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... MON X def X X X e X x y z X x y z x y z x X e x x x e e The type former is a generalisation of the cartesian product to dependent types and corresponds under the propositionsastypes analogy to existential quanti cation. An object of the above generalised signature thus consists of an object of type ...
ppt - UBC Computer Science
ppt - UBC Computer Science

Teach Yourself Logic 2016: A Study Guide
Teach Yourself Logic 2016: A Study Guide

Syntax and Semantics of Dependent Types
Syntax and Semantics of Dependent Types

... view becomes important if one wants to see type theory as a foundation of constructive mathematics which accordingly is to be justi ed by a philosophical argument rather than via an interpretation in some other system, see (Martin-Lof 1975;(1984)). For us the distinction between canonical and nonca ...
Taming method in modal logic and mosaic method in temporal logic
Taming method in modal logic and mosaic method in temporal logic

Lecture Slides
Lecture Slides

...  Assume x is any rational number, y is any irrational number and that x+y is a rational number.  Then x+y = a / b for some aZ and some bZ+  Since x is rational, x = c /d for some cZ and some dZ+  Then (c /d ) + y = a / b  and y = (a / b) - (c /d ) = (ab – bc) / bd  Since ab – bc and bd are ...
The Foundations
The Foundations

Partial Grounded Fixpoints
Partial Grounded Fixpoints

... generalise them to points in the bilattice, while still maintaining the elegance and desirable properties of groundedness. For the case of logic programming, this generalisation boils down to extending groundedness to partial (or three-valued) interpretations. There are several reasons why it is imp ...
CS 512, Spring 2017, Handout 05 [1ex] Semantics of Classical
CS 512, Spring 2017, Handout 05 [1ex] Semantics of Classical

... Theorem (Completeness): If Γ |= ψ then Γ ` ψ . (Stronger than the statement of Completeness in [LCS, Corollary 1.39, p 53]: If ϕ1 , ϕ2 , . . . , ϕn |= ψ then ϕ1 , ϕ2 , . . . , ϕn ` ψ .) ...
1. Proof Techniques
1. Proof Techniques

... arbitrarily chosen, value. Prove that P(x) is true. Conclude that since P(x) is true for this particular x (which has no other special properties), it must be true for all x. An analogy: suppose you are asked to prove the statement “All CS students take CS1231”. You pick Tom, a typical CS student. N ...
A  THEOREM-PROVER FOR  A  DECIDABLE SUBSET OF  DEFAULT
A THEOREM-PROVER FOR A DECIDABLE SUBSET OF DEFAULT

The Development of Categorical Logic
The Development of Categorical Logic

... R. Diaconescu (1975) established the important fact, conjectured by Lawvere, that, in a topos, the axiom of choice implies that the topos is Boolean. This means that, in IZF, the axiom of choice implies the law of excluded middle. This latter formulation of Diaconescu’s result was refined by Goodman ...
Ground Nonmonotonic Modal Logics - Dipartimento di Informatica e
Ground Nonmonotonic Modal Logics - Dipartimento di Informatica e

1 Non-deterministic Phase Semantics and the Undecidability of
1 Non-deterministic Phase Semantics and the Undecidability of

Labeled Natural Deduction for Temporal Logics
Labeled Natural Deduction for Temporal Logics

Prolog 1 - Department of Computer Science
Prolog 1 - Department of Computer Science

Ribbon Proofs - A Proof System for the Logic of Bunched Implications
Ribbon Proofs - A Proof System for the Logic of Bunched Implications

... the level of propositions. It generalizes box proofs (as in Fitch[10]), which are essentially onedimensional, into two dimensions. The horizontal structure of the proof is used to model the resource-sensitive part of the logic. We will develop this system informally as an attractive graphical notati ...
Reductio ad Absurdum Argumentation in Normal Logic
Reductio ad Absurdum Argumentation in Normal Logic

Syllogisms
Syllogisms

The Emergence of First
The Emergence of First

... a logician used first-order logic and where, as more frequently occurred, he employed some richer form of logic. I have distinguished between a logician's use of first-order logic (where quantifiers range only over individuals), second-order logic (where quantifiers can also range over sets or relat ...
Sample pages 2 PDF
Sample pages 2 PDF

Hybrid Interactive Theorem Proving using Nuprl and HOL?
Hybrid Interactive Theorem Proving using Nuprl and HOL?

... We now describe the implementation of this idea for theory importation by describing the steps one takes to import a theory. The rst step is to run HOL, load the desired theory into it, and then execute a function that writes out a le containing a Nuprl-readable version, called the reference copy, ...
Decidability for some justification logics with negative introspection
Decidability for some justification logics with negative introspection

Logic and Sets
Logic and Sets

tbmk5ictk6
tbmk5ictk6

... propositions. Our argument then has the form: All S are M. All M are P. Therefore, all S are P. Any argument that follows this pattern, or form, is valid. Try it for yourself. Think of any three plural nouns; they do not have to be related to each other. For example, you could use submarines, candy ...
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Natural deduction

In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the ""natural"" way of reasoning. This contrasts with the axiomatic systems which instead use axioms as much as possible to express the logical laws of deductive reasoning.
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