Questions and Answers
What are the two types of inference rules in natural deduction?
Which type of logical connectives are used in propositional logic?
What is the role of introduction rules in natural deduction?
Which foundational language is essential for natural deduction systems?
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What is the primary purpose of direct proof in natural deduction?
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Which rule in natural deduction allows us to split a statement like A ∧ B
into A
and B
?
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How does indirect proof (reductio ad absurdum) work in natural deduction?
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What is the main function of assumptions in natural deduction?
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Which rule in natural deduction allows us to infer B
if we have A → B
and A
?
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What does hypothetical syllogism (→I) allow us to infer in natural deduction?
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Study Notes
Diving into Natural Deduction Systems: Propositional Logic, Inference Rules, Proof Strategies, Assumptions, and Implications
Natural deduction systems are a powerful and elegant way to reason in logic, particularly in propositional and predicate logics. They enable us to explore the depths of these systems using clear, stepbystep arguments that closely mirror the way human reasoning unfolds. Here, we'll explore the fundamental concepts of natural deduction, focusing on propositional logic, inference rules, proof strategies, assumptions, and implications.
Propositional Logic
Propositional logic is the foundational language of natural deduction, using atomic propositions (statements that can be either true or false) and logical connectives (like AND, OR, and NOT) to build more complex statements. These connectives follow specific rules of inference, which we'll explore next.
Inference Rules
In natural deduction, inference rules govern how we move from one statement to another. There are two types of inference rules: introduction rules and elimination rules.

Introduction rules allow us to introduce a new logical connective into an argument. For example, the rule for conjunction introduction (∧I) allows us to write
A
andB
separately, then combine them with→ (A ∧ B)
. 
Elimination rules allow us to remove a logical connective from an argument. For example, the rule for conjunction elimination (∧E) allows us to split an
A ∧ B
statement intoA
andB
.
Proof Strategies
Proof strategies are methods for organizing and structuring arguments to guarantee their validity. In natural deduction, we use two primary proof strategies: direct proof and indirect proof (also known as reductio ad absurdum or proof by contradiction).
 Direct proof relies on constructing arguments that build up from premises to a conclusion.
 Indirect proof shows that a statement is false by assuming it's true and deriving a contradiction.
Assumptions
Assumptions, or hypotheses, are statements we accept as true for the purpose of constructing an argument. In natural deduction, we must clearly indicate when we're making an assumption and must eventually discharge it.
Assumptions are written in a box to the left of the line they appear on, and we use a horizontal bar to separate assumptions from other lines in the proof. For example, assuming A
, we can write  A
and then use the assumption as needed.
Implications
Implications, or conditional statements, are fundamental to natural deduction. They help us understand cause and effect relationships and can be represented using the arrow symbol (→).
In natural deduction, we have two primary rules for implications:

Modus ponens (→E) states that if we have
A → B
andA
, then we can inferB
. 
Hypothetical syllogism (→I) states that if we have
A → B
andB → C
, then we can inferA → C
.
Natural deduction systems offer a rigorous, elegant, and intuitive approach to logic. By understanding the basics of propositional logic, inference rules, proof strategies, assumptions, and implications, we can create wellstructured and valid arguments in natural deduction systems. and do not contain information relevant to the topic of natural deduction systems.
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Description
Test your knowledge on natural deduction systems by exploring propositional logic, inference rules, proof strategies, assumptions, and implications. Learn about constructing valid arguments, using logical connectives, and deriving conclusions through clear stepbystep reasoning.