Modern Logic — A Survey pp 93-112 | Cite as

# Infinitary Logics

Chapter

## Abstract

We begin with two examples. The sentence “No one has more than a finite number of ancestors” might be written symbolically as
which would be read: for every person then we could no longer agree, for one could always conceive of man some show surviving for another

$$forall x[{A_0}(x)\nu {A_1}(x)\nu {A_2}(x)\nu \ldots etc.]$$

*x*(∀*x*) either*x*has no ancestors (A_{0}(*x*)) or (*v*) he has one ancestor (*A*_{1}(x)), or he has two ancestors etc. Even if we interpret ‘person’ to mean past, present or future person, most of us would consider this sentence as being true. On the other hand, if one were to try to put an*a priori*upper bound*n*on the number of possible ancestors a person might have, and assert$$forall x[{A_0}(x)\nu {A_1}(x)\nu {A_2}(x)\nu \ldots \nu {A_n}(x)]$$

*n*=1 generations. Thus what we have is an infinitary sentence which is not logically equivalent to any of its finite approximations.## Keywords

Test Sentence Negation Normal Form Provable Formula Abstract Model Theory Regular Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Bibliography

- For ease of referencing, we have referred to all papers by their author and year of publication. This differs slightly from the dating system used in [15] below.Google Scholar
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## Added in Proof

- 1.Wilfred Seig has drawn my attention to a little known anti-formalist paper of Zermelo which studies the propositional part of
*L*ω_{, ω}: ‘Grundlagen einer allgemeinen Theorie der mathematischen Satzsysteme’,*Fundamenta Mathematica***25**(1935), 136–146.Google Scholar - 2.A technical version of this paper will appear as ‘The Role of the Omitting Types Theorem in Infinitory Logic’, in
*Arch. f. math. Logik u. Grundl. Forschung*.Google Scholar

## Copyright information

© D. Reidel Publishing Company 1981