Problem 46 - Thinking Mathematically

Question

Write an example of an argument with two quantified premises that is invalid but that has a true conclusion.

Step-by-Step Solution

Answer

One possible invalid argument with two quantified premises and a true conclusion is: First premise - 'All dogs are animals', Second premise - 'All cats are birds', and the Conclusion - 'Dogs are not birds'. Despite the premises being true, the conclusion doesn't follow logically from them, making the argument invalid.

Step 1: Fabricating an Invalid Argument

To fabricate an invalid argument, one can put forward two premises and a conclusion such that the truth of the conclusion does not depend on the truth of the premises. For example, if it's discussed about a certain population of animals:

Step 2: Establishment of the First Premise

The first premise - 'All dogs are animals.' This premise is a positive universal proposition, which immediately proves it to be true as all dogs come under the category of animals.

Step 3: Establishment of the Second Premise

The second premise - 'All cats are birds.' This is a false proposition, as not all cats can be classified as birds as they belong to a different category.

Step 4: Creating a True Conclusion

Establish a conclusion that is true but does not follow the sequence of the premises provided. For this example the conclusion could be: 'Dogs are not birds.' This conclusion is true but is an invalid conclusion for the given premises.

Key Concepts

Invalid Argument

An invalid argument in logic is like a broken bridge; it doesn't successfully take you from the premises to the conclusion. Understanding invalid arguments is crucial so you don’t mistakenly believe something is true just because it sounds compelling. Imagine you're given two statements: 'All dogs are animals.' which is true, and 'All cats are birds.' which is obviously false. Now, if the conclusion drawn from these premises is that 'Dogs are not birds.', it might be true, but the argument itself is flawed.

The reason it's flawed is that the truth of the conclusion isn't supported by the premises. This is a classic case of an argument that doesn't logically follow - what we call an 'invalid argument'. Knowing how to spot such logical inconsistencies will save you from building castles on sand when it comes to reasoning through problems.

Propositional Logic

Propositional logic, the backbone of logical reasoning, is all about joining or modifying statements to express more complex ideas. Think of it like constructing sentences using words. In propositional logic, the 'words' are simple statements, which can be either true or false. These can then be combined using logical connectors like 'and', 'or', and 'not', to construct more elaborate statements, which we call 'propositions'.

For instance, you can have a simple statement 'today is sunny', and another one 'I will go on a picnic'. Propositional logic allows you to combine these to say 'If today is sunny, then I will go on a picnic.' Learning to use propositional logic is like learning a new language that helps you to clearly articulate and understand complex ideas in a structured way.

Logical Fallacies

Watch Out for Logical Traps

Logical fallacies are the sneaky errors in reasoning that can easily mislead you. They're like optical illusions for the mind - they seem to make sense at first glance, but under scrutiny, they fall apart. In constructing arguments, it’s important to avoid these traps to ensure your reasoning is sound.

An example of a logical fallacy would be the 'appeal to authority', where we assume something must be true because an expert said so, rather than because of the evidence. Knowing about logical fallacies empowers you to question and critically evaluate arguments, rather than just accepting them at face value.

Deductive Reasoning

Deductive reasoning is the Sherlock Holmes of thinking - it's all about drawing specific conclusions from a general rule. If you know a broad principle to be true, you can deduce that it applies to particular cases.

In our animal example, if we use the correct premise 'All birds are animals', we can deduce that 'All sparrows are animals' because sparrows are a type of bird. This is valid deductive reasoning. Learning this technique is essential, as it helps you apply general knowledge to specific instances with confidence, knowing that your conclusions are logically sound, provided the initial premises are true.