Propositional Logic

Logic deals with the formalization of natural language and reasoning methods.

A proposition is a sentence that is either true or false, but not both.

Propositional logic is the study of the structure and meaning of (simple and complex) propositions.

Key questions are:

• How is the truth value of a complex proposition obtained from the truth value of its simpler components?
• Which propositions represent correct reasoning arguments?

Examples

Example simple propositions include:

• I passed the proficiency exam.
• 5+1 = 6
• 426 > 1721
• It is 52 degrees outside right now.

Some complex propositions are:

• John is five and Mary is six.
• I passed the proficiency exam or I did not pass it.

Sentences which are not propositions include:

• Did Bill get an A on the 113 exam?
• Go away!

Propositional Formulas

In studying properties of propositions we represent them by expressions called proposition forms or formulas.

Formulas are built from propositional variables, which represent simple propositions, and symbols representing logical connectives such as and, or, not, etc.

We use the letters p, q, r, s, to denote propositional variables and the symbols such as , , and to denote the standard logical connectives.

Proposition:

I passed the exam or I did not pass it.

Formula:

The formula expresses the logical structure of the proposition, where p is essentially an abbreviation for the simple proposition ``I passed the exam.''

Proposition:

If I do not pass the exam I will fail the course.

Formula:

Note. The term propositional logic also often refers to the formal language of propositional formulas.

Negation

We use the symbol to denote negation.

Formalization (syntax): If is a formula, then is also a formula. We say that the second formula is the negation of the first.

For instance, p, , and are all formulas.

The similarity in the structure of a formula and its negation reflects a relationship between the meaning of propositions of this form.

Meaning (semantics): If a proposition is true, then its negation is false. If it is false, then its negation is true.

Example:

• John went to the store yesterday.
• John did not go to the store yesterday.

At the formula level we express the connection via a so-called truth table:

If p is true, then is false.

If p is false, the is true.

Conjunction

We use the symbol to denote conjunction.

Syntax: If and are formulas, then is also a formula.

Semantics: If is true and is true, then is true. In all other cases, is false.

Truth table:

Example:

1. Bill went to the store.
2. Mary ate cantaloupe.
3. Bill went the store and Mary ate cantaloupe.

If p and q abbreviate the first and second sentence, then the third is represented by the conjunction .

Inclusive Disjunction

We use the symbol to denote (inclusive) disjunction.

Syntax: If and are formulas, then is also a formula.

Semantics: If is true or is true or both are true, then is true. If and are both false, then is false.

Truth table:

Example:

• John works hard.
• Mary is happy.
• John works hard or Mary is happy.

Exclusive Disjunction

We use the symbol to denote exclusive disjunction.

Syntax: If and are formulas, then is also a formula.

Semantics: An exclusive disjunction is true if, and only if, one of or is true, but not both.

Truth table:

Example:

• Either John works hard or Mary is happy.

Evaluation of Formulas

The semantics of logical connectives determines how propositional formulas are evaluated depending on the truth values assigned to variables.

Each possible truth assignment or valuation for the variables of a formula yields a truth value. The different possibilities can be summarized in a truth table.

Example: (read ``p and not q'')

Example: (read ``p and, in addition, q or r'')

Note that it is usually necessary to evaluate all subformulas as indicated by the respective rows.

Truth Tables

A truth table for a formula lists all possible ``situations'' of truth or falsity, depending on the values assigned to the variables of the formula.

Example:

If p, q, r are the propositions ``Peter [Quincy, Richard] will lend Sam money,'' then Sam can deduce, not too surprisingly but logically correct, that he will be able to borrow money whenever one of his three friends is willing to lend him some.

Each row in the truth table corresponds to one possible situation of assigning truth values to p, q and r.

How many rows are there in a truth table with n variables? For n=1, there are two rows, for n=2, there are four rows, for n=3, there are eight rows, and so on. Do you see a pattern?

Constructing Truth Tables

There are two choices (true or false) for each of n variables, so in general there are rows for n variables.

How can we construct these rows systematically?

• Count in binary - 000, 001, 010, 011, 100, ...
• Make two copies of the table with n-1 rows with different first columns.

A systematic procedure (an algorithm) is necessary to make sure you construct them all without duplicates.

The rightmost column must be computed as a function of all the truth values in the row.

Because it is clumsy and time-consuming to build large explicit truth tables, we will be interested in more efficient logical evaluation procedures.

Syntax of Formulas

The formal language of propositional logic can be specified by grammar rules.

The syntactic structure of a complex logical expression (i.e., its parse tree) must be unambiguous.

Parentheses are needed to avoid ambiguities. For example, the expression can be interpreted in two different ways:

Without parentheses the meaning of the formula is not clear! The same problem arises in arithmetic: does mean or ?

Simplified Syntax

In arithmetic one often specifies a precedence among operators (say, times ahead of plus) to eliminate the need for some parentheses in certain programming languages. The same can be done for the logical connectives, though deleting parentheses may cause confusion.

The properties of the logical connectives can also be exploited to simplify the notation.

For instance, disjunction is commutative,

and associative,

We will therefore ambiguously write to denote either or . The ambiguity is usually of no consequence, as both formulas have the same meaning.

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