Intro to Logic: Arguments, Deductive & Inductive Reasoning

• Logic is the study of methods for evaluating arguments.
• Logic is also defined as the science of reasoning.

Arguments

• An argument is a set of statements where some statements are intended to support another.
• Arguments consist of premises and a conclusion.
• Premises are statements intended to support the conclusion.
• The conclusion is the statement being supported.
• A statement is a declarative sentence that is either true or false

Deductive Arguments

• Premises intend to guarantee the conclusion.
• Example: All philosophers like logic, Ned is a philosopher, so Ned likes logic.

Inductive Arguments

• Premises intend to make the conclusion probable.
• Example: Most philosophers like logic, Ned is a philosopher, so Ned likes logic.
• Some Americans work in business, Donald Trump is an American, so Donald Trump works in business.

Valid Arguments

• Valid arguments are deductive arguments where the premises guarantee the conclusion.
• If the premises are true, the conclusion must be true.
• There is a necessary connection between the premises and the conclusion.
• Truth of the conclusion is guaranteed by the truth of the premises.

Soundness

• Valid arguments are either sound or logical.
• Soundness: Premise and conclusion are true, the argument is valid.
• Logical: Premise is true or false, guarantees the conclusion, the argument is valid.

Examples of Valid Arguments

• True premise and true conclusion: If Harry loved Dumbledore, Harry was sad when Dumbledore died, so Harry was sad when Dumbledore died.
• False premise and false conclusion: All sharks are birds, all birds are politicians, so all sharks are politicians.

Invalid Arguments

• The premises do not guarantee the conclusion.
• Conclusion can be true but is not guaranteed by the premises.
• Example: Some Americans work in business, Donald Trump is an American, so Donald Trump works in business

Sound Arguments

• Sound arguments are valid arguments with all true premises.
• Valid + All Premises True = Sound

Unsound Arguments

• Unsound arguments are either invalid or valid with at least one false premise.

Categories of Unsound Arguments

• Category 1: Valid but has at least one false premise.
• Category 2: Invalid, but all premises are true.
• Category 3: Invalid and has at least one false premise.

Conclusions

• Arguments are neither true nor false; statements are either true or false.
• Arguments can be valid, invalid, sound, or unsound statements cannot be valid, invalid, sound, or unsound.
• Premises and conclusions can be either true or false, but cannot be valid, invalid, sound, or unsound.
• Deductive arguments should be valid with all true premises, which are sound arguments where validity focuses on the conclusion is linked to the premise.

How to make a good argument

• Logic makes arguments using: Deductive reasoning and Inductive Reasoning.

Good deductive arguments

• Identify the argument's forms
• Compare the forms to the 5 most used common forms
• Procedure is called the Famous Valid Forms Method.

Conditional Statements

• It is an "if-then" statement, often called a "conditional".
• The "if" clause of a conditional is its antecedent.
• The "then" clause of a conditional is its consequent.
• Example: If Tom goes to school (antecedent), then Tom gets an education (consequent).
• Stylistic variants of an "if it is raining, then the ground is wet" statement:
  • Given that it is raining, the ground is wet.
  • Assuming that it is raining, the ground is wet.
  • The ground is wet if it is raining.
  • The ground is wet given that it is raining.
  • The ground is wet assuming that it is raining.
  • It is raining only if the ground is wet.

Argument Form

• A pattern of reasoning.
• A group of sentence forms where all substitution instances are valid arguments.
• A substitution instance of an argument form is an argument that results from uniformly replacing the variables in that form with statements (or terms).

Summary of Valid Forms

• Modus Ponens: If A, then B. A. So, B.
• Modus Tollens: If A, then B. Not B. So, Not A.
• Hypothetical Syllogism: If A, then B. If B, then C. So, if A, then C.
• Disjunctive Syllogism (two versions):
  • Either A or B. Not A. So, B.
  • Either A or B. Not B. So, A.
• Constructive Dilemma: Either A or B. If A, then C. If B, then D. So, either C or D.

Valid Argument Form

• There is no substitution instances that are invalid.
• If the premises are true, the conclusion is also true.
• An argument with one invalid argument is an invalid argument form

Famous Forms Method Steps

• Step 1: Identify component statements in the argument, and Uniformly label each with a capital letter
• Step 2: Rewrite the argument using capital letters instead of English statements and eliminate any stylistic variants
• Step 3: Check if the pattern of reasoning is taken from the list of famous forms, then the argument is valid.

Modus Ponens

• Modus Ponens affirms the antecedent
• The form: If A then B, A, therefore, B.
• It is a common pattern of deductive reasoning.
• Both arguments have the valid form Modus Ponens, and so both are valid.

Modus Tollens

• Modus Tollens Denies the Consequent (Denying the Consequent)
• Takes this form: If A then B, Not B, so not A.
• If the form is valid, the individual argument is valid due to the valid form.

Hypothetical Syllogism (HS)

• A conditional statement is also called a “hypothetical statement ".
• Takes this form: If A then B, If B then C, Therefore, if A then C

Disjunctive Syllogism (DS)

• Disjunctive Syllogism can take two forms
• Either A or B, Not A, so B is the 1st form
• Either A or B, Not B, so A is the 2nd form
• The statement “A or B" is called the "disjunction " of the simpler statements “A” and “B”.
• The simpler statements are called the “disjuncts ".
• The disjunction “A or B” means “A or B or both" in its inclusive sense.
• disjunction “A or B” means “A or B but not both” in the exclusive sense.
• The argument form, disjunctive syllogism, is valid for both senses of the disjunction.
• In this unit, use "or" in its inclusive sense, unless specified.

Constructive Dilemma examples

• If A then C, and If B then D statement, so the argument is A or B.

Counterexample & Invalidity

• Valid forms method may not help identify many invalid arguments, but you can use counterexamples to help determine validity.
• Counter-example: Proof of invalidity

Counterexample

• An invalid argument form has some invalid substitution instances.
• A counterexample is a substitution instance with the premises being true and the conclusion being false.
• A good counterexample is one where the premises are well-known truths, and the conclusion is a well-known falsehood.

The Counterexample Method

• Step 1: Identify the most logically sensitive form of the argument. Use capital letters to stand for statements or terms.
• Step 2: Find English statements or terms that, if they are substituted for the capital letters in the conclusion of the argument form; this will produce a well-known falsehood.
• Step 3: Substitute these English statements or terms for the relevant capital letters uniformly throughout the argument form.
• Step 4: Find English statements or terms that if substituted uniformly for the remaining capital letters in the argument form, produce premises that are well-known truths.
• Step 5: Check your work. If you have succeeded, you have shown the argument to be invalid.