Arguments and Conditions

Arguments and Conditions

• Arguments contain premises that provide evidence or rational support for a conclusion.
• If the moon is made of green cheese, then cows jump over it. The moon is made of green cheese; Therefore, cows jump over the moon is an example of a set of sentences.
• This cooler contains 30 cans. 29 randomly selected cans are all Cokes and probably all the cans are Cokes are all example sentences.
• A necessary condition P for Q means Q cannot be true without P being true, represented as (Q->P).
• A sufficient condition P for Q means if P is true, Q must be true as well, (P->Q).
• If Professor Singer can vote, he is a citizen.
• Being a citizen is a necessary condition for voting.
• Being able to vote is a sufficient condition to know you are a citizen.

Deductive Argument Goodness

• Inductive arguments have premises that provide probable support for the conclusion.
• Good inductive arguments are strong if the premises are true, the conclusion is probably true.
• A cogent argument is strong with true premises.
• Deductive arguments have conclusions that necessarily follow from premises, and are truth-preserving.
• A valid argument is impossible for the conclusion to be false while the premises are true.
• A sound argument is valid with true premises, guaranteeing the conclusion.

Argument Form

• Formal validity is the hypothesis that an argument's validity can be determined by examining its form alone.
• If an argument has a valid form, then it is valid.
• If an argument has only invalid forms, then it is invalid.
• The counterexample method demonstrates an argument's invalidity, offering an example with true premises and obviously false conclusion.

Metatheory: Expressive Adequacy, Soundness, Completeness

• Expressive Adequacy is the set of connectives that can express all possible sentences in a language.
• Soundness means all provable sentences are true, indicating good proofs.
• Completeness means all true sentences are provable.

Truth-Functional Logic: Translations

• Tricky phrases and sentence constructions such as but, still, etc can be expressed as AND.
• "A if B" translates to (B->A), while "A only if B" translates to (A->B).
• "Unless" is represented as an inclusive OR.
• "Exclusive or" means A or B, but not both.
• "Neither nor" means none of them.
• "Not both A and B" translates to -(A/B).
• Example: "If John makes supper only if Mary works late, then John is very hungry"
• P = John makes supper; S = Mary works late; T = John is very hungry.
• Neither taxes nor interest rates will rise if the deficit is reduced, but if the deficit is not reduced, then both taxes and interest rates will rise.
• P = Interest rates will rise; U = The deficit will be reduced; V = Taxes are raised.

Truth-Functional Logic: Truth-Tables

• Completing a comprehensive truth table is essential.
• The number of rows is determined by 2^n, where n equals the amount of variables.
• For the first variable, alternate after 2^(n-1). With three variables, alternate 4T and 4F.
• For the next variable, divide that number in half. Examples, 2T2F2T2F
• The last variable varies by 1. Example, TFTFTFTF
• Consistency can be confirmed if there is a row where all sentences are true.
• Validity is confirmed when there is no row with all premises true concluding with a false conclusion.

Truth-Functional Logic: Natural Deduction

• Use textbook notation.
• Know the rules and how to write justifications
• Distinguish between comma vs hyphen, using hyphens for VE, ->I, <->I, IP, and -I.
• When the conclusion is a conditional, start with a subproof assuming the antecedent, repeating for nested conditionals.
• Consider a contradiction when stuck, avoiding confusion between -E, -I, and IP.
• X doesn't get you out of a subproof! It should align with the contradiction.
• Assume strategically: For a conclusion of -(A/B), assume A/B over - -(A/B) and get a negation intro at the end.

Translating into First-Order Logic

• Distinguish when to use names (a-r) vs. variables (s-z).
• Introduce quantifiers carefully, paying attention to scope.
• "There is a philosopher (∃xF(x))" versus "Idil is a philosopher (Fi)".
• The four basic forms include: All Ps are Qs, Some Ps are Qs, No Ps are Qs, Some Ps aren't Qs.