Chapter 3 - The Logic Of Quantified Statements

In Chapter 2 we discussed the logical analysis of compound statements those made of simple statements joined by the connectives.

Predicates. (In grammar)

In grammar, the word predicate refers to the part of a sentence that gives information about the subject.

For example: In the sentence “James is a student at Bedford College,” the word James is the subject and the phrase is a student at Bedford College is the predicate.

The predicate is the part of the sentence from which the subject has been removed.

Predicates. (In logic)

• A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.

• The domain of a predicate variable is the set of all values that may be substituted in place of the variable.

• The set of all such elements that make the predicate true is called the truth set of the predicate.

Predicate’s usage

If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x.

The truth set of P(x) is denoted. { x ∈ D | p(x)} → (“the set of all x in D such that P(x).”)

Quantifiers

Quantifiers are words that refer to quantities such as “some” or “all” and tell for how many elements a given predicate is true.

The Universal quantifier

∀ is called the universal quantifier. Depending on the context, it is read as “for every,” “for each,” “for any,” “given any,” or “for all.”

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Universal statements

• Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form “∀x element of D, Q(x).”

• It is defined to be true if, and only if, Q(x) is true for each individual x in D. It is defined to be false if, and only if, Q(x) is false for at least one x in D.

• A value for x for which Q(x) is false is called a counterexample to the universal statement.

The Existential Quantifier

∃ is read as “there exists, there is a, we can find a, there is at least one, for some, and for at least one.” and is called the existential quantifier.

Let Q(x) be a predicate and D the domain of x. An existential statement is a state ment of the form “∃ x elements of D such that Q(x).”

It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D.

Bound variables & Scope

• Bound variables are placeholders for elements in a set.

• Scope defines the range where a bound variable is valid.

• Quantifiers (like "for every" or "there exists") bind variables.

• Example: In "For every integer x, x^2 ≥ 0," x is bound by "for every" and its scope is the entire statement.

• Variable reuse: The same variable can have different meanings in different statements.

• Computer programming analogy: Variables in programming also serve as placeholders and can be bound by declarations or scopes.

Implicit Quantification

Implicit quantification is a type of quantification in which the quantifier is implied by the context rather than being explicitly stated. It's often used in natural language and mathematics to avoid redundancy or make statements more concise.

Key points about implicit quantification:

• It's a common practice in both formal and informal language.

• It can make statements more concise and easier to understand.

• It can be ambiguous if the context is unclear.

• It's important to be able to identify implicit quantifiers to accurately interpret statements.

Implicit Quantification usage

• The notation P(x) ⇒ Q(x) means that every element in the truth set of P(x) is in the truth set of Q(x), or, equivalently, ∀x, P(x) → Q(x).

• The notation P(x) ↔ Q(x) means that P(x) and Q(x) have identical truth sets, or, equivalently, ∀x, P(x) ↔ Q(x).

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Negation of Quantified Statements

• The negation of universal statement is Existential statement, and vice versa.

• The negation of Universal Conditional Statement is Existential and Conjunction statement (with the negation of the second part).

Statements with multiple quantifiers (of Uni/Exi)

When a statement contains more than one type of quantifier, the order in which they appear determines the order of actions.

pick whatever element x in D, and then you must find an element y in E that “works” for that particular x.

Example: "For all x in D, there exists y in E such that P(x, y)."

Flexibility: You can choose a different y for each x you're given.

Statements with multiple quantifiers (of Exi/Uni)

"There exists x in D such that for all y in E, P(x, y)."

Find one particular x in D that will “work” no matter what y in E any one might choose to challenge you with.

Order of Quantifiers.

In a statement containing both ∀ and Ǝ, changing the order of the quantifiers can significantly change the meaning of the statement.

Universal Instantiation

If a property is true of everything in a set, then it is true of any particular thing in the set.

If we know that a statement is true for all members of a group, we can apply it to any individual member of that group.

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Universal Modus Ponens

The rule of universal instantiation can be combined with modus ponens to obtain the valid form of argument called Universal Modus Ponens.

Universal Modus Ponens Rule: f something is true for everything in a group (e.g., "For all x, if P(x) then Q(x)") and it's true for one specific thing (e.g., "P(a)"), then it must be true for that specific thing (e.g., "Q(a)").

Universal Modus Tollens

Another crucially important rule of inference is Universal Modus Tollens. Its validity results from combining universal instantiation with modus tollens.

Using diagrams to Test for Validity

Testing for validity using diagrams can be a helpful visual method to understand logical arguments. Here's a simplified approach:

1. Identify the Premises and Conclusion: Write down the premises and the conclusion of the argument.

2. Draw Diagrams for Each Premise: Use Venn diagrams or other visual tools to represent each premise. For example, if a premise states "All A are B," draw a circle for A inside a circle for B.

3. Combine the Diagrams: Overlay or combine the diagrams to see if the conclusion logically follows from the premises. If the conclusion is represented in the combined diagram, the argument is valid.

4. Check for Consistency: Ensure that the combined diagram does not contradict any of the premises.

This method helps to visually verify if the conclusion necessarily follows from the premises, making it easier to spot logical errors.

Using Diagrams to Show Invalidity

Using diagrams to show invalidity can be a powerful visual tool to understand why an argument doesn't hold up. Here's a simplified approach:

1. Identify the Premises and Conclusion: Write down the premises and the conclusion of the argument.

2. Draw Diagrams for Each Premise: Use Venn diagrams or other visual tools to represent each premise. For example, if a premise states "All A are B," draw a circle for A inside a circle for B.

3. Combine the Diagrams: Overlay or combine the diagrams to see if the conclusion logically follows from the premises. If the conclusion is not represented in the combined diagram, the argument is invalid.

4. Check for Inconsistencies: Ensure that the combined diagram does not contradict any of the premises. If it does, the argument is invalid.

This method helps to visually verify if the conclusion does not necessarily follow from the premises, making it easier to spot logical errors.

Converse & Inverse Error (Quantified Form)

One reason why so many people make converse and inverse errors is that the forms of the resulting arguments would be valid if the major premise were a biconditional rather than a simple conditional.

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Universal Transitivity

Universal Transitivity, a logical rule derived from combining transitivity with universal instantiation.

• Transitivity Rule (General):
  Ifp→q and q→r, then p→r