## Mathematical LogicAssuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optional sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in Logic, Mathematics, Philosophy, and Computer Science. |

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argument arity atomic formula bijection calculate cardinality clauses closed terms complexity compositional deﬁnition conclusion constant symbol countable cr-structure dandah deﬁne Deﬁnition derivation diﬀerent diophantine domain equation equivalence relation example Exercise expression F F F F T F ﬁnd ﬁnite ﬁrst ﬁrst-order logic formula of LP free occurrence Hintikka set inﬁnite injective function isomorphism left label linearly ordered set logically equivalent LR(cr mathematical n-tuple Natural Deduction Rule natural numbers notation p2 A p3 parsing tree pi A p2 proof calculi propositional logic propositional symbol qf formula qf sentences quantiﬁer real numbers recursion relation symbol right-hand label satisﬁes Section Sequent Rule set of formulas set of sentences signature statement structure substitutable Suppose syntactically consistent Theorem true truth function truth function symbols truth table truth value undischarged assumptions variable write Γ σ ψ