different combinations of truth values for the variables. The "method of truth table", originates with Peirce, enables us to recongnize whether a given proposition, from a certain set (so-called domain of the propositional logic) is true and therefore, the given proposition can be counted among the laws of logic.
Mathematical Systems
A formal mathematical system ∑ is determined by
1. A Given set of undefined symbols (or undefined expressions).
We can roughly divide these symbols in four categories:
a. Objects of the
systems represented by letters and numerals.
b. Arithmetic
symbols represented by +,
c. Grammatical symbols severs as
verbs such as =, ,
etc.
d.
Connective symbols severs as modifiers of whole sentences such as
.
2. A set of rules for forming propositions. A set of rules determining which finite strings of symbols qualify as statements. Any proposition formed in accord with these rules is called a "proposition of the system" or a "proposition of ∑" whether it is provable, disprovable, or undecided.
3. A set of propositions called postulates. These must qualify as propositions according to the rules. Along with the postulate we can include definitions, subject to special restrictions.
Scope Conventions
we shall refer to the five symbols ~,
,
, →, and ↔
as sentential operators (or logical connectives). The sentential operator, ~, is
a monary (or unary) operator, affecting a single sentential variable (or
statement). Whereas, each of the other (,
, →, ↔) involving
two sentential variables for its use. The negation symbol will have minimum
scope. Thus, ~p
q will mean (~p)
q. Among the binary operators,
conjunction and disjunction have the smallest scope. So, p
q → r will
mean (p q) → r, not p
(q → r).
Note that a formula such as p q
r is not
permitted since it is ambiguous without brackets.
Thus, the full hierarchy of operators for the five sentential operators (or logical connectives) can be summarized as follows:
|
Order of Operations |
|
| 1. | ~ |
| 2. |
,
|
| 3. | →, ↔ |
For practice, we will construct a dictionary of sentential variable, breaking it down as far as possible with respect to sub-statements. Then write the corresponding sentential formula, using the scope conventions where appropriate.
Example 1. Statement: x > 4 and y2 > x, then y > 2 or y is not positive.
Dictionary:
p: x > 4
q: y2 > x
r: y > 2
s: y is positive.
Formula: p
q →
r ~y
Example 2. Statement: If ABC is a triangle and a line segment, (AD), bisects angle A, then (BD) / (CD) = (AB) / (AC) and (AD)2 = (AC)(AB) – (AD)(CD).
Dictionary:
p: ABC is a triangle.
q: (AD) bisects angle A.
r: (BD) / (CD) = (AB) / (AC)
s: (AD)2 = (AC)(AB) – (AD)(CD)
Formula: p
q →
r s
Example 3. Statement: |sin2x1 – sin2x2| < k when |x1 – x2| < h, if k > 0 and h < k/2.
Dictionary:
p: k > 0
q: h < k/2
r: |sin2x1
– sin2x2|
< k
s: |x1 – x2| < h
Formula: p
q →
r → s
Example 4. If lim f = a and lim g = b, then
lim fg = ab, and ,
provided that b 
0,
lim f/g = a/b.
Dictionary:
p: lim f = a
q: lim g = b
r: lim f/g = a/b
s: b
0
t: lim f/g = a/b
Formula: p
q →
r (s → t)
Example 5. Statement: The sum of the numbers x and y is rational only if both are rational or both are irrational.
Dictionary:
p: x + y is a rational
q: x is a rational
r: y is a rational
Formula: p →
(q r)
(~q
~r)
----------------------------------------------------------