NOTES on BACKGROUND LOGIC 1. Notation the Logical Notation Used

NOTES ON BACKGROUND LOGIC Holger Leuz

1. Notation The logical notation used by Fitting and Smullyan [5] is a mixture of contemporary standard notation and an older style of logical notation, which was prevalent in English-speaking countries approximately from the 1940s through the 1960s. (A period which is sometimes called Mid-Century, often in relation to art and design.) Most noticeably, Fitting and Smullyan used the horseshoe, , to symbolize implication. The following table explains their logical notation. For the sake of comparison, the older “Mid- Century”-notation is also shown.

Fitting/Smullyan Standard Mid-Century

It is not the case that A  A  A ~ A If A then B A  B A  B A  B A if and only if B A  B A  B A  B A and B A  B A  B A & B A or B (inclusively) A  B A  B A v B For every x (x) x (x) For at least one x (x) x (x)

Fitting and Smullyan also used redundant bracketing in non-identity-statements, for example:  (x = y). Such brackets are not used in modern standard notation, since “ x“ is a syntactically incomplete string of symbols, leaving the brackets redundant.

2. Material implication vs logical implication In logic, the symbol  (or ) denotes material implication. This means that A  B is true if and only if A is false or B is true (or equivalently: it is not the case that A is true and B is false). No further relationship between statements A, B is required by material implication. The symbol  does not have a single standard interpretation in logic. Some authors even use it to denote material implication. It is customary, however, to let A  B denote that B follows logically from A, or that the statement A  B is valid, which is a stronger condition than the truth of A  B. If A  B means that A  B is valid, then the symbol  denotes what is called logical implication or logical consequence. That a statement is valid means that it is true (or satisfied) in every model structure. In standard predicate logic, a model structure, or shortly a model, is a pair (U, I) consisting of a non-empty set U, called the domain or universe of discourse of the model, and an interpretation function I, which assigns extensions to the constants and predicates of the language of predicate logic such that for each sentence of the

1 language of predicate logic a unique truth value is recursively determined by given rules of interpretation and the aforementioned extensions. Every constant is assigned a unique element of U as its extension and every n-place predicate is assigned a subset of Un as its extension. That a sentence is satisfied in a given model structure means that the value true is determined for the sentence in that model structure. (For more details, see [7] or [8]) Mathematical axioms need not be valid in the present sense of being satisfied in all model structures. For example, the axioms of Peano Arithmetic are not satisfied in any model structure with a finite domain. However, all logical consequences of those axioms are satisfied in all model structures where the Peano Axioms are satisfied.

3. Historical note: Strict implication Historically, the modal logics which we encounter in [5], originate from a study of strict implication, a notion introduced by the US-logician C.I. Lewis. Lewis introduced a one-place sentence-operator , where A is supposed to mean it is logically possible that A. One can now define it is logically necessary that A as A, symbolized as □A. A strictly implies B is then defined as □(A  B), or equivalently (as one can easily verify) (A  B). Lewis did not give a precise characterization of logical possibility. In model-theoretic terms one would nowadays say that A is logically possible if and only if A is satisfied in at least one model structure.

4. References

(Numbering continues the numbering found in the course description)

[5] R.M. Smullyan, M. Fitting. Set theory and the continuum problem, revised edition, Dover, 2010. [7] W. Rautenberg. Einführung in die Mathematische Logik: Ein Lehrbuch, 3. Auflage, Vieweg+Teubner, 2008. Kapitel 2. [8] H. Leuz. Einführung in die moderne Logik. Vorlesungsskript, 2018. Kapitel 6.