Tag: logic

Having established both syntax and semantics, we now delve into the essential methods used to formally prove statements in predicate logic. Specifically, we’ll explore Natural Deduction and Hilbert-style systems, illustrating their application to real mathematical reasoning.

Natural Deduction in Predicate Logic

Natural deduction is a formal proof system that extends naturally from propositional logic, incorporating quantifiers. The fundamental idea is to use inference rules to construct proofs directly, step by step.

Extension of Propositional Logic Rules

Predicate logic inherits all propositional inference rules (such as conjunction, implication, negation) and adds rules for quantifiers.

Introduction and Elimination Rules for Quantifiers

Two new pairs of rules are introduced to handle quantifiers:

• Universal Quantifier (\(\forall\)):
  • Introduction (\(\forall I\)): If you can derive \(\varphi(x)\) for an arbitrary element \(x\), you may conclude \(\forall x \varphi(x)\).
  • Elimination (\(\forall E\)): From \(\forall x \varphi(x)\), you may infer \(\varphi(a)\) for any specific element aa.
• Existential Quantifier (\(\exists\)):
  • Introduction (\(\exists I\)): From \(\varphi(a)\) for a particular element \(a\), conclude \(\exists x \varphi(x)\).
  • Elimination (\(\exists E\)): If \(\exists x \varphi(x)\) holds, and from the assumption \(\varphi(a)\) (for an arbitrary \(a\)), you derive a conclusion \(\psi\) independent of \(a\), then you can conclude \(\psi\).

Examples of Formal Proofs (Natural Deduction)

Example: Prove the statement “If everyone loves pizza, then someone loves pizza.”

Formally: \(\forall x \text{LovesPizza}(x) \rightarrow \exists y \text{LovesPizza}(y)\)

Proof Sketch:

1. Assume \(\forall x \text{LovesPizza}(x)\).
2. From this assumption, choose any specific individual, say \(a\), then we have \(\text{LovesPizza}(a)\) by \(\forall E\).
3. Now apply existential introduction (\(\exists I\)): From \(\{\text{LovesPizza}(a)\}\), conclude \(\exists y \text{LovesPizza}(y)\).

This example illustrates the intuitive connection between universal and existential quantification.

Hilbert-Style Proof Systems

Hilbert-style proof systems build from axioms and inference rules, similar to propositional logic but augmented for quantification.

Typical Axioms involving Quantifiers

Hilbert-style systems typically add axioms to propositional logic to handle quantifiers, such as:

• Universal Instantiation: \(\forall x \varphi(x) \rightarrow \varphi(a)\), where \(a\) is any term.
• Existential Generalization: \(\varphi(a) \rightarrow \exists x \varphi(x)\).

These axioms enable formal manipulation of quantified statements without explicitly referring to inference rules.

Examples and Subtleties Introduced by Quantification

Quantifiers add subtleties not present in propositional logic:

• Order of quantifiers matters significantly: \(\forall x \exists y P(x,y)\) differs importantly from \(\exists y \forall x P(x,y)\).
• Careful treatment of variables (bound vs. free) is crucial to avoid ambiguities.

Real Mathematical Example

Statement: “For every integer \(x\), there exists an integer \(y\) such that \(y = x + 1\).”

Formally: \(\forall x \in \mathbb{Z}, \exists y (y = x + 1)\)

Proof (Natural Deduction Sketch):

• Consider an arbitrary integer \(x\).
• Let \(y = x + 1\). Clearly, \(y\) is an integer.
• Thus, \(\exists y(y = x + 1)\).
• Since \(x\) was arbitrary, we conclude \(\forall x \exists y(y = x + 1)\).

Complexities and Subtleties Introduced by Quantification

Quantification introduces significant complexity, especially regarding variable scope and substitutions. Consider the formula: \(\forall x \exists y (x < y)\)

It is intuitively clear that “for every integer, there is always a larger one.” Yet, rearranging quantifiers drastically changes the meaning: \(\exists y \forall x (y > x)\)

Now, the statement claims a single largest integer, which is false in standard integer arithmetic.

Exercises

1. Using natural deduction, prove formally:

\(\forall x (Even(x) \rightarrow \exists y (y = x + 2 \wedge Even(y)))\)

2. Demonstrate why the following statement is not logically valid:

\(\exists x \forall y (x > y)\)

Provide a counterexample to illustrate your reasoning clearly.

3. Translate and formally prove the following informal statement using natural deduction:

• “If some integer is even, then not all integers are odd.”

Summary

Proof techniques in predicate logic enable precise mathematical reasoning through formalized methods. While natural deduction closely mirrors intuitive mathematical reasoning, Hilbert-style systems rely on powerful axioms and simple inference rules to build proofs systematically. Understanding quantification’s subtleties is critical for rigorous mathematical reasoning.

Mastering these proof methods provides the essential skillset required to handle advanced mathematical and logical concepts effectively.

In the previous posts, we’ve discussed the syntax of predicate logic, outlining how terms, predicates, and formulas are formed. Now, we’ll explore semantics, explaining how meaning is formally assigned to these formulas.

Structures and Interpretations

The meaning of formulas in predicate logic is given by structures (or interpretations). Intuitively, a structure assigns concrete meanings to symbols, transforming abstract formulas into meaningful statements about specific objects.

Formal Definition of a Structure

Formally, a structure \(\mathcal{M}\) consists of:

• A non-empty set \(D\), the domain of discourse.
• An interpretation for each constant symbol, associating it with an element of \(D\).
• An interpretation for each function symbol \(f\) of arity \(n\), assigning a function: \(f^{\mathcal{M}}: D^n \rightarrow D\)
• An interpretation for each predicate symbol \(P\) of arity \(n\), assigning a subset of \(D^n\) indicating exactly when \(P\) holds true.

Formally, we represent a structure as: \(\mathcal{M} = (D, I)\)

where \(I\) gives the interpretations of all symbols.

Examples of Mathematical Structures

Example 1: Structure of integers with standard arithmetic

• Domain \(D = \mathbb{Z}\).
• Constants: integers like \(0, 1\).
• Functions: Arithmetic operations (addition \(+\), multiplication \(\times\)).
• Predicate “<” interpreted as: \(\{(x,y) \in \mathbb{Z}^2 \mid x < y \text{ as integers}\}\). Intuitively, the set defined by “<” includes all pairs of integers where the first integer is strictly less than the second.

Truth in a Model

Given a structure \(\mathcal{M}\), truth for formulas is defined recursively:

Atomic Formulas

An atomic formula \(P(t_1, \dots, t_n)\) is true in \(\mathcal{M}\) precisely when the tuple \((t_1^{\mathcal{M}}, \dots, t_n^{\mathcal{M}})\) lies in the subset defined by \(P\).

This aligns intuitively with our everyday notion of truth: “Even(2)” is true because 2 indeed satisfies our usual definition of evenness.

Extending Truth Definitions to Quantified Statements

For quantified formulas, truth is defined as follows:

• Universal Quantification (\(\forall x \varphi\)): True if, when we substitute any element from the domain for \(x\), the formula \(\varphi\) remains true.
  • Formally, \(\mathcal{M} \models \forall x \varphi\) means for every \(a \in D\), the formula \(\varphi[a/x]\) (formula \(\varphi\) with \(x\) replaced by \(a\)) is true in \(\mathcal{M}\).
• Existential quantification (\(\exists x \varphi\)) is true if there exists at least one element in the domain for which \(\varphi\) becomes true when \(x\) is interpreted as this element.
  • Formally, \(\mathcal{M} \models \exists x \varphi\) means there is at least one \(a \in D\) such that \(\varphi[a/x]\) is true in \(\mathcal{M}\).

In these definitions, \(\mathcal{M} \models \varphi\) means “formula \(\varphi\) is true in the structure \(\mathcal{M}\).”

Models and Logical Validity

• Model: A structure \(\mathcal{M}\) is a model for formula \(\varphi\) (written \(\mathcal{M} \models \varphi\)) if \(\varphi\) is true in \(\mathcal{M}\).
• Satisfiability: A formula is satisfiable if it has at least one model.
• Logical Validity: A formula is logically valid if it is true in every possible structure. We write this as: \(\models \varphi\)

Intuitively, a logically valid formula represents a statement that holds universally, regardless of how its symbols are interpreted.

Exercises

1. Describe explicitly a structure that makes the formula \(\exists x (x^2 = 4) \) true.
2. Consider the structure \(\mathcal{M}\) with domain \(\mathbb{N}\) and predicate \(Divides(x,y)\) meaning “\(x\) divides \(y\)”. Determine the truth value of:
   \(\forall x \exists y \; Divides(x,y)\)
3. Give an example of a logically valid formula and one that is satisfiable but not logically valid. Clearly explain why each has the given property.

This foundational understanding of semantics will enable us to move forward into exploring proof techniques and deeper logical analysis in predicate logic.

In the previous post, we saw why propositional logic is not sufficient to express general mathematical statements, and why predicate logic is required.
In this post, we’ll explore the formal syntax of predicate logic, detailing how statements are constructed using terms, predicates, quantifiers, and formulas.

Terms

In predicate logic, terms represent objects within a given domain. Terms can be:

• Variables (usually denoted by lowercase letters such as \(x\),\(y\),\(z\).
• Constants (symbolizing specific, fixed objects, e.g., \(0\),\(1\),\(a\),\(b\).
• Function symbols applied to terms (e.g., \(f(x)\),\(g(x,y)\)).

How Terms Are Built

Formally, terms are constructed recursively:

1. Any variable is a term.
2. Any constant symbol is a term.
3. If \(f\) is an \(n\)-ary function symbol and \(t_1, t_2, \dots, t_n\) are terms, then \(f(t_1, t_2, \dots, t_n)\) is also a term.

Examples of Terms:

• Variables: \(x\),\(y\)
• Constants: \(0\),\(1\)
• Function terms: \(f(x)\), \(+(x,y)\), \(\sin(z)\)

Predicates

Predicates express properties of terms or relations between them. A predicate \(P\) applied to terms \(t_1, t_2, \dots, t_n\) is written as \(P(t_1, t_2, \dots, t_n)\).

Examples:

• \(Even(x)\) — “\(x\) is even.”
• \(Prime(y)\) — “\(y\) is prime.”
• \(Greater(x,y)\) — “\(x\) is greater than \(y\).”

Predicates are crucial in clearly describing properties and relationships in logic.

Quantifiers

Quantifiers allow us to express statements involving generality or existence:

• Universal quantifier (\(\forall\)): means “for all” or “every.”
  • Example: \(\forall x, Even(x) \vee Odd(x)\) — “Every number is either even or odd.”
• Existential quantifier (\(\exists\)): means “there exists.”
  • Example: \(\exists y, Prime(y) \wedge y > 2\) — “There exists some \(y\) that is prime and greater than 2.”

Quantifiers bind variables within their scope, turning predicates into full propositions.

Forming Formulas

Predicate logic uses the following rules to form formulas:

1. Atomic Formulas: If \(P\) is an \(n\)-ary predicate and \(t_1, t_2, \dots, t_n\) are terms, \(P(t_1, t_2, \dots, t_n)\) is an atomic formula.
2. Complex Formulas:
   • If \(\varphi\) and \(\psi\) are formulas, then so are \((\varphi \wedge \psi)\), \((\varphi \vee \psi)\), \((\varphi \rightarrow \psi)\), \((\varphi \leftrightarrow \psi)\), \(\neg \varphi\).
   • If \(\varphi\) is a formula and \(x\) a variable, then \(\forall x \varphi\) and \(\exists x \varphi\) are also formulas.

Well-Formed Formulas (WFFs)

Formulas constructed following the above rules are called well-formed formulas. They represent meaningful mathematical statements.

Examples:

• \(\forall x (Prime(x) \rightarrow x > 1)\)
• \(\exists y (y^2 = 2)\)
• \(\forall x \exists y (y > x)\)

Exercises

1. Identify the terms, predicates, and quantifiers in the following formula: \(\forall x (Even(x) \rightarrow \exists y (y = x + 2 \wedge Even(y)))\)
2. Construct a predicate logic formula stating: “Every positive integer has a successor.”
3. Are the following expressions terms or formulas? Explain why.
   • \(f(x,g(y))\)
   • \(P(x,y,z)\)
   • \(\forall x, g(x)\)
4. Write a predicate logic formula to express: “There exists a number that is less than all other numbers.”

This post lays out the precise structure needed to form meaningful mathematical statements, setting the stage for exploring semantics and proofs in predicate logic.

Why Predicate Logic?

In our journey through formal mathematics, we’ve explored propositional logic—a powerful yet limited tool. Propositional logic allows us to reason about the truth of statements built from simpler ones using logical connectives. However, it falls short when we need to express statements about generalizations, properties, or specific relationships involving objects.

Predicate logic (also known as first-order logic) extends propositional logic by enabling precise expressions involving objects, their properties, and relationships between them. It’s an essential tool in mathematics, computer science, and logic because it captures a richer class of statements we frequently encounter.

Motivation and Limitations of Propositional Logic Revisited

Consider the statement:

“All prime numbers greater than 2 are odd.”

Using propositional logic alone, this statement can’t be properly captured. While propositional logic handles statements like “A number is prime,” it doesn’t support quantifying over numbers or expressing relationships like “greater than.”

This is where predicate logic comes into play, providing the necessary expressive power.

The Need for Expressing Generalizations, Properties, and Relationships

In mathematics and formal reasoning, we regularly encounter statements such as:

• “For every integer \(x\), \(x^2 \ge 0\).”
• “There exists a prime number greater than 1000.”
• “Every continuous function on a closed interval is bounded.”

Predicate logic gives us the tools—quantifiers, predicates, and terms—to rigorously state and analyze these types of claims.

What is Predicate Logic (First-Order Logic)?

Predicate logic enhances propositional logic by introducing:

• Terms: Variables, constants, and functions denoting mathematical objects.
• Predicates: Properties or relationships about objects.
• Quantifiers: Statements of generality or existence (“for all,” “there exists”).

These new elements significantly expand the expressive capability of logic, making it possible to formalize complex mathematical reasoning fully.

Historical Context and Significance

Predicate logic was formalized in the late 19th and early 20th centuries by logicians like Gottlob Frege, Bertrand Russell, and Alfred Tarski. It quickly became foundational, underpinning modern mathematics through the axiomatization of set theory.

In computer science, predicate logic provides the theoretical backbone for automated reasoning, databases, logic programming (like Prolog), and formal verification of software and hardware.

Comparison with Propositional Logic

Propositional logic deals with statements that can only be true or false without internal structure. Predicate logic, on the other hand, deals explicitly with the internal structure of statements, using quantification and detailed internal references to objects and their properties.

For example, propositional logic treats the statement “x is prime” as a simple true or false proposition. Predicate logic allows us to clarify precisely what “x” is and to reason explicitly about varying values of “x.”

Example Illustrating the Difference:

• Propositional logic:
  • “P: 2 is prime.”
  • “Q: 3 is odd.”
• Predicate logic:
  • “\(\forall x (\text{Prime}(x) \land x > 2 \rightarrow \text{Odd}(x))\)”

Here, predicate logic explicitly quantifies over numbers and specifies conditions clearly, something impossible in propositional logic.

Exercises

1. Provide examples of three mathematical statements that can be expressed clearly in predicate logic but not in propositional logic.
2. Can predicate logic express the following statement? Explain why or why not:
   • “There are infinitely many prime numbers.”
3. Rewrite the following informal statement in formal predicate logic:
   • “Every integer divisible by 4 is even.”

This foundational introduction sets the stage for exploring the syntax, semantics, and proof methods of predicate logic in greater detail.

Before we delve deeper into predicate logic, it’s important to clarify a few essential concepts from set theory. Predicate logic itself relies on some basic set-theoretic notions for its formal definitions and interpretations. This short introduction provides the minimal set theory you’ll need.

Introduction to Sets

A set is a collection of distinct objects, called elements, considered as a single entity.

• Examples:
  • Set of integers \(\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}\)
  • Set of real numbers \(\mathbb{R}\)

Membership and Subsets

• Membership: If an object aa belongs to a set \(A\), we write \(a \in A\).
  • Example: \(3 \in \mathbb{Z}\), \(\pi \in \mathbb{R}\).
• Subsets: A set \(A\) is a subset of another set \(B\) (written \(A \subseteq B\)) if every element of \(A\) is also in \(B\).
  • Example: The set of integers \(\mathbb{Z}\) is a subset of real numbers \(\mathbb{R}\), written as \(\mathbb{Z} \subseteq \mathbb{R}\).

Cartesian Product

The Cartesian product \(A \times B\) of sets \(A\) and \(B\) is the set of all ordered pairs where the first element is from \(A\) and the second from \(B\): \(A \times B = \{(a,b) \mid a \in A, b \in B\}\)

• Example: If \(A = \{1,2\}\) and \(B = \{x,y\}\), then: \(A \times B = \{(1,x), (1,y), (2,x), (2,y)\}\)

Relations and Functions

• A relation between sets \(A\) and \(B\) is a subset of their Cartesian product \(A \times B\).
  • Example: “Less than” relation on integers, represented as: \(\{(x,y) \mid x,y \in \mathbb{Z}, x<y\}\)
• A function from a set \(A\) to set \(B\) assigns exactly one element of \(B\) to each element of \(A\).
  • Formally: \(f: A \rightarrow B\).
  • Example: The square function on integers \(f(x)=x^2\) takes an integer \(x\) and maps it to its square in \(\mathbb{Z}\).

Relations as Subsets

In predicate logic, predicates are interpreted as subsets of Cartesian products. For instance, the predicate “\(<\)” (less than) on integers is the subset of all integer pairs \((x,y)\) where \(x<y\).

---

Exercises

1. Define the set \(A \times A\) explicitly, given \(A = \{0,1\}\).
2. Let \(A = \{1,2,3\}\). Write explicitly the subset defined by the predicate “greater than.”
3. Given sets \(A=\{a,b\}\), \(B=\{1,2\}\), and \(C=\{x\}\), determine \(A\times B\times C\).

These basic set-theoretic concepts are foundational to clearly understanding the semantics of predicate logic, enabling us to rigorously discuss structures and interpretations in logic.

In the previous posts, we’ve extensively discussed propositional logic, exploring its syntax, semantics, and proof techniques. Propositional logic is powerful and foundational; however, it has significant limitations in its expressiveness. Recognizing these limitations is essential to understanding why more advanced logical systems, such as predicate logic, are necessary.

Expressiveness Limitations

Propositional logic deals exclusively with entire statements (propositions) as indivisible units. It does not analyze the internal structure of these statements. Consequently, propositional logic cannot express statements that involve quantification, generalizations, or relationships between individual objects. It lacks the capability to handle statements that refer explicitly to particular individuals or properties that objects can possess.

This lack of expressiveness restricts propositional logic to very simple assertions, leaving many important mathematical and philosophical statements beyond its reach. To overcome this, predicate logic introduces the concepts of variables, quantifiers (such as “for all” and “there exists”), predicates, and functions, allowing for richer and more precise expression of complex ideas.

Examples of Statements Propositional Logic Cannot Express

To illustrate these limitations clearly, consider the following examples that propositional logic cannot adequately capture:

1. Generalizations:
   • “All humans are mortal.”
   • “Every even number greater than 2 is the sum of two primes.”
   Propositional logic cannot represent general statements involving “all” or “every,” since it cannot quantify over a set or category of objects.
2. Existential Statements:
   • “There exists an integer solution to the equation \(x^2 – 2 = 0\).”
   • “Some cats are black.”
   Statements involving existence or nonexistence of certain elements are beyond the scope of propositional logic since it has no concept of individual objects or variables.
3. Relational Statements:
   • “Alice is taller than Bob.”
   • “Paris is the capital of France.”
   These statements explicitly describe relationships between specific entities or individuals. Propositional logic treats such statements as atomic and provides no way to express the underlying structure or relationships explicitly.

In propositional logic, each of these statements would have to be represented by a single, unanalyzable symbol, losing all internal structural information.

Practical Implications

The expressiveness limitations of propositional logic have practical consequences, particularly in areas such as mathematics, computer science, and artificial intelligence.

• Complex Mathematical Reasoning: Propositional logic is insufficient for expressing and reasoning about even basic algebraic or geometric properties explicitly. For example, expressing and proving statements about arithmetic or geometric relationships requires the ability to quantify and reason about specific objects or numbers.
• Logical Reasoning in Computer Science: In database queries, rule-based systems, and software verification, propositional logic quickly reaches its limits. Queries such as “List all employees who have a salary greater than their manager” or verifying software correctness with quantified conditions necessitate the richer structure provided by predicate logic.

These practical scenarios underscore why moving beyond propositional logic is not just beneficial but essential for rigorous reasoning in more complex domains.

Transition to Predicate Logic

To address these limitations, we introduce predicate logic, also known as first-order logic. Predicate logic extends propositional logic by allowing:

• Variables and Quantification: Variables represent individuals or objects, and quantifiers such as “for all” (\(\forall\)) and “there exists” (\(\exists\)) allow us to state general or existential claims explicitly.
• Predicates and Relations: These represent properties of objects or relationships between objects, allowing for structured expressions such as “\(x\) is mortal” or “\(x\) is greater than \(y\).”
• Functions: Functions permit explicit expression of operations on objects, enhancing the expressiveness even further.

For instance, the statement “All humans are mortal” can be precisely expressed in predicate logic as:

\[\forall x (H(x) \rightarrow M(x))\]

meaning “for every object \(x\), if \(x\) is human (\(H(x)\)), then \(x\) is mortal (\(M(x)\)).”

In the upcoming posts, we will dive deeply into predicate logic, exploring its syntax, semantics, proof methods, and applications. This advancement will enable us to capture more sophisticated mathematical and philosophical concepts and significantly expand our logical toolkit.

In previous posts of this thread, we introduced formal proof techniques in propositional logic, discussing natural deduction, Hilbert-style proofs, and the fundamental concepts of soundness and completeness. Now, we turn to advanced proof strategies that enhance our ability to construct and analyze proofs efficiently. In particular, we will explore proof by contradiction and resolution, two powerful techniques frequently used in mathematics, logic, and computer science.

Proof by Contradiction

Proof by contradiction (also known as reductio ad absurdum) is a fundamental method in mathematical reasoning. The core idea is to assume the negation of the statement we wish to prove and show that this leads to a contradiction. If the assumption results in an impossible situation, we conclude that our original statement must be true.

Formalization in Propositional Logic

Proof by contradiction can be expressed formally as:

\((\neg P \vdash (Q \land \neg Q)) \vdash P\).

This means that if assuming \(\neg P\) leads to a contradiction (\(Q\land \neg Q\)), then \(\neg P\) must be false, so \(P\) holds. This formulation captures the essence of proof by contradiction: by demonstrating that an assumption results in a logical impossibility, we conclude that the assumption must have been incorrect.In propositional logic, suppose we wish to prove a formula \(P\).

Proof by contradiction consists of the following steps:

1. Assume \(\neg P\) (i.e., assume that \(P\) is false).
2. Using inference rules, derive a contradiction—i.e., derive a formula of the form \(Q \land \neg Q\), where \(Q\) is some proposition.
3. Since a contradiction is always false, the assumption \(\neg P\) must also be false.
4. Therefore, \(P\) must be true.

This follows from the principle of the excluded middle in classical logic, which states that for any proposition \(P\), either \(P\) or \(\neg P\) must be true.

Example in Propositional Logic

Let us prove that if \(P \rightarrow Q\) and \(\neg Q\) hold, then \(\neg P\) must also hold:

1. Assume the negation of the desired conclusion: Suppose \(P\) is true.
2. Use the given premises:
   • We know that \(P \rightarrow Q\) is true.
   • By Modus Ponens, since \(P\) is true, we must have \(Q\) as true.
   • However, we are also given that \(\neg Q\) is true, meaning that \(Q\) must be false.
3. Contradiction: Since \(Q\) is both true and false, we reach a contradiction.
4. Conclusion: Since our assumption \(P\) led to a contradiction, we conclude that \(\neg P\) must be true.

This establishes the validity of Modus Tollens: If \(P→Q\) is true and \(\neg Q\) is true, then \(\neg P\) is also true.

Applied Example

To illustrate how proof by contradiction works in an applied setting, consider proving that \(2\sqrt{2}\) is irrational.

We define the following propositions:

• \(R\): “\(2\sqrt{2}\) is irrational.”
• \(E_p\): “\(p\) is even.”
• \(E_q\): “\(q\) is even.”

1. Assume the opposite: Suppose that \(R\) is false, meaning \(2\sqrt{2}\) is rational and can be written as a fraction \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are integers with no common factors other than \(1\).
2. Square both sides: \(2 = \frac{p^2}{q^2}\), which implies \(2q^2 = p^2\).
3. Conclude that \(p^2\) is even: Since \(2q^2 = p^2\), \(p^2\) is divisible by \(2\), which means \(p\) must also be even. That is, \(E_p\) holds.
4. Write \(p\) as \(p=2k\) for some integer \(k\), then substitute: \(2q^2 = (2k)^2 = 4k^2\), so \(q^2 = 2k^2\).
5. Conclude that \(q^2\) is even, which implies that \(q\) is even, i.e., \(E_q\) holds.
6. Contradiction: Both \(p\) and \(q\) are even, contradicting the assumption that \(\frac{p}{q}\) was in lowest terms. That is, we have derived \(E_p \land E_q\), which contradicts the assumption that \(\neg (E_p \land E_q)\) held under \(R\).
7. Conclusion: Since assuming \(\neg R\) led to a contradiction, we conclude that \(R\) must be true. Therefore, \(2\sqrt{2}\) is irrational.

Proof by contradiction is a widely used technique, particularly in theoretical mathematics, number theory, and logic.

Resolution

Resolution is a proof technique commonly used in automated theorem proving and logic programming. It is based on the idea of refutation: to prove that a statement is true, we assume its negation and derive a contradiction using a systematic process.

Resolution operates within conjunctive normal form (CNF), where statements are expressed as a conjunction of disjunctions (i.e., sets of clauses). The resolution rule allows us to eliminate variables step by step to derive contradictions.

The Resolution Rule:

If we have two clauses:

• \(P \lor A\)
• \(\neg P \lor B\)

We can resolve them to infer a new clause:

• \(A \lor B\)

By eliminating \(P\), we combine the remaining parts of the clauses.

Example:

Suppose we have the following premises:

1. “Alice studies or Bob is happy.” \(S \lor H\)
2. “Alice does not study or Bob goes to the gym.” \(\neg S \lor G\)
3. “Bob does not go to the gym.” \(\neg G\)

We wish to determine whether Bob is happy (i.e., prove \(H\)).

Step 1: Apply Resolution

• From (2) and (3), resolve on \(G\): \(\neg S \lor G\) and \(\neg G\) produce \(\neg S\).
• From (1) and \(\neg S\), resolve on \(S\): \(S \lor H\) and \(\neg S\) produce \(H\).

Thus, we have derived \(H\), proving that Bob is happy.

Summary

• Proof by contradiction is a classical method that assumes the negation of a statement and derives a contradiction, proving that the statement must be true.
• Resolution is a formal proof technique used in logic and computer science, particularly in automated reasoning.

Both methods are powerful tools in mathematical logic, each serving distinct purposes in different areas of theoretical and applied reasoning.

Next Steps

Now that we have covered fundamental and advanced proof techniques in propositional logic, in the next post of this thread I will talk about the Limitations of Propositional Logic.

In the previous post, we explored the semantics of propositional logic using truth tables to determine the truth values of logical expressions. While truth tables are useful for evaluating small formulas, they become impractical for complex logical statements. Instead, formal proof techniques allow us to establish the validity of logical statements using deductive reasoning. This post introduces key proof methods in propositional logic, compares different proof systems, and discusses the fundamental notions of soundness and completeness.

Deductive Reasoning Methods

Deductive reasoning is the process of deriving conclusions from a given set of premises using formal rules of inference. Unlike truth tables, which exhaustively list all possible cases, deductive reasoning allows us to derive logical conclusions step by step.

A valid argument in propositional logic consists of premises and a conclusion, where the conclusion logically follows from the premises. If the premises are true, then the conclusion must also be true.

Common rules of inference include:

1. Modus Ponens (MP): If \(P \rightarrow Q\) and P are both true, then \(Q\) must be true.
   • Example:
     • Premise 1: If it is raining, then the ground is wet. (\(P \rightarrow Q\))
     • Premise 2: It is raining. (\(P\))
     • Conclusion: The ground is wet. (\(Q\))
2. Modus Tollens (MT): If \(P \rightarrow Q\) is true and \(Q\) is false, then \(P\) must be false.
   • Example:
     • Premise 1: If it is raining, then the ground is wet. (\(P \rightarrow Q\))
     • Premise 2: The ground is not wet. (\(\neg Q\))
     • Conclusion: It is not raining. (\(\neg P\))
3. Hypothetical Syllogism (HS): If \(P \rightarrow Q\) and \(Q \rightarrow R\) are true, then \(P \rightarrow R\) is also true.
4. Disjunctive Syllogism (DS): If \(P \lor Q\) is true and \(\neg P\) is true, then \(Q\) must be true.

These inference rules form the basis of formal proofs, where a conclusion is derived using a sequence of valid steps.

Formal Notation for Proofs

When working with formal proofs, we often use the notation (\(\vdash\)) to indicate that a formula is provable from a given set of premises. Specifically, if \( S \) is a set of premises and \( P \) is a formula, then:

\[
S \vdash P
\]

means that \( P \) is provable from \( S \) within a proof system.

It is important to distinguish between \(\vdash\) and \(\rightarrow\), as they represent fundamentally different concepts:

• The symbol \( P \rightarrow Q \) is a propositional formula that asserts a logical relationship between two statements. It states that if \( P \) is true, then \( Q \) must also be true.
• The symbol \( S \vdash P \) expresses provability: it states that \( P \) can be derived as a theorem from the premises \( S \) using a formal system of inference rules.

In other words, \( \rightarrow \) is a statement about truth, while \( \vdash \) is a statement about derivability in a formal system.

For example, Modus Ponens can be expressed formally as:

\[
P, (P \rightarrow Q) \vdash Q.
\]

This notation will be useful in later discussions where we analyze formal proofs rigorously.

Natural Deduction vs. Hilbert-Style Proofs

There are multiple systems for structuring formal proofs in propositional logic. The two primary approaches are Natural Deduction and Hilbert-Style Proof Systems.

Natural Deduction

Natural Deduction is a proof system that mimics human reasoning by allowing direct application of inference rules. Proofs in this system consist of a sequence of steps, each justified by a rule of inference. Assumptions can be introduced temporarily and later discharged to derive conclusions.

Key features of Natural Deduction:

• Uses rules such as Introduction and Elimination for logical connectives (e.g., AND introduction, OR elimination).
• Allows assumption-based reasoning, where subproofs are used to establish conditional statements.
• Proofs resemble the step-by-step reasoning found in mathematical arguments.

However, natural language statements remain ambiguous, which can lead to confusion. For instance, “If John studies, he will pass the exam” might not specify if passing the exam is solely dependent on studying. Later, when dealing with mathematical statements, we will ensure that all ambiguity is removed.

Example proof using Natural Deduction:

1. Assume “If the traffic is bad, I will be late” (\(P \rightarrow Q\))
2. Assume “The traffic is bad” (\(P\))
3. Conclude “I will be late” (\(Q\)) by Modus Ponens.

Hilbert-Style Proof Systems

Hilbert-style systems take a different approach, using a minimal set of axioms and inference rules. Proofs in this system involve applying axioms and the rule of detachment (Modus Ponens) repeatedly to derive new theorems.

Key features of Hilbert-Style Proofs:

• Based on a small number of axioms (e.g., axioms for implication and negation).
• Uses fewer inference rules but requires more steps to construct proofs.
• More suitable for metamathematical investigations, such as proving soundness and completeness.

Example of Hilbert-style proof:

1. Axiom: “If it is sunny, then I will go to the park” (\(P \rightarrow Q\))
2. Axiom: “If I go to the park, then I will be happy” (\(Q \rightarrow R\))
3. Using Hypothetical Syllogism: “If it is sunny, then I will be happy” (\(P \rightarrow R\))

While Hilbert-style systems are theoretically elegant, they are less intuitive for constructing actual proofs. Natural Deduction is generally preferred in practical applications.

Soundness and Completeness

A well-designed proof system should ensure that we only derive statements that are logically valid and that we can derive all logically valid statements. The concepts of soundness and completeness formalize these requirements and play a fundamental role in modern logic.

Soundness guarantees that the proof system does not allow us to derive false statements. If a proof system were unsound, we could deduce incorrect conclusions, undermining the entire logical structure of mathematics. Completeness, on the other hand, ensures that the proof system is powerful enough to derive every true statement within its domain. Without completeness, there would be true logical statements that we could never formally prove.

These properties are especially important in mathematical logic, automated theorem proving, and computer science. Soundness ensures that logical deductions made by computers are reliable, while completeness ensures that all provable truths can be algorithmically verified, given enough computational resources.

Since this is an introductory course, we will not formally define these concepts. However, informally we can state them as follows:

1. Soundness: If a formula can be proven in a formal system, then it must be logically valid (i.e., true in all possible interpretations).
   • This ensures that our proof system does not prove false statements.
   • Informally, if a statement is provable, then it must be true.
2. Completeness: If a formula is logically valid, then it must be provable within the formal system.
   • This guarantees that our proof system is powerful enough to prove all true statements.
   • Informally, if a statement is true in all interpretations, then we should be able to prove it.

Gödel’s Completeness Theorem states that propositional logic is both sound and complete—everything that is true can be proven, and everything that can be proven is true. However, the proof of this theorem is beyond the scope of this course.

Next Steps

Now that we have introduced formal proof techniques in propositional logic, the next step is to explore proof strategies and advanced techniques, such as proof by contradiction and resolution, which are particularly useful in automated theorem proving and logic programming.

In the previous post of this thread, we examined the syntax of propositional logic, focusing on how logical statements are constructed using propositions and logical connectives. Now, we turn to the semantics of propositional logic, which determines how the truth values of logical expressions are evaluated. This is achieved using truth tables, a fundamental tool for analyzing logical statements.

Truth Tables for Basic Connectives

A truth table is a systematic way to display the truth values of a logical expression based on all possible truth values of its atomic propositions. Each row of a truth table corresponds to a possible assignment of truth values to the atomic propositions, and the columns show how the logical connectives operate on these values.

It is important to emphasize that the truth tables for the basic logical connectives should be understood as their definitions. In the previous post, we introduced these connectives in natural language, but their precise meaning is formally established by these truth tables.

Below are the truth tables that define the basic logical connectives:

1. Negation (NOT, \(\neg P\)):

   \( P \)	\( \neg P \)
   T	F
   F	T

2. Conjunction (AND, \(P \land Q\)):

   \( P \)	\( Q \)	\( P \land Q \)
   T	T	T
   T	F	F
   F	T	F
   F	F	F

3. Disjunction (OR, \(P \lor Q\)):

   \( P \)	\( Q \)	\( P \lor Q \)
   T	T	T
   T	F	T
   F	T	T
   F	F	F

4. Implication (IMPLIES, \(P \rightarrow Q\)): Note: Implication is often misunderstood because it is considered true when the antecedent (P) is false, regardless of Q. This is due to its interpretation in classical logic as asserting that “if P is true, then Q must also be true.”

   \( P \)	\( Q \)	\( P \rightarrow Q \)
   T	T	T
   T	F	F
   F	T	T
   F	F	T

5. Biconditional (IF AND ONLY IF, \(P \leftrightarrow Q\)): The biconditional is true only when PP and QQ have the same truth value.

   \( P \)	\( Q \)	\( P \leftrightarrow Q \)
   T	T	T
   T	F	F
   F	T	F
   F	F	T

Tautologies, Contradictions, and Contingencies

Using truth tables, we can classify logical statements based on their truth values under all possible circumstances:

1. Tautology: A statement that is always true, regardless of the truth values of its components.
   • Example: \(P \lor \neg P\) (The law of the excluded middle)
2. Contradiction: A statement that is always false, no matter what values its components take.
   • Example: \(P \land \neg P\) (A proposition and its negation cannot both be true)
3. Contingency: A statement that is neither always true nor always false; its truth value depends on the values of its components.
   • Example: \(P \rightarrow Q\)

Logical Equivalence and Important Identities

Two statements A and B are logically equivalent if they always have the same truth values under all possible truth assignments. We write this as \(A \equiv B\).

Many logical identities can be proven using truth tables. As an example, let us prove De Morgan’s first law:

• Statement: \(\neg (P \land Q) \equiv \neg P \lor \neg Q\)

\( P \)	\( Q \)	\( P \land Q \)	\( \neg (P \land Q) \)	\( \neg P \)	\( \neg Q \)	\( \neg P \lor \neg Q \)
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

Since the columns for \(\neg (P \land Q)\) and \(\neg P \lor \neg Q \) are identical, the equivalence is proven.

Other important logical identities include:

1. Double Negation: \(\neg (\neg P) \equiv P\)
2. Implication as Disjunction: \(P \rightarrow Q \equiv \neg P \lor Q\)
3. Commutative Laws: \(P \lor Q \equiv Q \lor P\), \(P \land Q \equiv Q \land P\)
4. Associative Laws: \((P \lor Q) \lor R \equiv P \lor (Q \lor R)\)
5. Distributive Laws: \(P \land (Q \lor R) \equiv (P \land Q) \lor (P \land R)\)

The remaining identities can be verified using truth tables as an exercise.

Exercises

1. Construct the truth table for \(P \rightarrow Q \equiv \neg P \lor Q\) to prove their equivalence.
2. Use truth tables to verify De Morgan’s second law: \(\neg (P \lor Q) \equiv \neg P \land \neg Q\).
3. Prove the associative law for disjunction using truth tables: \((P \lor Q) \lor R \equiv P \lor (Q \lor R)\).

Next Steps

Now that we understand the semantics of propositional logic through truth tables and logical equivalence, the next step is to explore proof techniques in propositional logic, where we formalize reasoning through structured argumentation and derivations.

In the previous post of this thread, we introduced propositional logic and its purpose: to provide a formal system for analyzing and evaluating statements using logical structures. Now, we turn to the syntax of propositional logic, which defines the fundamental building blocks of this system.

Propositions and Atomic Statements

At the heart of propositional logic are propositions, which are statements that are either true or false. These propositions serve as the basic units of reasoning, forming the foundation upon which logical structures are built. The need for propositions arises because natural language can be ambiguous, making it difficult to determine the validity of arguments. By representing statements as precise logical symbols, we eliminate ambiguity and ensure rigorous reasoning.

Atomic statements are the simplest propositions that cannot be broken down further. These statements capture fundamental mathematical facts or real-world assertions. In mathematics, statements such as “5 is a prime number” or “A function is continuous at x = 2” are examples of atomic statements. In everyday language, sentences like “The sky is blue” or “It is raining” serve as atomic statements.

By introducing atomic statements, we create a standardized way to express truth values and establish logical relationships between different facts, allowing us to construct more complex reasoning systems.

Logical Connectives

While atomic statements provide the basic building blocks, more complex reasoning requires combining them. This is where logical connectives come into play. Logical connectives allow us to form compound statements from atomic ones, preserving precise meaning and facilitating logical deductions.

The primary logical connectives are:

1. Negation (NOT, \(\neg\)): Negation reverses the truth value of a proposition. If a statement is true, its negation is false, and vice versa.
   • Example: If \(P\) represents “It is raining,” then \(\neg P\) means “It is not raining.”
2. Conjunction (AND, \(\land\)): The conjunction of two propositions is true only if both propositions are true.
   • Example: \(P \land Q\) means “It is raining AND it is cold.”
3. Disjunction (OR, \(\lor\)): The disjunction of two propositions is true if at least one of them is true.
   • Example: \(P \lor Q\) means “It is raining OR it is cold.”
4. Implication (IMPLIES, \(\rightarrow\)): Implication expresses a logical consequence. If the first proposition (antecedent) is true, then the second (consequent) must also be true. This is often misunderstood because an implication is still considered true when the antecedent is false, regardless of the consequent.
   • Example: \(P \rightarrow Q\) means “If it is raining, then the ground is wet.” Even if it is not raining, the implication remains valid as long as there is no contradiction.
   • A common confusion arises because people often think of implication as causation, but in formal logic, it represents a conditional relationship rather than a cause-effect mechanism.
5. Biconditional (IF AND ONLY IF, \(\leftrightarrow\)): A biconditional statement is true when both propositions have the same truth value.
   • Example: \(P \leftrightarrow Q\) means “It is raining if and only if the ground is wet.” This means that if it is raining, the ground must be wet, and conversely, if the ground is wet, it must be raining.

Well-Formed Formulas (WFFs)

A well-formed formula (WFF) is a syntactically correct expression in propositional logic. The rules for forming WFFs include:

• Every atomic proposition (e.g., \(P, Q\)) is a WFF.
• If \(\varphi\) is a WFF, then \(\neg \varphi\) is also a WFF.
• If \(\varphi\) and \(\psi\) are WFFs, then \(\varphi \land \psi\), \(\varphi \lor \psi\), \(\varphi \rightarrow \psi\), and \(\varphi \leftrightarrow \psi\) are WFFs.
• Parentheses are used to clarify structure and avoid ambiguity (e.g., \((P \lor Q) \land R\)).

Conventions and Precedence Rules

To simplify expressions, we often omit unnecessary parentheses based on operator precedence. The order of precedence for logical operators is as follows:

1. Negation (\(\neg\)) has the highest precedence.
2. Conjunction (\(\land\)) comes next, meaning \(P \land Q\) is evaluated before disjunction.
3. Disjunction (\(\lor\)) follows, evaluated after conjunction.
4. Implication (\(\rightarrow\)) has a lower precedence, meaning it is evaluated later.
5. Biconditional (\(\leftrightarrow\)) has the lowest precedence.

For example, \(\neg P \lor Q \land R\) is interpreted as \((\neg P) \lor (Q \land R)\) unless explicitly parenthesized otherwise. Similarly, \(P \lor Q \land R \rightarrow S\) is evaluated as \(P \lor (Q \land R) \rightarrow S\) unless parentheses dictate otherwise.

Understanding these precedence rules helps avoid ambiguity when writing logical expressions.

Next Steps

Now that we understand the syntax of propositional logic, the next step is to explore truth tables and logical equivalence, which provide a systematic way to evaluate and compare logical expressions.