Logical Systems

Mid-Term ReviewPeter Suber, Philosophy Department, Earlham College Here I try to list all the important terms, distinctions, symbols, and results from the first half of the course. It wouldn't help much as a study guide if I listed everything we covered. You can get that by re-reading Hunter. I've tried to limit the list to what's most important. Believe it or not, I've omitted a lot.

This list covers the introductory unit on infinite sets and the unit on the metatheory of truth-functional propositional logic (TFPL).

The topics are only roughly in the order in which we encountered them. I've adjusted this order when I thought it important to cluster related topics together. I've put in some blank lines to separate clusters from one another.

I believe that all the technical terms on this list are defined in my glossary. If any are not, please let me know and I'll revise the glossary.

For a review sheet for the second half of the course, on predicate logic, see the final exam review.

- set notation

- = {A, B, C} (set gamma consists of members A, B, and C)
- A (A is a member of set gamma)
- (set delta is a subset of set gamma)
- (set delta is a proper subset of set gamma)
- (the union of set delta and set gamma)
- Ø (the null set)

- formal language
- alphabet / grammar (what Hunter calls the set of symbols and formation rules)
- well-formed formula, or wff
- why formal languages are "formal"
- formal language / metalanguage
- metalanguage variables that range over wffs of the formal language
- deductive apparatus
- axioms
- rules of inference (what Hunter calls transformation rules)
- formal system
- why formal systems are "formal"
- abbreviation, "iff"
- theorem / metatheorem
- proof within a system (proof of a theorem) / proof about a system (proof of a metatheorem)
- model theory / proof theory
- semantics / syntax

- effective method
- decidable set
- one-to-one correspondence of sets
- cardinality of a set
- notation, |S| (not used in Hunter)
- subset / proper subset
- power set
- notation, *S
- set / sequence
- enumeration / effective enumeration
- enumerable set / effectively enumerable set

- finite / countable / denumerable / uncountable
- natural numbers / integers / rational numbers / real numbers
- 11.2, the power set of the natural numbers is uncountable
- diagonalization
- Cantor's theorem (the cardinality of an abitrary set A is greater than the cardinality of A) (Hunter 24-25)
- notation,
_{0},_{1 },_{3}...- negative proof (also called indirect proof, apagogical proof)
- 14.1 - 14.3, informal proof of the incompleteness of any finitary formal system of number theory (Hunter 28-30)
- arithmetization
- why there are only countably many strings of symbols of finite length (hence countably many wffs, theorems, names, sentences, descriptions, books...)
- proof that the set of rational numbers is countable (not in Hunter)
- notation, c (for the cardinality of the continuum)
- A4, the set of real numbers is uncountable
- A7, the power set of the set of natural numbers has cardinality c
- A9, the set of points inside a square has cardinality c
- A16, the power set of the set of natural numbers has the cardinality 2^
_{0}- continuum hypothesis / generalized continuum hypothesis
- why the claim c =
_{1}presupposes the continuum hypothesis

- function
- domain / range of a function
- 1-place function / n-place functions
- total / partial functions
- computable / uncomputable functions
- truth functions
- truth tables
- truth values
- connectives
- truth-functional connectives
- truth-functional propositional logic (what I've been calling TFPL)

- interepretation / model
- model of a wff / model of a set of wffs
- truth for an interpretation (or truth for I)
- why truth for I must be defined separately for every connective in the language
- tautology / contingency / contradiction
- logically valid formula
- notation, A (A is a tautology or logically valid formula)
- notation, A (A is a semantic consequence of the wffs in set )
- model-theoretic consistency (m-con) / model-theoretic inconsistency (m-incon)
- disjunctive normal form (DNF)
- why every wff of language P can be expressed in DNF
- sets of connectives adequate to express all truth functions
- 21.1, that the set {~, } is adequate (this is the set used in language P)
- 21.5, that the dagger function alone is adequate
- 21.6, that the stroke function alone is adequate
- 21.12, that the stroke and dagger functions are the only dyadic connectives adequate by themselves

- language P / system PS
- axiom / axiom schema
- modus ponens (the only rule of inference for PS)
- proof / derivation
- why proofs and derivations are finite by definition
- why proofs and derivations are defined syntactically (without reference to truth)
- why axioms are theorems
- notation, A (A is a theorem)
- notation, A (A is a syntactic consequence of the wffs in set )
- semantic consequence () / syntactic consequence ()
- why we need and in the metalanguage when we already have in the formal language

- m-consistency / p-consistency
- why one model is enough for m-consistency
- 23.2, if A, then A (what I've been calling monotonicity) (variation on 20.2, for )
- 23.4, if A and AB, then B (what I've been calling meta-modus-ponens) (variation on 20.4, for )
- simple consistency / absolute consistency
- 24.1, simple con implies absolute con (for most systems)
- 24.2, absolute con implies simple con (for most systems)
- why contradictions imply everything (for most systems) (not in Hunter)
- Post's model-theoretic proof of the consistency of TFPL (Hunter 79)
- Lukasiewicz's proof-theoretic proof of the consistency of TFPL (Hunter 81)
- logical hereditary (not Hunter's term)
- dilemma for consistency proofs: circularity v. relativity (not in Hunter)

- mathematical induction
- basis / induction step / induction hypothesis
- 26.1, the deduction theorem for PS
- conditional proof (not in Hunter)

- semantic completeness
- 28.3, if A, then A
- why truth tables are an effective method for testing tautologousness
- why, given AB, all models of A are models of B
- why all the models of a set of wffs are models of an arbitrary subset of wffs
- lemma
- 31.14, lemma for Kalmar's proof of semantic completeness (that all compound wffs can be expressed as derivations)
- 31.15, if A, then A (Kalmar's proof)
- 32.15, if A, then A (Henkin's proof)
- maximal p-consistent set
- Lindenbaum's lemma for PS (that every p-con set of wffs is a subset of some maximal p-con set of wffs)
- 32.7, {~A} is p-inconsistent iff A
- 32.11, enumeration theorem for language P
- 32.6, m-consistency implies p-consistency
- 32.13, p-consistency implies m-consistency
- 28.4, if A, then A
- 32.14, if A, then A (strong completeness)

- syntactic completeness
- decidable wff / decidable system
- decidable system / system with an effective proof procedure
- 33.1, PS is syntactically complete
- 33.2, if ~A is a wff of P and a non-theorem of PS, then A can be added to PS as an axiom without creating inconsistency (what I've been calling the non-standardness theorem for PS)
- 34.1, PS is decidable

- independence of axioms (and axiom schemata)
- standard / non-standard interpretations
- 36.1 - 36.3, the axiom schemata of PS are independent

This file is an electronic hand-out for the course, Logical Systems.

Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.

Peter Suber,
Department of Philosophy,
Earlham College, Richmond, Indiana, 47374, U.S.A.

*peters@earlham.edu*. Copyright © 1999-2002, Peter Suber.