Mathematical Logic Statements and Notations

Mathematical logic uses formal notation to represent statements, determine their truth values, and reason about them systematically. The key building blocks are propositions, predicates, well-formed formulas, and quantifiers.

Proposition

A proposition is a declarative statement that has either a truth value "true" or a truth value "false". A proposition consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.

Some examples of propositions −

• "The sun rises in the east" − True
• "12 + 5 = 20" − False
• "x + 2 = 5" − Not a proposition (depends on the value of x)

Predicate

A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.

The following are some examples of predicates −

• Let E(x, y) denote "x = y"
• Let X(a, b, c) denote "a + b + c = 0"
• Let M(x, y) denote "x is married to y"

Well Formed Formula

A Well Formed Formula (wff) is a syntactically valid expression in predicate logic. A predicate is a wff if it satisfies any of the following rules −

• All propositional constants and propositional variables are wffs.
• If x is a variable and Y is a wff, then ∀x Y and ∃x Y are also wffs.
• Truth values (true and false) are wffs.
• Each atomic formula is a wff.
• All connectives (∧, ∨, ¬, →, ⇔) connecting wffs produce wffs.

Quantifiers

The variables of predicates are quantified by quantifiers. There are two types of quantifiers in predicate logic −

Universal Quantifier (∀)

The universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀.

∀x P(x) is read as "for every value of x, P(x) is true."

Example − "Man is mortal" can be written as ∀x P(x) where P(x) denotes "x is mortal" and the universe of discourse is all men.

Existential Quantifier (∃)

The existential quantifier states that the statements within its scope are true for at least one value of the specific variable. It is denoted by the symbol ∃.

∃x P(x) is read as "there exists some x for which P(x) is true."

Example − "Some people are dishonest" can be written as ∃x P(x) where P(x) denotes "x is dishonest" and the universe of discourse is all people.

Nested Quantifiers

When a quantifier appears within the scope of another quantifier, it is called a nested quantifier. The order of quantifiers matters −

• ∀a ∃b P(a, b) where P(a, b) denotes "a + b = 0" − means "for every a, there exists some b such that a + b = 0."
• ∀a ∀b ∀c P(a, b, c) where P(a, b, c) denotes "a + (b + c) = (a + b) + c" − means addition is associative for all values.

Note − The order of different quantifiers cannot be swapped: ∀a ∃b P(a, b) ≠ ∃a ∀b P(a, b). The first says "for every a, some b exists" while the second says "there is one specific a that works for all b."

Conclusion

Propositions are fixed true/false statements, predicates become propositions when variables are assigned values or quantified, and quantifiers (∀ and ∃) specify whether a predicate holds for all or some values. Together, these form the notation system of mathematical logic.