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Non-Classical Logics; 2 {A} An example of logical metatheory would be giving a proof, Preliminaries in English, that a certain logical system is sound and com- plete, i.e., that every inference its deductive system allows is valid according to its semantics, and vice versa. Since the proof is about a logical system, the proof is not given A.1 Some Vocabulary within that logical system, but within the metalanguage. An object language is a language under discussion or be- Though it is not our main emphasis, we will be doing a fair ing studied (the “object” of our study). amount of logical metatheory in this course. Our metalan- guage will be English, and to avoid confusion, we will use A metalanguage is the language used when discussing an English words like “if”, “and” and “not” rather than their object language. corresponding object language equivalents when work- In logic courses we often use English as a metalanguage ing in the metalanguage. We will, however, be employing when discussing an object language consisting of logical a smattering of symbols in the metalanguage, including symbols along with variables, constants and/or proposi- generic mathematical symbols, symbols from set theory, tional parameters. and some specialized symbols such as “” and “ ”. ` As logical studies become more advanced and sophisti- cated, it becomes more and more important to keep the A.2 Basic Set Theory object language and metalanguage clearly separated. A logical system, or a “logic” for short, typically consists of We shall use these signs metalanguage only. (In another three things (but may consist of only the rst two, or the logic course, you might nd such signs used in the object rst and third): language.) 1. A syntax, or set of rules specifying what expressions A set is a collection of entities for which it is determined, are part of the language of the system, and how they for every entity of a given type, that the entity either may be combined to form more complex expres- is or is not included in the set. sions/statements (often called “formulæ)”. An urelement is a thing that is not a set. 2. A semantics, or set of rules governing the meanings An entity A is a member of set Γ i it is included in that or possible meanings of expressions, and how the set. meaning, interpretation, evaluation and truth value of complex expressions depend on the meaning or We write this as: “A Γ”. We write “A / Γ” to 2 2 interpretation of the parts. mean that A is not a member of Γ. 3. A deductive system, or set of rules governing what Sets are determined entirely by their members: for sets makes for an acceptable or endorsed pattern of rea- Γ and ∆, Γ = ∆ i for all A, A Γ i A ∆. 2 2 soning within in the system. A singleton or unit set is a set containing exactly one mem- For example, in your rst course on propositional or sen- ber. tential logic, you learned that the language consisted of “ A ” means the set containing A alone. Gener- propositional parameters or variables, p , p , etc., sym- f g 0 1 ally, “ A1, ... , An ” means the set containing all of bols such as “ ”, “ ”, and that they be combined to form f g A1, ... , An, but nothing else. _ : complex formulæ such as “ (p1 p0)”. This was syntax. You also learned that these: formulæ_ : may be true or false The members of sets are not ordered, so from and that truth-functions, such as those reected in truth A, B = C, D one cannot infer that A = C, only f g f g tables, were used in determing the truth or falsity of com- that either A = C or A = D. plex formulæ. That was semantics. Finally, you learned a If ∆ and Γ are sets, ∆ is said to be a subset of Γ, written set of rules for constructing proofs or derivations, such “∆ Γ”, i all members of ∆ are members of Γ; and as modus ponens: that was a deductive system. ∆ is⊆ said to be a proper subset of Γ, written “∆ Γ”, ⊂ Logical metatheory is the use study of a logical system or i all members of ∆ are members of Γ, but not all logical object language using the logical resources members of Γ are members of ∆. of a metalanguage. 1 If ∆ and Γ are sets, the union of ∆ and Γ, written “∆ Γ”, If R is an equivalence relation then, the R-equivalence is the set that contains everything that is a member[ class on A, written “[A]R”, is the set of all B such that of either ∆ or Γ. A, B R. h i 2 The intersection of ∆ and Γ, written “∆ Γ”, is the set A function (in extension) is a binary relation which, for all that contains everything that is a member\ of both A, B and C, if it includes A, B then it does not also ∆ and Γ. contain A, C unless B =h C. i h i The relative complement of ∆ and Γ, written “∆ Γ”, is So if F is a function and A is in its domain, then the set containing all members of ∆ that are− not there is a unique B such that A, B F; this unique members of Γ. B is denoted by “F(A)”. h i 2 The empty set or null set, written “?”, “Λ” or “ ”, is the An n-place function is a function whose domain con- set with no members. f g sists of n-tuples. For such a function, we write “F(A1, ... , An)” to abbreviate “F( A1, ... , An )”. If Γ and ∆ are sets, then they are disjoint i they have no h i members in common, i.e., i Γ ∆ = ?. An n-place operation on Γ is a function whose domain is n \ Γ and whose range is a subset of Γ. An ordered n-tuple, written “ A1, ... , An ”, is something somewhat like a set,h except thati the elements If F is a function, then F is one-one i for all A and B in are given a xed order, so that A1, ... , An = the domain of F, F(A) = F(B) only if A = B. h i B1, ... , Bn i Ai = Bi for all i such that 1 i n. h i ≤ ≤ If Γ and ∆ are sets, then they are equinumerous, written An ordered 2-tuple, e.g., A, B is also called an ordered “ ”, i there is a one-one function whose do- Γ ∼= ∆ pair. An entity is identiedh i with its 1-tuple. main is Γ and whose range is ∆. If Γ and ∆ are sets, then the Cartesian product of Γ and ∆, Sets Γ and ∆ have the same cardinality or cardinal number written “Γ ∆”, is the set of all ordered pairs A, B if and only if they are equinumerous. such that A× and B . Generally, “ n” ish usedi Γ ∆ Γ If and are sets, then the cardinal number of is said to represent2 all ordered2n-tupes consisting entirely Γ ∆ Γ 2 to be smaller than the cardinal number of ∆ i there of members of Γ. Notice that Γ = Γ Γ. × is a set Z such that Z ∆ and Γ = Z but there is no ⊆ ∼ An n-place relation (in extension) on set Γ is any subset of set W such that W Γ and W = ∆. n. A 2-place relation is also called a binary relation. ⊆ ∼ Γ If is a set, then A is denumerable i is equinumer- Binary relations are taken to be of sets of ordered Γ Γ ous with the set of natural numbers 0, 1, 2, 3, 4, ... , pairs. A 1-place relation is also called (the extension (and so on ad inf.) . f of) a property. g Aleph null, also known as aleph naught, written “ ”, is If R is a binary relation, then the domain of R is the set of 0 the cardinal number of any denumerable set.@ all A for which there is an B such that A, B R. h i 2 If is a set, then is nite i either or there is some If R is a binary relation, the range of R is the set of all B Γ Γ Γ = ? positive integer n such that is equinumerous with for which there is an A such that A, B R. Γ h i 2 the set 1, ... , n . The eld of R is the union of the domain and range of R. f g A set is innite i it is not nite. If R is a binary relation, R is reexive i A, A R for all A set is countable i it is either nite or denumerable. A in the eld of R. h i 2 Homework If R is a binary relation, R is symmetric i for all A and B in the eld of R, A, B R only if B, A R. Assuming that Γ, ∆ and Z are sets, R is a relation, and A h i 2 h i 2 If R is a binary relation, R is transitive i for all A, B and and B are any entities, informally verify the following: C in the eld of R, if A, B R and B, C R then 1. A B i A = B A, C R. h i 2 h i 2 2 f g h i 2 2. if Γ ∆ and ∆ Z then Γ Z A binary relation R is an equivalence relation i R is sym- ⊆ ⊆ ⊆ metric, transitive and reexive. 3. if Γ ∆ and ∆ Γ then Γ = ∆ ⊆ ⊆ 4.( Γ ∆) (Γ ∆) = Γ \ [ − 2 R R 5.
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