Philosophical Introduction to Set Theory (Dover Books on Mathematics) by Stephen Pollard

Author:Stephen Pollard [Pollard, Stephen]
Language: eng
Format: azw
Publisher: Dover Publications
Published: 2015-07-20T00:00:00+00:00

The annotation on each line cites the rule of inference we have used to obtain the sentence on that line. It also lists the line numbers of the sentences to which we have applied the rule (if there are any such sentences). The annotation on line 3 says that we obtained ‘¬∀x x∈x’ by applying the rule of negation introduction (abbreviated ‘¬ I’) to the sentences on lines 1 and 2. Our derivation has not established the truth of ‘¬∀x x∈x’–it has established merely that ‘¬∀x x∈x is derivable from the sentences on lines 1 and 2. We make a record of this deductive relation by writing ‘1,2’ in the premise number column on line 3. The premise numbers on the first two lines play a somewhat different role: they indicate that the sentences on those lines were merely assumed rather than derived. (Rule A is our rule of assumption.) All of this should make a bit more sense when we have familiarized ourselves with the rules of inference.

Suppose φ, ψ are formulas, α is a variable, and β,γ are terms in which no variable occurs free.

Rule A (Assumption): Any sentence may appear on a line as long as the premise number on that line is the same as the line number.

Rule ¬ I (Negation Introduction): ¬φ may be written on a line n if (φ → ψ) and (φ → ¬ ψ) appear on earlier lines m and m′; write all the premise numbers from m and m′ in the premise number column on n.

Rule ¬ E (Negation Elimination): φ may be written on a line n if ¬ ¬ φ appears on an earlier line m; write all the premise numbers from m in the premise number column on n.

Rule →I (Conditional Introduction): (φ → ψ) may be written on a line n if ψ appears on an earlier line m; write all the premise numbers from m in the premise number column on n omitting at most those numbers which represent φ. (A premise number p is said to represent a sentence φ if φ occupies the sentence column on line p.)

Rule ∀I (Universal Introduction): Suppose β is an individual constant which does not occur in φ. ∀αφ may be written on a line n if φα/β appears on an earlier line m and no premise number on m represents a sentence in which β occurs; write all the premise numbers from m in the premise number column on n.

Rule ∀E (Universal Elimination): φα/β may be written on a line n if ∀αφ appears on an earlier line m; write all the premise numbers from m in the premise number column on n.

Rule = I (Identity Introduction): β = β may be written on any line; leave the premise number column on that line empty.

Rule =E (Identity Elimination): Suppose ψ is the result of replacing one or more occurrences of β in φ by occurrences of γ. ψ may be written on a line n if φ

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