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What Is Mathematical Logic? A Survey

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Proof, Computation and Agency

Part of the book series: Synthese Library ((SYLI,volume 352))

Abstract

What is mathematical logic? Mathematical logic is the application of mathematical techniques to logic. What is logic? I believe I am following the ancient Greek philosopher Aristotle when I say that logic is the (correct) rearranging of facts to find the information that we want. Logic has two aspects: formal and informal. In a sense logic belongs to everyone although we often accuse others of being illogical. Informal logic exists whenever we have a language. In particular Indian Logic has been known for a very long time. Formal (often called, “mathematical”) logic has its origins in ancient Greece in the West with Aristotle. Mathematical logic has two sides: syntax and semantics. Syntax is how we say things; semantics is what we mean.

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Notes

  1. 1.

    A list of the symbols used is included in the appendix.

  2. 2.

    Some people avoid using negation, ¬. They employ a constant ⊥ for the false formula. Then they use the formula \((A\rightarrow \perp)\) instead of ¬A.

  3. 3.

    The phrase “x is free/not free in [some formula]” is a technical condition that avoids misunderstandings.

  4. 4.

    Some people take this as the definition of a sub-group but other examples can be given, see [21] or [5].

  5. 5.

    But not as much by computer scientists.

  6. 6.

    This is not a question of the rule being right or wrong, it is a question of what one can say about what computers do. There are certainly problems which a compute cannot decide (see below, Section 1.5, so the computer does not necessarily “know” whether A is true or ¬A is true.

  7. 7.

    In fact he even showed that there is no finite complete system of axiom schemes for formal arithmetic.

  8. 8.

    To be precise, attention actually focussed on partial functions, those that may not be defined for all arguments.

  9. 9.

    At least as far as we can tell. It seems obvious that John von Neumann used Turing’s ideas but there is no record of him admitting to that! See Martin Davis [11].

  10. 10.

    The Continuum Hypothesis says that there are no infinite cardinal numbers between the smallest infinite cardinal number, that of the set of natural numbers and the cardinal number of the set of all subsets of the natural numbers.

  11. 11.

    The square brackets indicate that A can be discharged, i.e. is not needed for the proof of B, though it is for the proof of B, of course.

  12. 12.

    In fact he only needed transfinite induction up to ε 0, see Section 1.6, in order to prove his result.

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Author information

Authors and Affiliations

  1. School of Information Technology, Monash University, Clayton, VIC, 3800, Australia

    John N. Crossley

Authors
  1. John N. Crossley
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Correspondence to John N. Crossley .

Editor information

Editors and Affiliations

  1. , Institute for Logic, Language and, University of Amsterdam, Science Park 904, Amsterdam, 1098 XH, Netherlands

    Johan van Benthem

  2. Adi Shankaracharya Marg, Powai, Mumbai, 400076, India

    Amitabha Gupta

  3. Graduate Center, Department of Computer and, City University of New York, Fifth Avenue 365, New York, 10016, New York, USA

    Rohit Parikh

Appendix: A Brief Guide to the Symbols

Appendix: A Brief Guide to the Symbols

Symbol

Read

∀

forall

→

implies

\(\frac{A\hspace* {5mm} (A \rightarrow B)} {B}\)

From A and \((A \rightarrow B)\) infer B

∧

and

T

True

F

False

∨

or

¬

not

∃

there exists

(→-E)

→-elimination

(∨-E)

∨-elimination

(∧-E)

∧-elimination

(∃-E)

∃-elimination

(∀-E)

∀-elimination

(→-I)

→-introduction

(∨-I)

∨-introduction

(∧-I)

∧-introduction

(∃-I)

∃-introduction

(∀-I)

∀-introduction

⊥

False(the false proposition)

□

necessarily

◇

possibly

∈

is a member of

∉

is not a member of

a proof of A

a proof of B from A

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Crossley, J.N. (2011). What Is Mathematical Logic? A Survey. In: van Benthem, J., Gupta, A., Parikh, R. (eds) Proof, Computation and Agency. Synthese Library, vol 352. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0080-2_1

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