Predicate Logic, Universal and Existential Quantifiers


Discrete Mathematics I

6th lecture, Oct. 27, 2017

Martin J. Dürst


© 2005-17 Martin J. Dürst Aoyama Gakuin University

Today's Schedule


Schedule for the Next Few Weeks


Summary of Last Lecture


Tautology and Contradiction


Types of Symbolic Logic


Limitations of Propositions

With propositions, related statements have to be made separately

Today it is sunny. Tomorrow it is sunny. The day after tomorrow, it is sunny.
2 is even. 5 is even.

We can express "If today is sunny, then tomorrow will also be sunny." or "If 2 is even, then 3 is not even".

But we cannot express "If it's sunny on a given day, it's also sunny on the next day." or "If x is even, then x+2 is also even.".



The problem with propositions can be solved by introducing predicates.

In the same way as propositions, predicates are objectively true or false.

A predicate is a function (with 0 or more arguments) that returns true or false.

If the value of an argument is undefined, the result (value) of the predicate is unknown.

A predicate with 0 arguments is a proposition.


Examples of Predicates

sunny(today), sunny(tomorrow), sunny(yesterday), even(2), even(5), ...

Generalization: sunny(x), even(y), ...

Using predicates, we can express new things:

Similar to propositions, predicates can be true or false.

But predicates can also be unknown/undefined, for example if they contain variables.

Also, even if a predicate is undefined (e.g. even(x)),
a formula containing this predicate can be defined
(true, e.g. even(y) → even(y+2), or false, e.g. odd(z) → even(z+24))


First Order Predicate Logic


Universal Quantifier

Example: ∀n∈ℕ: even (n) → even(n+2)


General form: ∀x: P (x)

∀ is the A of "for All", inverted.

Readings in Japanese:


Universal Quantifier for the Empty Set

x∈{}: P(x) = T

Reason: We have to check P(x) for all elements x in the set.
If we find even one x where P(x) is false, the overall statement is false.
But we cannot find any x where P(x) is false.

Application example:

All students in this room from Hungary are over 50 (years old).

See: Vacuous truth,


Existential Quantifier

Example: ∃n∈ℕ: odd (n)


General form: ∃y: P (y)

∃ is the mirrored form of the E in "there Exists".

Readings in Japanese:


More Quantifier Examples

n∈ℕ: n + n + n = 3n

n∈ℕ: n2 = n3

n∈ℕ: n2 < 50n < n3

m, n∈ℕ: 7m + 2n = 2n + 7m


Peano Axioms in Predicate Logic

Peano Axioms (Guiseppe Peano, 1858-1932)

  1. 1∈ℕ
  2. a∈ℕ: s(a)∈ℕ
  3. ¬∃x∈ℕ: s(x) = 1
  4. a, b∈ℕ: abs(a) ≠ s(b)
  5. P(1) ∧ (∀a∈ℕ: (P(a)→P(s(a)))) ⇒ ∀a∈ℕ: P(a)


Laws for Quantifiers

  1. ¬∀x: P(x) = ∃x: ¬P(x)
  2. ¬∃x: P(x) = ∀x: ¬P(x)
  3. (X≠{}∧∀xX: P(x)) → (∃x: P(x))
  4. (∀x: P(x)) ∧ Q(y) = ∀x: P(x)∧Q(y)
  5. (∃x: P(x)) ∧ Q(y) = ∃x: P(x)∧Q(y)
  6. (∀x: P(x)) ∨ Q(y) = ∀x: P(x)∨Q(y)
  7. (∃x: P(x)) ∨ Q(y) = ∃x: P(x)∨Q(y)
  8. (∀x: P(x)) ∧ (∀x: R(x)) = ∀x: P(x)∧R(x)
  9. (∀x: P(x)) ∨ (∀x: R(x)) ⇒ ∀x: P(x)∨R(x)
  10. (∃x: P(x)) ∨ (∃x: R(x)) = ∃x: P(x)∨R(x)
  11. (∃x: P(x)) ∧ (∃x: R(x)) ⇐ ∃x: P(x)∧R(x)
  12. (∃y: ∀x: P(x, y)) ⇒ (∀x: ∃y: P(x, y))
  13. P(x) is a tautology ⇔∀x: P(x) is a tautology


This Week's Homework

Deadline: November 9, 2017 (Thursday), 19:00.

Format: A4 single page (using both sides is okay; NO cover page; an additional page is okay if really necessary, but staple the pages together at the top left corner), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Problem 1: Show that the Wolfram axiom of Boolean logic is a tautology (you can use either a truth table or formula manipulation).

Problem 2: For ternary (three-valued) logic, create truth tables for conjunction, disjunction, and negation. The three values are T, F, and ?, where ? stands for "unknown" (in more words: "maybe true, maybe false, we don't know").

Hint: What's the result of "?∨T"? ? can be T or F, but in both cases the result will be T, so ?∨T=T.

Problem 3: For each of the laws 1, 5, 8, 11, and 12 of "Laws for Quantifiers", imagine a concrete example and explain it. For laws 11 and 12, give examples for both why the implication works one way and why the implication does not work the other way.



predicate logic
恒真 (式)、トートロジー
恒偽 (式)
symbolic logic
multi-valued logic
fuzzy logic
first-order predicate logic
temporal logic
binary logic
higher-order logic
universal quantifier
全称限量子 (全称記号)
existential quantifier
存在限量子 (存在記号)