Metatheory - University of Cambridge

Introduction
This book is an introduction to the metatheory of truth-functional logic, also
known as the propositional calculus.
This book does not itself contain an account of truth-functional logic, but
instead assumes a prior understanding. More specifically, this book assumes
a thorough understanding of the syntax, semantics and proof-system for TFL
(truth-functional logic) that is presented in forallx :Cambridge 2013-4 (for
brevity, I henceforth refer to that book as ‘fx C’). fx C can be obtained, free
of charge, at www.nottub.com/forallx.shtml. There is nothing very surprising
about the syntax or semantics of TFL, and the proof-system is a fairly standard
Fitch-style system.
This book does not, though, presuppose any prior understanding of
(meta)mathematics. Chapter 1 therefore begins by explaining induction from
scratch, first with mathematical examples, and then with metatheoretical examples.
Each of the remaining chapters presupposes an understanding of the
material in Chapter 1.
One idiosyncrasy of this book is worth mentioning: it uses no set-theoretic
notation. Where I need to talk collectively, I do just that, using plural-locutions.
Although this is unusual, I think Metatheory benefits from being free of unnecessary
epsilons and curly brackets. So, where some books say:
{A 1 , A 2 , . . . , A n } is inconsistent
to indicate that no valuation makes all of A 1 , A 2 , . . . , A n true, I say:
A 1 , A 2 , . . . , A n are jointly inconsistent
I use upper-case Greek letters for plural terms, so I might equally say:
Γ are jointly inconsistent
Inconsistency is therefore a (collective) property of sentences, rather than a
property of sets (of sentences). Semantic entailment also comes out as a (collective)
property of sentences. To indicate that every valuation that makes all
of Γ true also makes C true, we write:
Γ ⊨ C
where, of course, Γ are some sentences. If there is a valuation that makes all
of Γ true but C false, we write:
Γ ⊭ C
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