Download Concepts of Proof in Mathematics, Philosophy, and Computer by Dieter Probst, Peter Schuster PDF

By Dieter Probst, Peter Schuster

This booklet presents the reader with learn bobbing up from the Humboldt-Kolleg 'Proof' held in Bern in fall 2013, which accumulated best specialists actively concerned with the concept that 'proof' in philosophy, arithmetic and laptop technology. This quantity goals to do justice to the breadth and intensity of the topic and offers proper present conceptions and technical advances that includes 'proof' in these fields.

Show description

Read Online or Download Concepts of Proof in Mathematics, Philosophy, and Computer Science PDF

Best logic books

An Introduction to Symbolic Logic and Its Applications

A transparent, finished, and rigorous remedy develops the topic from user-friendly suggestions to the development and research of particularly advanced logical languages. It then considers the appliance of symbolic good judgment to the explanation and axiomatization of theories in arithmetic, physics, and biology.

Errors of Reasoning. Naturalizing the Logic of Inference

Mistakes of Reasoning is the long-awaited continuation of the author's research of the common sense of cognitive structures. the current concentration is the person human reasoner working less than the stipulations and pressures of actual existence with capacities and assets the flora and fauna makes to be had to him.

Quanta, logic and spacetime

During this extended variation of Quanta, good judgment and Spacetime, the logical base is drastically broadened and quantum-computational elements of the procedure are delivered to the fore. the 1st elements of this variation may perhaps certainly be considered as supplying a self-contained and logic-based beginning for — and an creation to — the firm referred to as quantum computing.

Knowledge Representation and Reasoning Under Uncertainty: Logic at Work

This quantity is predicated at the foreign convention common sense at paintings, held in Amsterdam, The Netherlands, in December 1992. The 14 papers during this quantity are chosen from 86 submissions and eight invited contributions and are all dedicated to wisdom illustration and reasoning lower than uncertainty, that are middle problems with formal man made intelligence.

Extra info for Concepts of Proof in Mathematics, Philosophy, and Computer Science

Sample text

If M |= ϕ[s], we shall say that M satisfies ϕ on assignment s or that ϕ is true in M on assignment s. We will often write M ϕ[s] if it is not the case that M |= ϕ[s]. Also, if Γ is a set of formulas of L, we shall take M |= Γ[s] to mean that M |= γ[s] for every formula γ in Γ and say that M satisfies Γ on assignment s. Similarly, we shall take M Γ[s] to mean that M γ[s] for some formula γ in Γ. 1. The key clause is 5, which says that ∀ should be interpreted as “for all elements of the universe”.

The idea is that every element of the universe which Σ proves must exist is named, or “witnessed”, by a constant symbol in C. Note that if Σ ¬∃x ϕ, then Σ ∃x ϕ → ϕxc for any constant symbol c. 8. 11. Suppose Γ and Σ are sets of sentences of L, Γ ⊆ Σ, and C is a set of witnesses for Γ in L. Then C is a set of witnesses for Σ in L. 2. Let LO be the first-order language with a single 2place relation symbol, <, and countably many constant symbols, cq for each q ∈ Q. Let Σ include all the sentences (1) cp < cq , for every p, q ∈ Q such that p < q, (2) ∀x (¬x < x), (3) ∀x ∀y (x < y ∨ x = y ∨ y < x), (4) ∀x ∀y ∀z (x < y → (y < z → x < z)), (5) ∀x ∀y (x < y → ∃z (x < z ∧ z < y)), (6) ∀x ∃y (x < y), and (7) ∀x ∃y (y < x).

Observe that any first-order language L has countably many logical symbols. It may have uncountably many symbols if it has uncountably many non-logical symbols. Unless explicitly stated otherwise, we will 1It is possible to formalize almost all of mathematics in a single first-order language, like that of set theory or category theory. However, trying to actually do most mathematics in such a language is so hard as to be pointless. 2Specifically, to countable one-sorted first-order languages with equality.

Download PDF sample

Rated 4.56 of 5 – based on 40 votes