What is a proof? Continue learning about logic and arguments by reading the latest installment of the TIW Logic Course. You can use a proof to persuade others to your way of thinking.
Logic Course Review
So far, we’ve talked about understanding beliefs and changing core beliefs. We’ve also started discussing argument analysis, the process of evaluating the arguments people use for persuasion. considered premise and conclusion indicators and how to write articles in standard form. We’ve examined an example of a sound argument and, most recently, the principle of charity.
Not every argument is valid. Not every valid argument is sound. Not every sound argument is a proof.
What is a Proof?
An argument is a proof when it (1) its validity is known and (2) it’s known to have all true premises. In other words, a proof is an argument that is known to be sound. Knowledge differentiates between sound arguments and proofs. A proof always has a valid conclusion.
People will try to persuade you into accepting their beliefs by presenting arguments as proofs:
- Man Made Climate Change. Politicians and activists will present climate change as “settled science” but will ignore its fraudulent premises. Climate change arguments cannot be a proof because its premises are not known to be true.
- Evolution. Teachers and “scientists” will refer to evolution as a proof, but ignore mountains of scientific evidence that dispute its validity. Evolution cannot be a proof because its premises are in doubt.
- Math. 2+2=4 is a proof, because practically everyone understands the validity of its premises and conclusion.
Criticizing an argument
Criticizing an argument that someone portrays as a proof requires a structured approach:
- Identify the argument and write it in standard form. For example in any case:
P1 [the first premise]
P2 [the second premise, followed by any other premises]
- Understand the implications of the argument:
*We know that the premises are true.
* We know the argument is valid.
- Choose your approach:
* Show that a premise is not known to be true (demonstrate that it is false).
* Show that the argument is invalid.
- Control yourself.
* Don’t attack the conclusion, even if you don’t like it.
* Don’t ridicule the arguer.
Disproving a proof’s premise
Challenge the premises of a supposed proof, use the following tools:
- Universal Propositions in the form of All S are P. <== All American Bobtails are cats.
No S are P. <== No dogs are cats. Can you show that an argument depends on a false universal proposition?
- Counterexamples – Can you use a counterexample to disprove a premise? A counterexample is something that proves that a proposition is false. A counterexample must be known to be true. EXAMPLE: All acts of piety involve prosecuting the unjust.
- Conditionals. If X, then Y. Argue that X could happen and Y not happen. EXAMPLE: If I fall from the top of a mobile home, I will die. Argue that news accounts show that people have fallen from the top of a mobile home or similar one-story building without dying. In other words, you must argue that the antecedent could be true and the consequent false.
- Create a counter argument that follows the same form as the original argument (with all true premises and a false conclusion).
- Proexamples – a statement that proves that a proposition is true. For example:
P1 All cows are mammals
P2 Some cows have 11 legs (supposedly true).
∴Some mammals have 11 legs.
Remember: Someone who asserts something has the burden to prove its truth.
Responding to False Proof Arguments
In cases where an argument seems unclear, respond by first considering it as an invalid argument and again as a valid argument. In a debate, concede the opposing argument as valid. If you choose to argue against a false proof, focus your attention on its premises. You don’t disprove an argument by attacking its conclusion.
Arguing well with true premises always leaves you with a true conclusion and a valid argument. Arguing poorly with true premises can end with either a true or false conclusion.
Our next step in studying logic involves recognizing patterns. For example, we can chain together arguments. Inferring a conclusion for one argument lets you use it as a premise for another. As you evaluate what someone says, look for premise and conclusion indicators. If you find two conclusions, two arguments must exist. Also, beware of hidden premises and conclusions.
Principles for Finding Missing Premises or Conclusions
- The missing premise or conclusion must make the argument valid.
- The missing proposition is the one that is most likely to be true. (without the required knowledge, this is the weakest premise or the strongest conclusion). If there’s a choice, choose the premise/conclusion that will most likely make the argument a proof.
- The missing premise or conclusion must fit in with all other propositions. Use all the material if possible, so missing premise / conclusion. Don’t ignore information supplied by the author.
- The missing premise or conclusion contributes to a critical discussion. Clarify: choose the missing premise / conclusion that makes the clearest argument.
Now that you understand proofs and know how to find missing premises and conclusions you can better understand the arguments that other’s make. You can also make better arguments that can persuade people to believe you.