Comprehensive Notes on Logic and Arguments

BASIC CONCEPTS IN LOGIC

• Logic is the science that evaluates arguments.

  • Tool for distinguishing correct from incorrect reasoning.

  • A science is an organized system or body of knowledge.

  • More formally: logic teaches us to develop a system of methods and principles to use as criteria for evaluating the arguments of others and to guide us in constructing arguments of our own.

  • Ranges from informal logic to symbolic logic.

LOGIC AND ARGUMENTS

• Logic concerns arguments: sets of statements where some (premises) are claimed to support another (the conclusion).

• Understanding arguments is central to evaluating reasoning across contexts.

LOGIC AND ITS PLACE IN PHILOSOPHY

• Philosophy = the love of wisdom; self-critical reflective thought; attempts to formulate, understand, and answer fundamental questions.

• Metaphysics: Appearance vs. Reality — What is real?

• Epistemology: Knowledge vs. Opinion — How do I know?

• Value/Aesthetics and Ethics (Axiology): Fact vs. Value — How do I act? What is good?

• Logic is part of epistemology (roughly): it helps distinguish how we know from what we merely believe or infer.

WHAT IS LOGIC?

• Logic: the science that evaluates arguments.

• Tool for distinguishing correct from incorrect reasoning.

• A science is an organized system of knowledge.

• Formal definition: logic teaches us to develop a system of methods and principles to use as criteria for evaluating the arguments of others and to guide us in constructing arguments of our own.

• Scope: informal logic to symbolic logic.

ARGUMENTS

• An argument is: “a group of statements, one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion).”

• Three main parts:

  • Premise: statement(s) that set forth reasons for believing something is true.

  • Conclusion: statement that the evidence (premise) is claimed to support.

  • Inference: implicit rule that connects the premises and the conclusion.

• Good vs. Bad arguments: Do the premises provide enough support? Is the support presented in the right ways?

A CLASSIC EXAMPLE

• “Man is by nature a political animal.”

• “I, a political animal, am therefore a man.”

• This may illustrate a non-valid syllogistic form in ordinary language; underscores the difference between everyday reasoning and valid logical structures.

• Attributed to ARISTOTLE (384 BC–345 BC).

STATEMENTS

• A statement is a sentence that is either true or false.

• Truth Value: truth, falsity, or logical undetermined.

• Truth values can be: true, false, or logically undetermined.

• Examples of statements vs. non-statements:

  • Cats are jerks. (statement)

  • Timothée Chalamet is a jerk. (statement)

  • Timothée Chalamet is a cat. (statement)

  • Oh no! Want to play a game? (non-statement)

  • Look at that hippo! (non-statement)

PREMISE AND CONCLUSION INDICATOR WORDS

• Indicator words help identify whether a passage expresses premises or conclusions.

• Conclusion indicators: therefore, thus, accordingly, etc.

• Premise indicators: since, because, given, that, etc.

• Note: “inference” and “proposition” can have broad versus narrow meanings; context matters for identifying the structure.

RECOGNIZING ARGUMENTS

• Arguments vs. Nonarguments:

  • At least one statement must claim to present evidence or reasons.

  • The alleged evidence must claim to support or imply something.

  • An argument aims to advance knowledge or demonstrate that something is the case.

• Explicit vs. Implicit Claims:

  • Explicit: usually use indicator words.

  • Implicit: inferential relationship between statements but with no explicit indicators.

• How to determine an argument: look for indicator words, inferential relationships, and whether the passage is a nonargument.

SIMPLE NONINFERENTIAL PASSAGES

• Types include:

  • Warning

  • Piece of advice

  • Statement of belief or opinion

  • Report

  • Loosely associated statements

EXPOSITORY PASSAGES

• Usually have a topic sentence followed by development of the topic.

• Proof vs. Elaboration:

  • Exposition can be an argument when used to prove something.

• Test: is the topic a claim that everyone agrees with?

ILLUSTRATIONS

• Intend to show what something means or how it is done.

• Be cautious of arguments from example.

EXPLANATIONS

• Attempts to shed light on something.

• Explanandum: fact or phenomena being explained.

• Explanans: statement(s) that claims to do the explaining.

• Can become an argument if it is also an explanation of a fact that is not widely accepted.

• Distinction: “Why something is the case” vs. “That something is the case.”

• Example: “Golf balls have a dimpled surface because dimples reduce air drag, causing the ball to travel farther.”

  • Explanations explain why something is the case.

  • Arguments try to prove that something is the case.

CONDITIONAL STATEMENTS

• Structure: “if antecedent then consequent” (A → C).

• Alternate phrasing: “consequent if antecedent.”

• Test: Look for a claim that presents evidence.

• A conditional statement by itself is not an argument.

• It is easy to confuse with an argument because conditional statements can be a premise or conclusion or both.

• Inferential content may be expressed as an argument.

NECESSARY AND SUFFICIENT CONDITIONS

• Sufficient condition: A is sufficient for B whenever the occurrence of A is all that is needed for the occurrence of B.

• Necessary condition: A is necessary for B when B cannot occur without A.

• Example:

  • If Fido is a dog, then Fido is an animal. (sufficient)

  • If Fido is not an animal, then Fido is not a dog. (necessary)

DEDUCTION AND INDUCTION

• Distinct inferential patterns:

  • Deduction: aims at necessity; conclusion follows with necessity from the premises.

  • Induction: aims at probability; conclusion follows with probability from the premises.

• Often illustrated with age-old examples like the witch scenario to contrast necessity vs. probability.

THE WITCH

• Slogan: "Burn the witch!" raises the question: how certain is the claim that she is a witch?

• Distinguishes between what follows with necessity (deduction) vs. what is probable (induction).

• Strength and weakness of inferential connection differentiate deduction from induction.

DEDUCTION

• In a deductive argument the arguer claims that it is impossible for the conclusion to be false if the premises are true.

• If premises are true and the argument is good, the conclusion must be true.

• Examples:

  • All triangles have 3 sides and angles that equal 180^{\circ}. This object has 3 sides and the angles equal 180^{\circ}. Therefore, this object is a triangle.

  • If something weighs as much as a duck, it is made of wood. Witches weigh as much as a duck. Therefore, a witch is made of wood.

INDUCTION

• In an inductive argument the arguer claims that it is improbable that the conclusion be false if the premises are true.

• If premises are true, the conclusion is probably true.

• Examples:

  • It usually snows in the winter. So this winter we can probably expect snow.

  • Most things made of wood burn. Witches burn. Therefore, witches are probably made of wood.

DISTINGUISHING BETWEEN INDUCTION AND DEDUCTION

• Look for:

  • (1) Indicator words

  • (2) Actual strength of the inference

  • (3) Argument forms

• Deductive indicator words: certainly, absolutely, necessary, never, etc.

• Inductive indicator words: probably, improbable, likely, reasonable to conclude, etc.

• Be wary of rhetorical tricks.

• Assess actual strength of the argument, not the intended strength.

• Even with true premises, conclusions may be false in incomplete ordinary-language arguments.

DEDUCTIVE ARGUMENT FORMS

• Arguments from Mathematics

• Arguments from Definition

• Categorical Syllogism: All …, No …, Some …

• Hypothetical Syllogism: if, then …

• Disjunctive Syllogism: either, or …

INDUCTIVE ARGUMENT FORMS

• Prediction: using knowledge of the past for predictions about the future.

• Argument from analogy: emphasizes similarity between two events.

• Generalization: argues from characteristics of a sample to a whole group.

• Argument from authority: assumes something is true because an authority says it is.

• Argument based on signs: from a sign to what it symbolizes.

• Causal inference: from knowledge of a cause to a claim about an effect, or vice versa.

• Note: There can be overlap among subspecies of inductive arguments.

SCIENTIFIC ARGUMENTS

• Can be either deductive or inductive.

• If aimed at discovery of a new law, likely inductive.

• If dependent on an established law, likely deductive.

GENERALS AND PARTICULARS

• Particular statements: make claims about particular members of a class.

• General statements: make claims about all members of a class.

• Aristotelian analysis: inductive arguments move from particulars to generals; deductive arguments move from generals to particulars.

• It is a useful guide but not always true in all cases.

VALIDITY, TRUTH, SOUNDNESS, STRENGTH, COGENCY

• These are key criteria for evaluating arguments.

IS THE ARGUMENT GOOD?

• Two main questions:

  • 1) Do the premises support the conclusion? Assess the argument’s inferential claim.

  • 2) Are all the premises true? Assess the argument’s factual claims.

VALIDITY

• Valid Deductive Argument: it is impossible for the conclusion to be false if the premises are true.

• Invalid Deductive Argument: it is possible for the conclusion to be false even if the premises are true.

SOME EXAMPLES (DEDUCTIVE AND NONDEDUCTIVE)

• Example 1: All sassy women living in Florida are Golden Girls. Sophia is a sassy woman living in Florida. Therefore, Sophia is a Golden Girl. (Illustrates a valid form, assuming premises are true.)

• Example 2: All dogs are cats. Nugget is a dog. Therefore, Nugget is a cat.

• Example 3: All cats have four legs. Nimbus has four legs. Nimbus is a cat.

IMPORTANT IMPLICATION

• Validity is determined by the form of the argument, not the actual truth value of the statements.

• The form is the relationship between the premises and the conclusion.

• Exception: when the premises are actually true and the conclusion is actually false then the argument is invalid (ALWAYS).

• The insight that an invalid argument can have true premises and a false conclusion is the cornerstone of deductive logic.

• A deductive argument is either valid or invalid; there is no middle ground.

SOUNDNESS

• A sound argument is one that is both valid and has all true premises.

• An unsound argument has false premises, is invalid, or both.

• A sound argument is a good deductive argument in the fullest sense.

• Note: all invalid arguments are also unsound.

• Soundness of an argument can be hard to determine; avoid confusing soundness with validity.

STRENGTH, COGENCY

• Strength: in inductive reasoning, a strong argument makes the conclusion highly probable given the premises.

• Weak: an inductive argument where the conclusion does not reasonably follow from the premises.

• Uniformity of nature: the principle that the future usually replicates the past; regularities tend to repeat across contexts.

• Most strong inductive arguments appeal to the uniformity of nature.

• Cogency: an inductive argument that is strong and has true premises.

• Uncogent: weak, has false premises, or fails total evidence, or any combination.

• Distinguish cogency from strength; they are related but not identical.

SOME EXAMPLES (CONTINUED)

• All dogs I have met have been kind and loyal to their masters. There’s a good chance that the next dog I meet will also be kind and loyal.

• All trees we cut down have iron in them. Therefore, the next tree we cut down will have iron in it.

• “For the past 50 years inflation has reduced the value of the American dollar. Therefore, industrial productivity will probably increase in years ahead.”

TOTAL EVIDENCE REQUIREMENT (INDUCTIVE ARGUMENTS)

• Premises of an inductive argument must be completely true in all relevant senses.

• Premises must not leave out important information that would change the truth of the conclusion.

• Strength and weakness of inductive arguments are a matter of degrees.

• Strong arguments should be better than about 50% probability (
  roughly).

COGENCY (RECAP)

• A cogent argument is an inductive argument that is strong and has true premises.

• An uncogent argument is weak, has one or more false premises, does not meet total evidence, or any combination of these.

• The assessment of cogency vs strength can be tricky; keep the distinction clear.

PRACTICE

• 1. All flowers are dogs. All poodles are flowers. Therefore, all poodles are dogs. (Illustrates deductive reasoning in a categorical/chain form.)

• 2. A few U.S. presidents were unmarried. Therefore, probably the next U.S. president will be unmarried. (Indicative of inductive probability.)

• 3. If George Washington was beheaded, then George Washington died. George Washington died. Therefore, George Washington was beheaded. (Invalid inference in standard form.)

• 4. The picnic scheduled in the park for tomorrow will most likely be cancelled. It’s been snowing for six days straight. (Inductive likelihood claim.)

• 5. If Prince Harry was born in Los Angeles, then he is a California native. Prince Harry is not a California native. Therefore, Prince Harry was not born in Los Angeles. (Evaluates contradiction in conditional reasoning.)

• 6. Since some fruits are green, and some fruits are apples, it follows that some fruits are green apples. (Analyzes conjunction and subset relations.)

• 7. When Neil Armstrong landed on the moon, he left behind a gold-plated Schwinn bicycle; he used to ride around on the moon’s surface. Probably that bicycle is still up there. (Pseudo-historical humor used to illustrate inference.)

• 8. Since John loves Nancy and Nancy loves Peter, it follows that necessarily that John loves Peter. (Questionable leap in relation and scope.)

THE COUNTEREXAMPLE METHOD: PROVING INVALIDITY

• A method to show arguments are invalid by producing a counterexample.

ARGUMENT FORMS

• Understanding and testing deductive validity via argument forms.

• An argument form is a template; its validity depends on the form, not the particular terms used.

• Example form given: All A are B. All C are A. Therefore, All C are B.

SUBSTITUTION

• We can replace the letters A, B, C with any terms to generate substitution instances.

• A substitution instance for a valid argument form is always valid.

• Example given: All dogs are mammals. All beagles are dogs. Therefore, All beagles are mammals.

• If the form is not apparent, look for missing premises.

INVALID FORMS

• In an invalid form the conclusion does not necessarily follow from the premises.

• Form example: All A are B; All C are B; All A are C.

• Substitution: A = cats, B = cuddly, C = dogs. Assess validity accordingly.

COUNTEREXAMPLE METHOD (STEP-BY-STEP)

• Step 1: Isolate the form of an argument.

• Step 2: Construct a substitution with true premises and a false conclusion.

• Note: This method cannot prove validity; it can only prove invalidity by counterexample.

USING THE METHOD WITH CATEGORICAL SYLLOGISMS

• Steps:
  1) Pick a set of terms (e.g., cat, dog, mammal, fish, animals).
  2) Begin with the conclusion; try to choose terms that make it false.
  3) The word “some” can be read as “at least one.” Example: Some students are not studious, and all freshmen are students. It follows that some freshmen are not studious.

REVIEW OF THE METHOD

• Follow these steps:
  1) Begin with an invalid argument.
  2) Extract the argument form; leave the form words (All, No, Some, are, are not).
  3) Substitute an instance with true premises and a false conclusion.
  4) Conclude that the initial argument is invalid because it has an invalid argument form.

• Requires practice, patience, and imagination.

ARGUMENT FORMS (DEDUCTIVE FORMS)

• Following the understanding of what makes an argument valid, we have a method for proving some deductive arguments invalid via forms.

• All A are B. All C are A. All C are B. (illustrates a valid form depending on consistency of terms)

SUBSTITUTION (DEDUCTIVE FORMS: EXAMPLE)

• Replace letters with terms to test validity.

• If the form is valid, substitution instances preserve validity.

INVALID FORMS (DEDUCTIVE)

• In invalid forms, conclusions do not necessarily follow; substitution preserves invalidity in most cases.

COUNTEREXAMPLE METHOD (DETAILED)

• Step 1: Isolate form.

• Step 2: Create substitution with true premises and false conclusion.

• This proves invalidity but not validity.

USING THE METHOD WITH CATEGORICAL SYLLOGISMS (DETAILED)

• Pick terms, begin with conclusion, choose terms to falsify the conclusion while premises stay true.

• Example: Some students are not studious, and all freshmen are students. It follows that some freshmen are not studious.

REVIEW OF THE METHOD (RECAP)

• Practical steps:

  • Start with an invalid argument.

  • Extract form and note quantifier words.

  • Build substitution to test validity.

  • Use results to judge invalidity.

• Practice and imagination are essential to mastering the method.