Arguments and Proofs

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Our logical language FOL will be both

• A logical precise language
• A tool for reasoning

Reasoning

• FOL can be a tool for reasoning because we define its elements unambiguously and precisely.
• This allows us to introduce inference rules which capture some form of reasoning that will preserve truth.
• Our first four rules:

  Repitition (R):
  
  P
  _____
  P
  
  Modus ponens (MP):
  
  (P-->Q)
  P
  _____
  Q
  
  Modus tollens (MT):
  
  (P-->Q)
  ~Q
  _____
  ~P
  
  Double Negation (DN):
  
  ~~P
  ______
  P
  
  P
  ______
  ~~P
  
  

Arguments

• Logicians do not decide whether particular claims are true (unless those claims are about logic or are claims that must be true).
• Logicians determine instead what must be true given other assumptions.
• We do this with arguments.
• A good argument is called a valid argument
• A valid argument is an argument where: if the premises (the assumptions we have made, if any) are true, then the conclusion must be true.
• A sound argument is an argument that is valid and the premises are true.
• We will create inference rules and proof methods designed to ensure that any argument we construct with them is guaranteed to be valid.

Proofs
All proofs have the following features.

• We call a list of symbolic sentences an argument. An argument that follows the rules below (and those rules for conditional, indirect, and universal derivation), we will call "proofs" or "derivations."
• Two kinds of things appear on this list:
  • "Show" lines: these are not assertions, but rather bookkeeping that we use to remind ourselves what we are trying to prove at any one time. (If we do prove the thing we aim to show, then we cross out the "Show" and this becomes an assertion of something proven.)
  • Assertions: these are sentences we are asserting, that we claim to be true. We assert in the proof only three kinds of sentences:
    1. Premise or other kind of assumption (we will make special kinds of assumptions when we do conditional proofs and indirect proofs);
    2. Sentences proven using another proof (this proof may be embedded within our proof, or it could be something we proved elsewhere, such as a theorem);
    3. Sentences derived from earlier assertions in the proof (that are not in a closed box) using inference rules.
  • A conclusion is some sentence we aim to demonstrate. Ideally, the conclusion is the last line of our proof.
  • When we've reached a line that is our conclusion, we box our argument and cross out the show line.

Direct Proof
A proof with no special assumptions is called a direct proof.

Conditional Derivation
A method for proving conditionals is to use conditional derivation: assume the antecendent, and then prove as your conclusion the consequent.

Indirect Derivation
To show that some sentence is true, we assume that it is false (we assume the negation of that sentence) and then show any contradiction. If a contradiction follows, something has gone wrong; and if we follow our rules of logic then the only thing that can have gone wrong was our assumption of the negation of our conclusion. So we must have been wrong to assume it is false, so it must be true.

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A reminder about validity: your book says that a valid argument is one that we can prove with our system. This is true but it is not a definition! An argument is valid just in case if the premises are true, then the conclusion must be true. That is our definition of "valid," and that alone.