Abstract
This book is about a formal approach to mathematical structures. Formal methods are by their very nature formal. When studying mathematical logic, initially one often has to grit ones teeth and absorb certain preliminary definitions on faith. Concepts are given precise definitions, and their meaning is revealed later after one has a chance to see their utility. We will try to follow a different route. Before all formalities are introduced, in this chapter, we will take a detour to see examples of mathematical statements and some elements of the language that is used to express them.
However treacherous a ground mathematical logic, strictly interpreted, may be for an amateur, philosophy proper is a subject, on one hand so hopelessly obscure, on the other so astonishingly elementary, that there knowledge hardly counts. If only a question be sufficiently fundamental, the arguments for any answer must be correspondingly crude and simple, and all men may meet to discuss it on more or less equal terms.
G. H. HardyMathematical Proof[10].
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Notes
- 1.
According to some conventions, zero is not a natural number. For reasons that will be explained later, we will count zero among the natural numbers.
- 2.
In our example we also used the ordering relation < , but in the domain of the natural numbers, the relation x < y can be defined in terms of addition, since for all natural numbers x and y, x is less than y if and only if there is a natural number z such that z is not 0 and x + z = y.
- 3.
In Poincaré’s time, ⊃ was used to denote implication.
- 4.
The phrase “if and only if” is commonly used in mathematics to connect two equivalent statements.
- 5.
We could actually start one level lower. We could say that the empty sequence, with no symbols at all, is a finite sequence of 0’s and 1’s.
- 6.
This unique readability of first-order formulas is not an obvious fact and requires a proof, which is not difficult, but we will not present it here.
- 7.
This is an example of mathematical pedantry. Of course, you would say, they are formulas. They are even atomic formulas! But when we defined atomic formulas, we defined a special kind of expression, and called expressions of this kind “atomic formulas.” When we did that, the formal concept of formula had not been defined yet. To know what a formula of first-order logic is one has to wait for a formal definition of the kind we gave here. To avoid this whole discussion we could have called atomic formulas atoms. If we did that, then clause (1.1) of the definition above would say “Every atom is a formula,” but since the term “atomic formula” is commonly used, we did not have that choice.
- 8.
This is another example of an informal abbreviation.
References
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Hardy, G. H. (1929). Mathematical proof. Mind. A Quarterly Review of Psychology and Philosophy, 38(149), 1–25.
Hodges, W. Tarski’s truth definitions. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2014 ed.). http://plato.stanford.edu/archives/fall2014/entries/tarski-truth/
Tarski, A. (1933). The concept of truth in the languages of the deductive sciences (Polish). Prace Towarzystwa Naukowego Warszawskiego, Wydział III Nauk Matematyczno-Fizycznych, 34, Warsaw; expanded English translation in Tarski, A. (1983). Logic, semantics, metamathematics, papers from 1923 to 1938. Edited by John Corcoran. Indianapolis: Hackett Publishing Company.
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Kossak, R. (2018). First-Order Logic. In: Mathematical Logic. Springer Graduate Texts in Philosophy, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-97298-5_1
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