The Logic of Atomic Sentences Flashcards

Vocabulary terms and definitions from the lecture notes on the logic of atomic sentences, formal proofs, and identity.

Argument

A piece of reasoning, consisting of a sequence of statements, attempting to establish a conclusion.

Conclusion

What the arguer is trying to establish, typically indicated by words such as therefore, so, hence, thus, or consequently.

Premises

The reasons given in support of the conclusion, typically preceded by words like because, since, or after all.

Validity

The property of an argument where it is impossible for its premises to be true and its conclusion false.

Soundness

The property of a valid argument that also has true premises.

Fitch bar notation

A notation for writing arguments where premises are listed above a horizontal line and the conclusion is written below it.

Proof

A step-by-step demonstration showing that a conclusion follows from the premises through a chain of intermediate conclusions.

Reflexive

A feature of the identity relation in which each thing is identical to itself (e.g., $b = b$).

Symmetrical

A feature of the identity relation stating that if $a = b$, then $b = a$.

Transitive

A feature of the identity relation stating that if $a = b$ and $b = c$, then $a = c$.

Indiscernibility of Identicals

The principle stating that if $a = b$, then whatever holds of $a$ also holds of $b$.

System F

The deductive system developed for writing up formal proofs.

Fitch (Program)

A user-friendly computer program version of the formal deductive system F.

Justification

A component of a formal proof that cites a specific rule and the lines to which the rule is applied to generate a new line.

Identity Introduction ($= Intro$)

A rule that allows you to enter a sentence of the form $n = n$ at any point, embodying the principle of reflexivity.

Identity Elimination ($= Elim$)

A rule that allows you to substitute $m$ for $n$ in a sentence $P(n)$, provided you have the sentence $n = m$, embodying the indiscernibility of identicals.

Int-elim rules

Pairs of rules in a system like F specifically consisting of an introduction rule and an elimination rule for each logical symbol.

Ana Con

A mechanism in Fitch that checks if a conclusion follows from premises by taking the meanings of predicates into account.

Analytic Consequence

An inference that is valid based on the meanings of predicates, even if it is not a first-order consequence (e.g., deriving $\text{Larger(a, b)}$ from $\text{Smaller(b, a)}$).

Counterexample

A possible situation or "world" in which the premises of an argument are true and the conclusion is false, used to demonstrate nonconsequence.

Tarski’s World

A program used to create counterexamples to demonstrate that an argument is invalid.

Deductive Reasoning

A form of reasoning used to determine with absolute certainty whether a conclusion follows from a set of premises.

Inductive Reasoning

A form of reasoning where premises make the truth of a conclusion more probable, but do not guarantee it.