Formal Logic Proofs

Engage with the precise machinery of propositional logic: conditionals, contrapositives, De Morgan's Laws, and valid inference forms. These exercises train you to distinguish between what an argument's structure guarantees and what it merely suggests. That is the foundation of mathematical reasoning, legal argumentation, computer programming, and any domain where precision of thought is non-negotiable.

Formal logic turns reasoning into something you can check the way you check arithmetic. Once you know the moves, meaning conditionals, contrapositives, and De Morgan's laws, an argument either follows or it doesn't, and you can show it step by step. This precision matters anywhere a single broken link produces a confident, completely wrong conclusion: math, law, contracts, code.

This exercise asks whether a conclusion genuinely follows from its premises, independently of whether it sounds convincing. You practice applying formal inference rules and their negations, which is what separates an argument that is structurally valid from one that merely arrives at a believable answer by an invalid route.

Background

Two forms are always valid: from 'if P then Q' plus P you get Q, and from 'if P then Q' plus 'not Q' you get 'not P.' Their evil twins, assuming P from Q or 'not Q' from 'not P', fool people because they look almost identical. Learning to tell the valid form from its look-alike is most of the skill.

The payoff is that you can check a tangled conclusion mechanically instead of by feel: write out the structure, apply the rules, and see whether it holds. That's exactly how proof-checking software works, scaled up. For the wider landscape of reasoning types, see Types of Reasoning.

Questions

0 of 5 answered

Question 1

Given these premises: (1) If P then Q, (2) If Q then R, (3) P is true. A student concludes R is true by applying modus ponens twice. Another student says you can simplify by first combining premises 1 and 2 into 'If P then R' and then applying modus ponens once. Which student is correct?

Question 2

Consider the statement: 'If it is raining, then the ground is wet.' Which of the following is the CONTRAPOSITIVE, and why is it the only one logically equivalent to the original?

Question 3

In propositional logic, 'P → Q' (If P then Q) is TRUE when P is false, regardless of Q's value. A student protests: 'How can a conditional be true when the condition doesn't hold? That seems wrong.' How would you explain why this convention makes logical sense?

Question 4

De Morgan's Laws state: NOT(P AND Q) = (NOT P) OR (NOT Q), and NOT(P OR Q) = (NOT P) AND (NOT Q). A company policy says: 'You cannot take vacation during December AND January.' Using De Morgan's Law, which interpretation is correct?

Question 5

Consider this argument: 'All squares are rectangles. Some rectangles are not squares. Therefore, there exist shapes that are rectangles but not squares.' Is this argument valid AND sound?

Keep going

Where to go after this exercise.

Next in this categoryClassic River CrossingWork through the famous farmer-fox-chicken-grain puzzle to build your constraint-satisfaction reasoning. You will learn to plan multiple steps ahead while keeping every constraint satisfied simultaneously. That skill transfers directly to project management, scheduling, and strategic planning.Beginner5q →Related skillSimple AnalogiesBuild foundational skills in analogical thinking by exploring how analogies work in science, everyday reasoning, and communication, and learn why even simple analogies carry hidden assumptions.Beginner5q →Read the matching lessonWhat Is Critical Thinking?Foundations module · Learn what critical thinking is, where it originated, and the simple building blocks behind every argument you encounter.Read lesson →