Introductory Logic: Claims, Arguments, Truth Tables, and Fallacies

Contingency

A formula with a mix of T and F under the major operator. Ex: p ⊃ q.

Valid argument

It's impossible for all premises to be true and the conclusion false. Depends on form. Ex: If P then Q. P. So Q.

Vagueness

A term with no precise cutoff for when it applies. Ex: heap, bald, tall.

p if q

The part after 'if' goes on the left: q ⊃ p. Ex: There are clouds if it is raining = R ⊃ C.

Inductive argument

Premises are meant to make the conclusion likely, not certain. Ex: 99.99% of flights land safely, so the next one probably will.

Tautology

A formula that is true in every row under the major operator. Ex: p ∨ ∼p (it's raining or it isn't).

Contradiction

A formula that is false in every row under the major operator. Ex: p • ∼p (it's raining and it isn't).

Invalid argument

There's a counterexample: all premises true, conclusion false. Ex: If P then Q. Q. So P (affirming the consequent).

Sound argument

A valid argument with all true premises, so its conclusion is always true. Ex: All humans are mortal. Socrates is human. So Socrates is mortal.

Can a valid argument have a false conclusion?

Yes, if at least one premise is false. Ex: If something is a horse, it's green. If green, plastic. So horses are plastic.

True premises + true conclusion = valid?

No, validity depends on form. Ex: If Bush was president, Cheney was VP. Cheney was VP. So Bush was president (true but invalid).

Subjectivist fallacy

Dismissing a claim as just someone's opinion. Ex: Evolution is just a theory, so there's no reason to believe it.

Appeal to ignorance

Treating a lack of evidence as proof that a claim is true or false. Ex: No one has proven ghosts don't exist, so they do.

When is appeal to ignorance OK?

When a thorough search with reliable methods found no evidence. Ex: Many studies found no link between the full moon and ER visits, so there probably isn't one.

Appeal to consensus (bandwagon)

Claiming something is true because many people believe it. Ex: Everyone knows we only use 10% of our brains.

Appeal to (unqualified) authority

Claiming something is true because a famous or non-expert person endorses it. Ex: Dr. Phil endorses this antidepressant, so it works.

When is appeal to authority OK?

When a relevant expert speaks within their field. Ex: Your cardiologist says you have heart disease.

False dilemma

Presenting only two options when more exist. Often valid but unsound. Ex: Either we ban all cars downtown or the city chokes on pollution.

Weasel words

A vague phrase that lets someone reinterpret a claim when challenged. Ex: Our toothpaste removes up to 50% plaque.

Equivocation

A word changes meaning partway through an argument. Ex: expect (what adults require) vs. expect (what adults predict).

Circular argument (begging the question)

The conclusion is assumed in the premises. Ex: The site is trustworthy because it tells the truth, and it tells the truth because it's trustworthy.

Presupposition failure

A claim or question assumes something false, so yes/no both fail. Ex: Have you stopped cheating on your homework?

Is-ought fallacy

Using only facts to reach a moral conclusion, with no moral premise. Ex: Humans have always eaten meat, so eating meat is right.

Red herring

Bringing up something irrelevant to change the subject. Ex: Asked about tuition, the dean talks about the new stadium.

Ad hominem

Attacking the person instead of their claim. Ex: Dr. Reyes's vaping study is worthless because he's been divorced twice.

Straw man

Distorting the opponent's view into a weaker, extreme version. Ex: She wants to cut police overtime, so she wants criminals running free.

Principle of charity (fidelity)

Interpreting an argument in its best light, filling in missing premises, and answering the strongest objection. Ex: Granting a reasonable missing premise before critiquing.

Simple claim

A claim that can't be divided into smaller claims, just a subject and a predicate. Ex: The cat is on the mat.

Compound (complex) claim

A claim that contains at least one full claim inside it (one is enough). Ex: The cat is not on the mat.

Contractions (isn't, didn't, won't)

A hidden negation, so the claim is compound and gets a tilde. Ex: The door isn't locked = ∼L (L = The door is locked).

Prefixes (un-, dis-, in-)

Negate only the predicate, so the claim stays simple with no tilde. Ex: Tom is unhappy.

Sufficient condition

A is enough to guarantee B, symbolized A ⊃ B. Ex: Having a daughter is sufficient for being a parent.

Necessary condition

B can't happen without A, symbolized B ⊃ A. Ex: Having a ticket is necessary for winning the lottery.

Conjunction (•)

Says both claims are true. Words: and, but, however, although, while. Ex: Jack is rude but loyal = R • L.

Disjunction (∨)

Says at least one claim is true. Words: or, either...or, unless. Ex: I'll fail unless I study = F ∨ S.

Conditional (⊃)

Says if the first is true, the second is true. False only when T ⊃ F. Ex: If it rains, the game is canceled = R ⊃ C.

Biconditional (≡)

Says both claims have the same truth value. Words: if and only if, just in case. Ex: Jones goes just in case he gets paid = J ≡ P.

Negation (∼)

Flips the truth value of a claim. Words: not, it is not the case that, -n't. Ex: Smith is not a good listener = ∼S.

p only if q

The part after 'only if' goes on the right: p ⊃ q. Ex: There are clouds only if it is raining = C ⊃ R.

Enthymeme

An argument with an unstated premise. OK to leave out only if it's common knowledge. Ex: Sam is a bachelor, so Sam is unmarried (missing: all bachelors are unmarried).

Deductive argument

Premises are meant to guarantee the conclusion (100% support). Ex: All humans are mortal. Socrates is human. So Socrates is mortal.

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