Predicate Logic Overview

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Questions and Answers

What is a predicate?

A predicate is a proposition defined with variables. The truth value of the predicate depends on the variables.

What is the domain of interpretation in predicate logic?

The domain of interpretation is the set of objects and values that the variables within a predicate can take from. It provides concrete meaning to symbols and terms used in a formal system.

What does the universal quantifier (∀) mean?

  • There are
  • For all (correct)
  • Some
  • At least one
  • There is

What does the existential quantifier (∃) mean?

<p>Some (A), There are (C), There is (D), At least one (E)</p> Signup and view all the answers

If the domain is finite, then ∀xP(x) is the same as P(x₁) ∧ P(x₂) ∧ ... ∧ P(xn).

<p>True (A)</p> Signup and view all the answers

What is the negation of 'all apples are sweet'?

<p>Some apples are not sweet (C)</p> Signup and view all the answers

What is the negation of 'Everybody loves somebody sometime'?

<p>Somebody hates everybody all the time</p> Signup and view all the answers

What is the negation of 'Some pictures are old or faded'?

<p>All pictures are neither old nor faded.</p> Signup and view all the answers

What is the negation of 'All people are tall and thin'?

<p>Some people are not tall or not thin.</p> Signup and view all the answers

∀x∀yP(x, y) is the same as ∀y∀xP(x, y).

<p>True (A)</p> Signup and view all the answers

∃xyP(x, y) is the same as ∃y∃xP(x, y).

<p>True (A)</p> Signup and view all the answers

What is the scope of a variable in a quantified expression?

<p>The scope of a variable in a quantified expression is the portion of the expression where that variable is bound by the quantifier. The quantifier controls the variable within its scope.</p> Signup and view all the answers

What is a free variable in a quantified expression?

<p>A free variable in a quantified expression is a variable that is not within the scope of any quantifier. Such variables might not have a truth value.</p> Signup and view all the answers

What is the truth value for the expression ∃x (A(x) ∧ ∀y(B(x, y) → C(y))) where A(x) is interpreted as x > 0, B(x, y) as x ≤ y, and C(y) as y ≤ 0, with the domain of both x and y as all integers?

<p>True</p> Signup and view all the answers

Explain why it doesn't make sense to have ∃x(P(x) → Q(x)).

<p>The implication (→) is typically used with a universal quantifier (∀) because it expresses a conditional statement that holds for all elements in the domain. The existential quantifier (∃) is more naturally used with a conjunction (∧), as it expresses the existence of an element satisfying both conditions.</p> Signup and view all the answers

How can we translate the phrase 'X loves only Y' into predicate logic using implication?

<p>X loves only Y = If X loves anything, then that thing is Y. This can be represented in predicate logic as ∀z(L(x, z) → (z = y)) where L(x, z) represents 'x loves z'.</p> Signup and view all the answers

How can we translate the phrase 'Only X loves Y' into predicate logic?

<p>Only X loves Y = If anything loves Y, then it is X. This can be represented in predicate logic as ∀z(L(z, y) → (z = x)) where L(z, y) represents 'z loves y'.</p> Signup and view all the answers

What is the formal logic representation of the statement 'All students are intelligent'?

<p>∀x(S(x) → I(x)) where S(x) represents 'x is a student' and I(x) represents 'x is intelligent'.</p> Signup and view all the answers

What is the formal logic representation of the statement 'Some intelligent students like music'?

<p>∃x(I(x) ∧ S(x) ∧ M(x)) where I(x) represents 'x is intelligent', S(x) represents 'x is a student', and M(x) represents 'x likes music'.</p> Signup and view all the answers

What is the formal logic representation of the statement 'Everyone who likes music is a stupid student'?

<p>∀x(M(x) → ¬(I(x) ∧ S(x))) where M(x) represents 'x likes music', ¬(I(x) ∧ S(x)) represents 'x is not both intelligent and a student'.</p> Signup and view all the answers

What is the formal logic representation of the statement 'All computers run all programs'?

<p>∀x(C(x) → ∀y(P(y) → R(x, y))) where C(x) represents 'x is a computer', P(y) represents 'y is a program', and R(x, y) represents 'x runs y'.</p> Signup and view all the answers

What is the formal logic representation of the statement 'Only computers run programs'?

<p>∀x, y(P(y) ∧ R(x, y) → C(x)) where P(y) represents 'y is a program', R(x, y) represents 'x runs y', and C(x) represents 'x is a computer'.</p> Signup and view all the answers

What is the negation of the statement 'Some students eat only pizza'?

<p>All students eat something that is not pizza.</p> Signup and view all the answers

What is the negation of the statement 'Only students eat pizza'?

<p>There exists a non-student who eats pizza.</p> Signup and view all the answers

Explain how the validity of a predicate argument differs from the validity of a propositional argument.

<p>A predicate argument is valid when the implication is always true regardless of the interpretation of the predicates, meaning it holds true for all possible domains and interpretations of the predicates. A propositional argument is valid when the implication is a tautology, meaning it's always true regardless of the truth values of the propositional variables. Predicate arguments are more complex because they incorporate quantifiers and domains, adding layers of interpretation.</p> Signup and view all the answers

What are the four new rules introduced for proofs in predicate logic?

<p>The four new rules are universal instantiation (ui), existential instantiation (ei), universal generalization (ug), and existential generalization (eg). These rules are used to manipulate quantifiers and derive conclusions in predicate logic proofs.</p> Signup and view all the answers

What are the key differences between using universal instantiation (ui) and existential instantiation (ei)?

<p>In ui, we have the power to choose an arbitrary instance or constant symbol, while in ei, we get whatever instance is provided and do not have control over selection. Universal instantiation (ui) allows choosing specific values from the domain, while existential instantiation (ei) involves working with existing instances and cannot choose the specific instance.</p> Signup and view all the answers

What are implicit quantifiers and why are they important in predicate logic?

<p>Implicit quantifiers are quantifiers that are not explicitly stated but are implied by the context of the statement. Identifying implicit quantifiers is crucial for correctly understanding and formalizing everyday language into logic.</p> Signup and view all the answers

What are implicit hypotheses in predicate logic arguments and why are they important?

<p>Implicit hypotheses are assumptions or background knowledge that are not explicitly stated in an argument but are necessary for the argument's validity. These hypotheses often involve definitions of concepts, properties, or relationships within the domain of discourse.</p> Signup and view all the answers

Flashcards

Predicate

A proposition that has variables and its truth value depends on the values assigned to those variables.

Domain of Interpretation

The set of values that variables in a predicate can take on. It gives meaning to the symbols in the predicate.

Universal Quantifier (∀)

It means "for all" or "every". It applies to a predicate, making the statement true for all values in the domain.

Existential Quantifier (∃)

It means "there exists" or "at least one". It applies to a predicate, making the statement true if at least one value in the domain satisfies it.

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∀xP(x) = 𝑃(𝑥1) ∧ 𝑃(𝑥2) ∧ ⋯ ∧ 𝑃(𝑥𝑛)

When the domain is a finite set of values, the universal quantifier is equivalent to the conjunction of the predicate applied to each value in the domain.

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∃xP(x) = 𝑃(𝑥1) ∨ 𝑃(𝑥2) ∨ ⋯ ∨ 𝑃(𝑥𝑛)

When the domain is a finite set of values, the existential quantifier is equivalent to the disjunction of the predicate applied to each value in the domain.

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∀xP(x) implies ∃xP(x)

It's not possible to find an interpretation of P(x) where ∀xP(x) is true but ∃xP(x) is false.

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∃xP(x) can be true while ∀xP(x) is false

Example 1: P(x): "x is yellow" (domain: all flowers). Example 3: P(x): x > 5 (domain: all integers).

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¬ ∀xP(x) ≡ ∃x¬P(x)

It's the same as saying "there exists an x such that P(x) is false".

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¬ ∃xP(x) ≡ ∀x¬P(x)

It's the same as saying "for all x, P(x) is false".

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¬ ∀xP(x) = Some not

The negation of "all apples are sweet" is the same as saying "some apples are not sweet".

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¬ ∃xP(x) = All not

The negation of "some apples are sweet" is the same as saying "all apples are not sweet".

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Negating Quantified Statements

The negation of "Everybody loves somebody sometime" is "Somebody hates everybody all the time".

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Negating Compound Predicates

The negation of "Some pictures are old or faded." is "All pictures are neither old nor faded".

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Negating Combined Predicates

The negation of "All people are tall and thin" is "Some people are not tall or not thin".

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Unary Predicate

Predicates that involve properties of a single variable. Example: P(x) - "x is a cat"

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Binary Predicate

Predicates that involve properties of two variables. Example: Q(x, y): "x is taller than y".

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Ternary, N-ary Predicates

Predicates involving properties of more than two variables. Example: R(x, y, z): "x is located between y and z".

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∀x∀yP(x, y) = ∀y∀xP(x, y)

∀x∀yP(x, y) is the same as ∀y∀xP(x, y) ≡ ∀x, yP(x, y).

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∃x∃yP(x, y) = ∃y∃xP(x, y)

∃x∃yP(x, y) is the same as ∃y∃xP(x, y) ≡ ∃x, yP(x, y).

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∀x∃yP(x, y) ≠ ∃y∀xP(x, y)

∀x∃yP(x, y) is not the same as ∃y∀xP(x, y).

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Scope of a Variable

The part of the statement where a quantifier applies. For example, in ∀x(P(x, y) → ∃yQ(x, y)), the scope of ∀x is the entire expression within parentheses, while the scope of ∃y is only Q(x, y).

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Free variable

A variable that is not within the scope of any quantifier in a statement. Example: In ∀x(P(x, y) → ∃yQ(x, y)), y in P(x, y) is a free variable.

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Valid Predicate Argument

An argument involving predicates where the implication derived from the premises is always true regardless of the interpretation of the predicates.

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Proofs in Predicate Logic

Formal rules for proving statements in predicate logic. Includes universal instantiation, existential instantiation, universal generalization, and existential generalization.

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Instantiation

A technique for replacing a quantified variable with a concrete value in a proof. For universal quantifiers, you can choose any value. For existential quantifiers, you must use a constant symbol not previously used in the proof

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Universal Generalization

A technique for deriving a universal quantifier from a statement that holds true for any arbitrary value. You can only generalize from universally quantified statements.

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Existential Generalization

A technique for deriving an existential quantifier from a statement that holds true for a specific value.

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Implicit Quantification

Statements where quantifiers are not explicitly stated, but are implied by the context.

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Implicit Hypotheses

Assumptions or background knowledge that are necessary to prove a statement in predicate logic.

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Study Notes

Predicate Logic Overview

  • Predicate logic extends propositional logic by introducing predicates and quantifiers.
  • Predicates are propositions with variables; their truth depends on the values of those variables.
  • Quantifiers specify the domain over which variables are defined (e.g., "for all," "there exists").

Predicates

  • A predicate is a statement that can be true or false depending on the values of its variables.
  • Predicates can have multiple variables (e.g., binary predicates).
  • Examples of predicates include "x is a month that has 31 days," "a > b," or "x is a cat."
  • The domain of a variable defines the set of values it can take.

Quantifiers

  • Quantifiers are symbols used to quantify variables in predicates.
  • The universal quantifier (∀) means "for all" or "for every." Examples: ∀xP(x)
  • The existential quantifier (∃) means "there exists" or "there is at least one." Examples: ∃xP(x)
  • Quantifiers bind variables; variables within their scope depend on the quantifier.

Expansion with Quantifiers

  • When the domain is finite (e.g. a set containing x₁, x₂, x₃, ... xₙ)
  • ∀x P(x) is equivalent to P(x₁)∧P(x₂)∧…∧P(xₙ).
  • ∃x P(x) is equivalent to P(x₁)∨P(x₂)∨…∨P(xₙ).

Negation of Quantified Predicates

  • De Morgan's Law applies:
    • ¬(∀x P(x)) ≡ ∃x ¬P(x)
    • ¬(∃x P(x)) ≡ ∀x ¬P(x)

Translation Exercises

  • Steps for translating verbal statements to predicate logic involve defining predicates and determining the correct quantifiers.
  • Examples include translating "all students are intelligent" or "some intelligent students like music."

Scope of a Variable

  • The scope of a quantified variable encompasses the part of the expression it affects.
  • Variables outside the scope of a quantifier are called free variables.

Validity of Predicate Arguments

  • A predicate argument is valid if the conclusion is true whenever the premises are true.
  • Valid arguments hold true under all interpretations.
  • Validity can be checked using instantiation and generalisation, similar to propositional logic. These proofs use the instantiation rules (UI = universal instantiation, EI= existential) and generalisation rules (UG = universal generalisation, EG = existential generalisation).

Proofs in Predicate Logic

  • New rules are needed to handle quantifiers, but other propositional logic rules apply.
  • Key rules include Universal Instantiation (UI), Existential Instantiation (EI), Universal Generalization (UG), and Existential Generalization (EG).

Implicit Quantification

  • Often, quantifiers are implied in statements.
  • For instance, "If an integer is divisible by 4, it is divisible by 2" implies that this is universally true for all integers.
  • A statement's implicit quantifier might indicate if it's generally true or true for a certain set.

Implicit Hypotheses

  • Implicit hypotheses are assumptions about entities in an argument.
  • Defining predicate logic statements requires implicit assumptions about the domain of variables.
  • Definitions of key terms (e.g. odd or even) might be necessary.

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