👁️‍🗨️Formal Logic I Unit 12 – Natural Deduction in Predicate Logic

What's Natural Deduction?

• System of formal logic used to prove the validity of arguments based on a set of inference rules
• Starts with premises and uses rules of inference to derive conclusions step-by-step
• Differs from other proof systems like truth tables or resolution by mimicking the natural reasoning process
• Proofs are constructed as a sequence of lines, each line being derived from previous lines using inference rules
• Allows for the introduction and elimination of logical connectives (and, or, if-then) and quantifiers (for all, there exists)
• Provides a foundation for understanding formal proofs in mathematics, philosophy, and computer science
• Can be applied to both propositional logic and predicate logic, with additional rules for handling quantifiers in predicate logic

Key Concepts and Terminology

• Premise: a statement that is assumed to be true for the purpose of the argument
• Conclusion: the statement that is derived from the premises using rules of inference
• Inference rule: a logical rule that allows new lines to be derived from previous lines in a proof
• Logical connectives: symbols used to combine propositions, such as $\wedge$ (and), $\vee$ (or), $\rightarrow$ (if-then), and $\neg$ (not)
• Quantifiers: symbols used to express the quantity of elements in a domain that satisfy a given predicate, such as $\forall$ (for all) and $\exists$ (there exists)
• Scope: the portion of a formula that a quantifier applies to, usually denoted by parentheses or brackets
• Free variable: a variable that is not bound by a quantifier and can take on any value from the domain
• Bound variable: a variable that is bound by a quantifier and ranges over the specified domain
• Domain: the set of elements that variables can take values from in a given context

Rules of Inference

• Modus Ponens (MP): if $P$ and $P \rightarrow Q$ are true, then $Q$ is true
• Modus Tollens (MT): if $P \rightarrow Q$ and $\neg Q$ are true, then $\neg P$ is true
• Hypothetical Syllogism (HS): if $P \rightarrow Q$ and $Q \rightarrow R$ are true, then $P \rightarrow R$ is true
• Disjunctive Syllogism (DS): if $P \vee Q$ and $\neg P$ are true, then $Q$ is true
• Constructive Dilemma (CD): if $(P \rightarrow Q) \wedge (R \rightarrow S)$ and $P \vee R$ are true, then $Q \vee S$ is true
• Conjunction Introduction (Conj): if $P$ and $Q$ are true, then $P \wedge Q$ is true
• Conjunction Elimination (Simp): if $P \wedge Q$ is true, then $P$ is true and $Q$ is true
• Disjunction Introduction (Add): if $P$ is true, then $P \vee Q$ is true for any $Q$
• Biconditional Introduction (Bicond): if $P \rightarrow Q$ and $Q \rightarrow P$ are true, then $P \leftrightarrow Q$ is true

Quantifiers and Their Role

• Universal quantifier ($\forall$): expresses that a predicate holds for all elements in a domain
  • Example: $\forall x (P(x) \rightarrow Q(x))$ means for all $x$, if $P(x)$ is true, then $Q(x)$ is true
• Existential quantifier ($\exists$): expresses that a predicate holds for at least one element in a domain
  • Example: $\exists x P(x)$ means there exists at least one $x$ such that $P(x)$ is true
• Quantifiers allow for reasoning about properties of entire sets or domains rather than just individual elements
• The order of quantifiers matters, as it determines the dependencies between variables
  • Example: $\forall x \exists y P(x, y)$ is different from $\exists y \forall x P(x, y)$
• Quantifier negation rules:
  • $\neg (\forall x P(x)) \equiv \exists x \neg P(x)$
  • $\neg (\exists x P(x)) \equiv \forall x \neg P(x)$
• Quantifier instantiation: replacing a universally quantified variable with a specific term or constant
• Existential generalization: introducing an existential quantifier when a statement holds for a specific element

Constructing Proofs

• Begin by clearly stating the premises and the conclusion to be proved
• Work backwards from the conclusion, looking for inference rules that can be applied to derive it
• Introduce assumptions as needed, using the premise lines or derived lines
• Apply inference rules to derive new lines from the premises and previously derived lines
• Use quantifier rules to instantiate variables or introduce new variables as needed
• Ensure that each line is properly justified by citing the inference rule and the line numbers it depends on
• Continue the process until the conclusion is derived, at which point the proof is complete
• Review the proof for clarity, correctness, and conciseness, making any necessary revisions
• Label the proof with a title or description for easy reference

Common Proof Strategies

• Direct proof: assume the premises and use inference rules to derive the conclusion directly
• Proof by contradiction: assume the negation of the conclusion and show that it leads to a contradiction with the premises or known facts
  • Also known as reductio ad absurdum
• Proof by cases: break the problem into distinct cases and prove the conclusion for each case separately
  • Useful when the conclusion depends on different conditions or scenarios
• Existential instantiation: when proving a statement with an existential quantifier, introduce a new constant to represent the element that satisfies the predicate
• Universal generalization: when a statement has been proved for an arbitrary element, conclude that it holds for all elements in the domain
• Hypothetical proof: assume an additional premise temporarily and use it to derive the conclusion, then discharge the assumption using the Deduction Theorem
• Proof by induction: prove a statement for a base case and then show that if it holds for one case, it must hold for the next case, concluding that it holds for all cases

Pitfalls and How to Avoid Them

• Forgetting to consider all possible cases or scenarios when using proof by cases
  • Ensure that the cases are exhaustive and mutually exclusive
• Misusing quantifiers or confusing their order, leading to incorrect statements
  • Pay close attention to the scope and order of quantifiers, using parentheses to clarify when needed
• Introducing unnecessary assumptions or variables that complicate the proof
  • Aim for a concise and straightforward proof, only introducing elements that are essential to the argument
• Failing to properly justify lines or cite the inference rules used
  • Each line should be clearly justified by citing the inference rule and the line numbers it depends on
• Confusing implication ($\rightarrow$) with biconditional ($\leftrightarrow$), or necessary conditions with sufficient conditions
  • Remember that implication is a one-way relationship, while biconditional is a two-way relationship
• Skipping steps or making leaps in reasoning that are difficult for others to follow
  • Break down the proof into smaller, more manageable steps to ensure clarity and understanding
• Misinterpreting the meaning of logical connectives or quantifiers in the context of the problem
  • Carefully consider the intended meaning of each symbol and how it relates to the problem at hand

Practical Applications

• Analyzing arguments in philosophy, law, and everyday reasoning to determine their validity and soundness
• Verifying the correctness of mathematical proofs and theorems
• Designing and validating algorithms in computer science and artificial intelligence
• Modeling and reasoning about complex systems in fields such as economics, biology, and social sciences
• Developing formal specifications and verifying the properties of software and hardware systems
• Creating and evaluating logical frameworks for decision-making and problem-solving in various domains
• Teaching critical thinking and logical reasoning skills to students in mathematics, philosophy, and other disciplines
• Contributing to the development of automated theorem provers and proof assistants for use in research and industry