Formal Logic Basics
Propositional Logic Symbols
| Symbol | Name | Example |
|---|---|---|
| ¬ | Negation (not) | ¬P — "It is not the case that P" |
| ∧ | Conjunction (and) | P ∧ Q — "P and Q" |
| ∨ | Disjunction (or) | P ∨ Q — "P or Q" |
| → | Conditional (if...then) | P → Q — "If P then Q" |
| ↔ | Biconditional (if and only if) | P ↔ Q — "P if and only if Q" |
Truth Table: P → Q
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
A conditional is false only when the antecedent is true and the consequent is false.
Truth Table: P ↔ Q
| P | Q | P ↔ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
A biconditional is true when both sides have the same truth value.
Valid Argument Forms
Modus Ponens
If P then Q. P. Therefore Q.
P → Q, P ∴ Q
Modus Tollens
If P then Q. Not Q. Therefore not P.
P → Q, ¬Q ∴ ¬P
Hypothetical Syllogism
If P then Q. If Q then R. Therefore if P then R.
P → Q, Q → R ∴ P → R
Disjunctive Syllogism
P or Q. Not P. Therefore Q.
P ∨ Q, ¬P ∴ Q
Constructive Dilemma
(If P then Q) and (If R then S). P or R. Therefore Q or S.
(P→Q)∧(R→S), P∨R ∴ Q∨S
Absorption
If P then Q. Therefore if P then (P and Q).
P → Q ∴ P → (P ∧ Q)
24 Informal Fallacies
Ad Hominem
+Definition: Attacking the person making the argument rather than the argument itself.
Example: "You can't trust his economic policy — he got a divorce last year."
Why it's fallacious: A person's character or personal life does not determine the truth or validity of their arguments.
Straw Man
+Definition: Misrepresenting someone's argument to make it easier to attack.
Example: "You want to cut military spending? So you want to leave our country defenseless."
Why it's fallacious: The distorted version does not reflect the actual position being argued.
Appeal to Authority
+Definition: Citing an authority figure as evidence, especially outside their expertise.
Example: "This diet must work — a famous actor endorses it."
Why it's fallacious: Authority in one field does not guarantee knowledge in another, and experts can be wrong.
False Dilemma
+Definition: Presenting only two options when more exist.
Example: "You're either with us or against us."
Why it's fallacious: Complex issues usually have multiple possible positions and solutions.
Slippery Slope
+Definition: Assuming one event will inevitably lead to an extreme consequence without justification.
Example: "If we allow students to redo one test, soon they'll expect to redo every assignment."
Why it's fallacious: The chain of consequences is assumed without evidence for each step.
Circular Reasoning
+Definition: Using the conclusion as a premise — the argument assumes what it tries to prove.
Example: "The Bible is true because it's the word of God, and we know it's the word of God because the Bible says so."
Why it's fallacious: No independent evidence is provided; the argument goes in a circle.
Hasty Generalization
+Definition: Drawing a broad conclusion from a small or unrepresentative sample.
Example: "I met two rude people from that city — everyone there must be rude."
Why it's fallacious: A few examples cannot justify a universal claim about a large group.
Red Herring
+Definition: Introducing an irrelevant topic to divert attention from the original issue.
Example: "Why worry about climate change when there are people starving in the world?"
Why it's fallacious: The new topic does not address the argument being made.
Appeal to Ignorance
+Definition: Claiming something is true because it hasn't been proven false, or vice versa.
Example: "Nobody has proven that ghosts don't exist, so they must be real."
Why it's fallacious: Lack of evidence for one position is not evidence for the opposite.
Begging the Question
+Definition: An argument whose conclusion is assumed in one of its premises (similar to circular reasoning).
Example: "Free speech is important because people should be allowed to say what they want."
Why it's fallacious: The premise simply restates the conclusion in different words.
Equivocation
+Definition: Using a word with different meanings in different parts of the argument.
Example: "The sign said 'fine for parking here,' so I parked — it must be fine."
Why it's fallacious: The shift in meaning creates a false sense of logical connection.
Composition
+Definition: Assuming that what is true of the parts must be true of the whole.
Example: "Every player on this team is excellent, so the team must be excellent."
Why it's fallacious: The whole may have emergent properties or problems that parts don't have individually.
Division
+Definition: Assuming what is true of the whole must be true of the parts (opposite of composition).
Example: "This university is prestigious, so every department must be prestigious."
Why it's fallacious: A whole can be excellent while some of its parts are mediocre.
Naturalistic Fallacy
+Definition: Assuming that what is "natural" is morally good, or what is "unnatural" is morally bad.
Example: "This behavior occurs in nature, so it must be morally acceptable."
Why it's fallacious: Natural occurrence does not determine moral value; many natural things are harmful.
Texas Sharpshooter
+Definition: Cherry-picking data clusters to fit a predetermined conclusion, ignoring contrary evidence.
Example: Drawing a target around the bullet holes after shooting at a barn wall.
Why it's fallacious: Patterns can appear random; coincidence does not prove causation.
No True Scotsman
+Definition: Redefining criteria to exclude counterexamples rather than addressing the argument.
Example: "No real philosopher would say that." "But this philosopher did." "Then they're not a real philosopher."
Why it's fallacious: Moving the goalposts avoids engaging with actual counterexamples.
Tu Quoque
+Definition: Deflecting criticism by accusing the accuser of the same behavior (appeal to hypocrisy).
Example: "You say I shouldn't smoke, but you smoke too."
Why it's fallacious: Whether someone is a hypocrite does not determine whether their advice is sound.
Appeal to Nature
+Definition: Arguing something is good because it is "natural" or bad because it is "unnatural."
Example: "Herbal remedies are better because they're natural, unlike synthetic drugs."
Why it's fallacious: Naturalness is not a reliable indicator of safety, efficacy, or moral value.
Genetic Fallacy
+Definition: Judging an idea based on its origin rather than its merit.
Example: "That idea came from a discredited source, so it must be wrong."
Why it's fallacious: The origin of an idea does not determine its truth or validity.
Bandwagon
+Definition: Arguing that something is true or good because many people believe it or do it.
Example: "Millions of people use this product, so it must be the best."
Why it's fallacious: Popular belief does not determine truth; majorities can be wrong.
Appeal to Emotion
+Definition: Manipulating emotions (fear, pity, anger) to win an argument instead of using logic.
Example: "If you don't support this policy, think of the children who will suffer."
Why it's fallacious: Emotional responses do not establish logical validity or factual truth.
Loaded Question
+Definition: Asking a question that contains an unjustified assumption.
Example: "Have you stopped cheating on your taxes?" (assumes you were cheating)
Why it's fallacious: The embedded assumption is not established and may be false.
Sunk Cost Fallacy
+Definition: Continuing an endeavor because of past investment (time, money, effort) rather than future value.
Example: "I've already watched three episodes — I might as well finish the season."
Why it's fallacious: Past costs are irrelevant to future decisions; only future outcomes matter.
Middle Ground
+Definition: Assuming the truth must be a compromise between two opposing positions.
Example: "One person says the earth is flat, another says it's round — the truth must be somewhere in between."
Why it's fallacious: One position can be entirely correct; compromise does not guarantee accuracy.