,
,
| and →, respectively.Proposition Operators
In this section, we will consider five operations (of
propositions) namely, negation, conjunction, disjunction, incompatibility, and
implication represented by sentential operators (or connectives) ~,
,
,
| and →, respectively.
1. Negation
The negation is the simplest operation of propositions. This is that operation (function) of proposition p which is true when p is false, and false when p is true. As Russell says, it is a lot more convenient to speak of the truth of a proposition, or its falsehood, as its "truth-value"; That is, truth is the "truth-value" of a true proposition, and falsehood is a false one. Note that the term, truth-value, is due to Frege and following Russell's advise, we shall use the letters p, q, r, s, ..., to denote variable propositions. The modern equivalent is the term "sentential variable" see above.
Formally,
| If p is a proposition variable, the negation of p is "not p" or "It is not the case that p." Symbolically, we shall denote the general negation of p by ~p. The prefixed symbol is called a tilde. Negation of p has opposite truth value form p. That is, if p is true, then ~p is false; if p is false, ~p is true. |
More commonly, we use internal negation to negate a sententials (proposition variables.) For instance, "b is a rational number" becomes "b is not a rational number," and "x = 5" becomes "x ≠ 5", and so on.
To infer the truth of a proposition, we must know that some other proposition is true, and there is some relation between the two (called implication; we shall define this relation shortly.) Or we may know that a certain other proposition is false, and that there is a "disjunction" relation between the two so that the knowledge that the one is false allows us to infer that the other is true. Again, our problem is to infer may be the falsehood of some proposition, not its truth. This may be inferred from the truth of another proposition, provided we know that the two are "incompatible," i.e. that if one true, the other false. It may also be inferred from the falsehood of another proposition, in just the same way in which the the truth of the other might have been inferred from the truth of the one; That is, from the falsehood of p we may infer the falsehood of q, when q implies p.
Using the above Russell's arguments, we can further formalize the negation as follow:
If p is true, then ~p is false; and if p is false, then ~ p is true.
The truth value of ~p may be defined equivalently by the table below. Thus the truth value of the negation of p is always the opposite of the truth of p.
| p | ~p |
| T | F |
| F | T |
In other word, ~p is a false proposition when p is true and true when p is false. The proposition ~p is called the negation of p.
As an example, consider the "Double Negative Property":
| p | ~p | ~(~p) |
| T | F | T |
| F | T | F |
It is important to note that the double negative property holds in a system of logic in which every statement is either true or false, only then we have ~(~p) has the same meaning as p, whatever statement p is. As an example, consider a system of logic in which statement can have one of three values say, "true, T", "false, F," and "maybe, M." Then, it would depend on what basic table is agreed upon for the operation ~. Suppose, we agreed upon the following table:
| p | ~p |
| T | M |
| M | F |
| F | T |
With the above table, ~(~p) and p would not be interchangeable (i.e., not equivalent), since ~(~p) would have the value F when p has the value T.
2. Conjunction
The joining of two or more propositions by the word "and" results in their so-called conjunction or logical product; the propositions joined in this manner are called the members of the conjunction or the factors of the logical product. The conjunction, "p and q", has truth for its truth-value when p and q are both true; Otherwise it has falsehood for its truth-value.
Formally,
| If p and q are proposition variables, the conjunction of p and q
is a compound proposition "p and q." We symbolize the logical conjunction of p and q by p
q. It is true when, and only when, both p and q are true. If either p or q is
false, or if both are false, p
q is false. |
Equivalently,
If p and q are true, then p
q is true; otherwise p
q is false.
The truth values for conjunction (or equivalently, the truth
table for the operator
) are depicted in a following table.
| p | q | p
q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
There are other English conjunction that have the same logical
meaning as "and," although their psychological content may be differ.
Famous examples are "but" and "however." Some additional
examples are "moreover," "nevertheless," and "in
addition to which." Whatever the verbalization may be, we symbolize the
logical conjunction of p and q by p
q.
3. Disjunction
The joining of two or more propositions by the word "or" results in their so-called disjunction or logical sum; the propositions joined in this manner are called the members of the disjunction or the summands of the logical sum. The disjunction, "p or q", has truth for its truth-value when p is true and also when q is true, but if falsehood when both p and q are false.
Formally,
| If p and q are proposition variables, the disjunction of p and q
is "p or q" and we symbolize the logical disjunction of p and q by p
q. It is true when at least one of p or q is true and is false only when both p
and q are false. |
Equivalently,
If p and q are false, then p
q is false; otherwise p
q
is true.
The truth values for disjunction (or equivalently, the truth
table for the operator
) are depicted in a following table.
| p | q | p
q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
In logic, we avoid possible ambiguity about the meaning of the word "or" by understanding it to mean the inclusive "and/or." In logic, we generally used "or" in the nonexclusive sense. For instance, to say "x is rational or it is negative" does not mean that it cannot be both rational and negative. We shall use consistently the interpretation that at least one of the choices must hold, and both may be hold.
4. Incompatibility
Russell defines incompatibility as the proposition whose truth-value is truth when p is false and likewise when q is false; its truth-value is falsehood when p and q are both true. We symbolize the logical incompatibility of p and q by p | q. Accordingly, the table for the operator | is shown in the following table.
| p | q | p | q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
Note that incompatibility is also the disjunction of the negation of p and q i.e. "~p or ~q."
5. Implication
We shall talk implication in a separate section, in depth. Here, we simply define and talk about the meaning in a general sense. We interpret the meaning of Implication, i.e. "p implies q," or "if p, then q" as "Unless p is false, q is true," or "either p is false or q is true," where p and q are proposition variables. That is to say, "p implies q" is to mean "~p or q"; its truth-value is to be truth if p is false, likewise if q is true, and is to be falsehood if p is true and q is false. The fact that "implies" is capable of other meanings does not concern us: For the time being this is our "meaning" and we are sticking to it.
Accordingly, the truth table for the "implies" or "p → q" is shown below:
| p | q | p → q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Needless to say we will introduce other operations of propositions too, for example, biconditional, joined falsehood, "~p and ~q," etc. But for now, the above five operations (or sentential operators) will suffice. It is easy to see that negation differ from other four in being a monary (or unary) operator, whereas each of the others is a binary operator, involving two propositions (not necessarily different) for its use. But all five operations agree on this, that their truth-value depends only upon that of the propositions which are their arguments. Given the truth or falsehood of p, or of p and q (as the case may be), we are given the truth or falsehood of the negation, disjunction, incompatibility, or implication. A proposition which has this property is called truth-function. We shall talk about the truth-function shortly.
Propositional Form
A propositional form (or statement form) is an expression made
up of sententials (statement variables) such as p, q, and r, and sentential
operators (or logical connectives) such as ~,
,
, that becomes a
sentential formula (or statement) when actual statements are substituted for the
compound statement variable. The truth table for a given statement propositional
form displays the truth values that correspond to the different combinations of
truth values for the sentential variables.
As an example, construct the truth table for propositional
form (p q)
~(p
q). As we have
said, when "or" is used in its exclusive sense, the statement "p
or q" means "p or q but not both" or "p or q and not both p
and q" which translate into (p
q)
~(p
q) and this
sometimes abbreviated (p
q) or (p XOR q) depending upon the context.
| p | q | p
q |
p
q |
~(p
q) |
(p
q)
~(p
q) |
| T | T | T | T | F | F |
| T | F | T | F | T | T |
| F | T | T | F | T | T |
| F | F | F | F | T | F |