Critical Thinking Toolkit: Inference to the Best Explanation

I’ve talked about deductive reasoning and inductive reasoning before. Both of these are important mechanisms we use for arriving at conclusions. The first makes use of the rules of logic – which we can essentially view as limiting ourselves to the definitions of words like true and false and not allowing ourselves any other resources.Continue reading “Critical Thinking Toolkit: Inference to the Best Explanation”

Critical Thinking Toolkit: The Fallacy of Arguing from Ignorance

When debating important topics, whether with your friends, online with strangers, or in a public debate with spectators, there is always a high emphasis placed on proving your case. People ask why you believe what you do, ask you to show your evidence, ask you to give them your arguments. There is nothing wrong withContinue reading “Critical Thinking Toolkit: The Fallacy of Arguing from Ignorance”

Critical Thinking Toolkit: False Dilemma

Choices are important. Every day, we make lots of decisions about what to do and what to say. When decisions are especially important is when two possible paths we might take are totally opposite another or grind against one another. There is a certain tension that arises in decisions that carry a great deal ofContinue reading “Critical Thinking Toolkit: False Dilemma”

Critical Thinking Toolkit: Inductive Reasoning

Deductive reasoning is a very useful way to gain knowledge, but it is also very limited. Deduction is a type of reasoning that intellectually compels you to believe something if you accept some number of other things. But very few aspects of reality are like this. In most situations, there are a range of availableContinue reading “Critical Thinking Toolkit: Inductive Reasoning”

Critical Thinking Toolkit: The Fallacy of Equivocation

This is one of many brief articles I am writing about how to avoid fallacious patterns of thinking. Here, we briefly discuss the fallacy of equivocation. Before I try to define it, it will be helpful to see an example of the fallacy in action. (I take this example out of the Wikipedia page forContinue reading “Critical Thinking Toolkit: The Fallacy of Equivocation”

Critical Thinking Toolkit: External and Internal Contradictions

When developing critical thinking skills, learning to recognize falsehood is as important as learning to recognize truth. This is important for many reasons. Recognizing falsehood helps you realize when you are making mistakes, when others might be making mistakes or using confusing language, and can help you find out the truth by process of elimination.Continue reading “Critical Thinking Toolkit: External and Internal Contradictions”

Critical Thinking Toolkit: Valid and Sound Arguments

This is the first post I am making in what will hopefully become a long and detailed series of posts about how to think more clearly about difficult questions. Since we ought to try to think clearly in every domain of life, we must begin the discussion at the broadest level, with the most importantContinue reading “Critical Thinking Toolkit: Valid and Sound Arguments”

Proof by Contrapositive

Sometimes when you are trying to solve a problem, you realize you really don’t have a lot of information to start with. One piece of good advice in problem-solving is to try to work backwards. That is, sometimes if you know what you want your solution to look like, you can backtrack to learn somethingContinue reading “Proof by Contrapositive”

Proof by Contradiction

The proof method that we will talk about here is quite different than many others. In his famous book A Mathematician’s Apology, the great mathematician G.H. Hardy made an analogy between this proof style, which we call a proof by contradiction, to a gambit in chess. So before I try to analyze what this proofContinue reading “Proof by Contradiction”

Types of Proofs in Math

Previously, I have talked about logic and some of the most important rules of logic. These are quite important and useful in doing mathematics. However, it is necessary to go further, because logic is not specific enough. Mathematics analyzes patterns that involve concepts like shape, number, repetition, and symmetry. Pure logic does not adequately handleContinue reading “Types of Proofs in Math”