Tips on Symbolizing Compound Propositions in the Propositional Calculus
Conjunction: The conjunction of two (or more) propositions (the "conjuncts") is the assertion of the truth of both (or all) of them. This relation is most commonly expressed in English using the "...and..." or, for emphasis, the "Both...and...." construction where the conjuncts that fill in for the "..." are truth functional independent clauses; i.e. each can be independently true or false.
There are many other ways of expressing this conjunctive relationship which differ in expressive or emotive tone from a simple "and" but which, with respect to truth value, also assert the truth of both conjoined clauses. The next most common after "and" is no doubt "but." Other common conjunction words include "therefore," "however," "moreover," "also," "additionally," "in addition to," "futhermore," "still," "yet," "nevertheless," and "although."
Negation: The negation of a proposition is the denial of the truth of that proposition, which, in a truth-value logic (like the propositional calculus) is equivalent to the assertion of the proposition's falsity. If a proposition is true its negation is false, if it is false, its negation is true. In English this logical relationship is most obviously expressed either by placing an expression such as "It is false that..." or "It is not the case that..." in front of the clause which asserts the proposition that is being negated or by inserting the word "not" inside the proposition.
There is no difficulty in symbolizing simple negations; the only problem here is likely to be simply careless reading which fails to notice a "not" at a crucial point. Always rescan the sentence you're symbolizing to check that you haven't overlooked a "not."
Negation and Conjunction: When negation and junction are combined, the negation may be either on the separate conjuncts individually or it may be on the whole conjunction; one denies the truth of the individual conjunct, the other denies the "bothness," so to speak, expressed by the conjunction, i.e. by the "dot." In the former case, where it is the individual conjunct being negated, the negation sign, ~, is placed directly in front of the symbol for the proposition. In the latter case, where what is being negated is the conjunction itself, the ~ is placed in front of the expression for the conjunction which must then be put into parentheses, brackets, or braces.~( P . Q) Failure to put the "punctuation" with the appropriate parentheses (or brackets or braces) makes the formula ambiguous thus making its truth value undefined. Generally the negation of the whole conjunction, where the ~ stands outside the parentheses, will be expressed by the "It is not the case that..." type of construction.
The negation of a conjunction is not logically equivalent to the conjunction of the conjuncts negated; i.e. ~( P . Q ) is not equivalent to (~P . ~Q). Failure to be sensitive to this fact is the greatest problem in symbolzing these propositions. [The negation of a conjunction is logically equivalent to the disjunction of the two former conjuncts, now disjuncts, negated; this is DeMorgan's Theorem.]
You must be extremely careful in reading the proposition to see what is being negated, i.e. whether it is a conjunt being negated or whether it is the conjunction which is negated, or possibly both. Here are the possibilities:
Negation of one conjunct only: "Not-p and q." (~p . q) or "p and not-q." (p . ~q)
Negation of both conjuncts individually: "Not-p and not-q." (~p . ~q)
Negation of only the conjunction: "It is not the case that both p and q." ~(p. q)
Negation of conjunction and negation of one conjunct: "It is not the case that not-p and q." ~(~p . q) or
"It is not the case that p and not-q." ~(p . ~q)
Negation of conjunction and both conjuncts: It is not the case that not-p and not-q." ~(~p . ~q)
[By De Morgan's this last expression is equivalent to (p v q)]
Disjunction: The relationship symbolized by inclusive disjunction asserts that one or the other or possibly both of the disjuncts are true. A disjunction is true then in three possible cases: the left disjunct is true, the right disjunct is true, or both dijuncts are true. A disjunction is false only in the case that both disjuncts are false.
Symbolizing disjunctions is normally easy because the word "or" almost always appears. Sometimes the disjunction is reinforced by "either," but the appearance of the word "either" is not essential in simple disjunctions. (In complex disjunctions where the disjuncts are themselves compound expressions, the word "either" may be necessary to avoid ambiguity.) The only means in English for expressing this relationship in which "or" does not appear uses the word "unless." (Like the word "or" the word "unless" is ambiguous; in many contexts it is normally assumed to assert the exclusive disjunction; in other contexts the inclusive; by arbitrary stipulation we rule here to interpret "unless," like ""or" as expressing the inclusive disjunction.)Disjunction and Negation: The combination of disjunction with negation allows two possibilities: negation of the disjuncts individually and/or negation of the disjunctive realtionship between them. In the former case the ~ is placed in front of the propsitional sumbol representing the disjunct which is being negated; in the latter case, the negation of the disjunction (their "eitherness") the negation stands in front of the expression for their disjunction which must be placed in parentheses (or brackets or braces). Failure to put the "punctuation" with the appropriate parentheses (or brackets or braces) makes the formula ambiguous thus making its truth value undefined.
In English the negation of a disjunction is almost always expressed by the "neither...nor" construction; do not misinterpret "neither...nor..." sentences as disjunctions of the two propositions negated. The negation of a disjunction is not logically equivalent to the disjunction of the disjuncts negated. Failure to be sensitive to this fact is the greatest problem in symbolzing these propositions. [The negation of a disjunction is logically equivalent to the conjunction of the two former disjuncts, now conjuncts, negated; this is DeMorgan's Theorem.]
You must be extremely careful in reading the proposition to see what is being negated, whether it is the disjuncts or the disjunction, or possibly both. Here are the possibilities:
Negation of one disjunct: "Either not-p or q." (~p v q)
"Either p or not-q." (p v ~q)
Negation of both disjuncts: "Either not-p or not-q." (~p v ~q)
Negation of only the disjunction: "Neither p nor q." ~(p v q)
Negation of disjunction and one disjunct: "Neither not-p nor q." ~(~p v q) or
"Neither p nor not-q." ~(p v ~q)
Negation of both disjuncts and disjunction: "Neither not-p nor not-q." ~(~p v ~q)
[By De Morgan's this last expression is equivalent to (p. q)]
Disjunction and Conjunction: Conjunctions and disjunctions composed of simple propositions may themselves be conjoined and disjoined, to yield complex propositions in which the conjuncts or dijuncts themselves have internal propositional relations. The "pure cases" in which conjunctions are thsemselves composed only of conjunctions, e.g. [(p . q) . (r . s)] or disjunctions composed only of disjunctions, e.g. [(p v q) v (r v s)] should give no problem because the grouping of the conjuncts or disjuncts is entirely arbitrary; in such pure cases. They may be grouped any way (this is the law of association for conjunction and disjunction).
However, the "mixed cases" in which conjunctions and disjunctions are combined must be grouped correctly by parentheses (or brackets or braces). Failure to group these mixed cases correctly is the primary cause for errors with propositions of this kind of logical structure.
In symbolizing these propositions be sure to pay careful attention to the placing of "either" and "both" as well as of commas and semi-colons.
The conjunction of two disjunctions is not logically equivalent to the disjunction of two conjunctions.
Here are the possibilities for cases with a mixture of conjunction and disjunction (i.e. of "dots" and "wedges"):
A conjunction in which one conjunct is a disjunction: "Both p, and either q or r." [p . (q v r)]
"Both either p or q, and r." [(p v q) . r]
A conjunction in which both conjuncts are disjunctions: "Both either p or q, and either r or s." [(p v q) . (r v s)]
A disjunction in which one disjunct is a conjunction: "Either p, or both q and r." [p v (q . r)]
"Either both p and q, or r." [(p . q) v r]
A disjunction in which both disjuncts are conjunctions: "Either both p and q, or both r and s." [(p . q) v (r . s)]