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The Pure Calculus of Entailment Author(s): Alan Ross Anderson and
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and

... logicians somewhat as follows: "The two-valued propositional calculus sanctions as valid many of the obvious and satisfactory inferences which we recognize intuitively as valid, such as (A--.B-TIC)-GAR-B-*.A-C,2 and A--B-.B--C-.A-C; it consequently suggests itself as a candidate for a formal analysi ...
1. Propositional Logic 1.1. Basic Definitions. Definition 1.1. The
1. Propositional Logic 1.1. Basic Definitions. Definition 1.1. The

... was an actual rule. The new rule is admissible: we could take any deduction in P0c and replace each instance of the L¬ rule with a short deduction in Pc , giving a deduction of the same thing in Pc . One might ask if all admissible rules are like this: if I is admissible, is it always because I is a ...
A Survey on Small Fragments of First-Order Logic over Finite
A Survey on Small Fragments of First-Order Logic over Finite

... McNaughton and Papert have considered definability in first-order logic for finite words and they showed that this coincides with the class of star-free languages [39]. Together with Schützenberger’s Theorem this gives decidability of the problem whether a regular language is definable in first-ord ...
John L. Pollock
John L. Pollock

... and concepts that are indispensable for advanced work in philosophy and to do so in a way that conveys the important concepts and techniques without becoming embroiled in unnecessary technical details. The most valuable technical tools are those provided by set theory and the predicate calculus. Kno ...
Functional Programming
Functional Programming

... same way as tail, except that safetail maps the empty list to the empty list, whereas tail gives an error in this case. Define safetail using: (i) a conditional expression; ...
Intuitionistic and Modal Logic
Intuitionistic and Modal Logic

... • Theorem For finite Γ, Γ ` IPC ϕ iff ϕ is valid in all finite Kripke models of Γ for IPC. • Proof. The proof can be done by filtration. We will not do that here. Or by reducing the whole discussion to the set of subformulas of Γ ∪ {ϕ} (a so-called adequate set, both in the definition of the (reduce ...
Integrating Linear and Dependent Types
Integrating Linear and Dependent Types

... of the G(F(A)). Syntactically, such a model then corresponds to a pair of lambda calculi, one intuitionistic, and one linear, which are related by a pair of modal operators F and G. In the linear/non-linear (LNL) calculus, we no longer have to view the intuitionistic function space P → Q through the ...
A Tableau Calculus for Minimal Modal Model Generation
A Tableau Calculus for Minimal Modal Model Generation

Chapter X: Computational Complexity of Propositional Fuzzy Logics
Chapter X: Computational Complexity of Propositional Fuzzy Logics

... There is a pattern in results presented in this chapter: for those decision problems whose complexity has been settled (the problems have been proved complete in some complexity class), the situation is analogous to the classical case: satisfiability is NP-complete, while tautologousness and consequ ...
Pragmatic Functional Programming in Dyalog
Pragmatic Functional Programming in Dyalog

... • Be pragmatic: Stay functional where you can • Use objects and mute state when you must • Languages like APL will be the solution to the next BIG problem after concurrency: ...
Reductio ad Absurdum Argumentation in Normal Logic
Reductio ad Absurdum Argumentation in Normal Logic

... current standard 2-valued semantics for Normal Logic Programs— the Stable Models Semantics [11] — has been around for almost 20 years now, and it is generally accepted as the de facto standard 2-valued semantics for NLPs. This thoroughly studied semantics, however, lacks some important properties am ...
Beginning Logic - University of Notre Dame
Beginning Logic - University of Notre Dame

... Some arguments, in English, are difficult to analyze because of the length and complexity of the sentences. It helps to break things into simpler parts, and focus on the way that the parts are put together. We will translate arguments from English into some formal languages. This process makes the f ...
An Introduction to Mathematical Logic
An Introduction to Mathematical Logic

... 2. the claim that this theorem is a logical consequence of other sentences (in this case: the axioms of equivalence structures) 3. the proof of the theorem More generally: we deal with 1. a set Φ of sentences (“axioms”), a sentence ϕ (“theorem”) 2. the claim that ϕ follows logically from Φ 3. the pr ...
Hybrid Interactive Theorem Proving using Nuprl and HOL?
Hybrid Interactive Theorem Proving using Nuprl and HOL?

... In addition, we prove that normal forms are unique. The formal mathematics in this example can be classi ed into three categories. Imported from HOL. In addition to the standard library, we have imported an HOL theory of combinatory logic and a theory of minimal intuitionistic logic [2]. These theo ...
pdf
pdf

... modeled is that in all the states an agent considers possible at a state s, fewer concepts may be defined than are defined at state s. Because a proposition p may be undefined at a given state s, the underlying logic in HMS is best viewed as a 3-valued logic: a proposition p may be true, false, or ...
Document
Document

... Languages (cont’d.) • xy is the concatenation of the two strings x and y; this is the basic operation on strings – If x = ab and y = bab then xy = abbab and yx = babab – For every string x, x = x = x – |xy| = |x| + |y| ...
Document
Document

... Languages (cont’d.) • xy is the concatenation of the two strings x and y; this is the basic operation on strings – If x = ab and y = bab then xy = abbab and yx = babab – For every string x, x = x = x – |xy| = |x| + |y| ...
Proof analysis beyond geometric theories: from rule systems to
Proof analysis beyond geometric theories: from rule systems to

... elimination of cuts is lost. What one can obtain is a generalized Hauptsatz that reduces all cuts in derivations to cuts on axioms, so analyticity is not preserved. If instead axioms appear as assumptions in the antecedent of sequents, full cut elimination is maintained, but analyticity in root-firs ...
Lecture #1, Sept. 30, 1996
Lecture #1, Sept. 30, 1996

Lisp and Scheme I
Lisp and Scheme I

... As a test, NIL means “false” T is usually used to mean “true,” but… …anything that isn’t NIL is “true” NIL is both an atom and a list – it’s defined this way, so just accept it ...
scheme1 - Department of Computer Science and Electrical
scheme1 - Department of Computer Science and Electrical

... • LISP is an acronym for LISt Processing language • Lisp is an old language with many variants – Fortran is only older language still in wide use – Lisp is alive and well today • Most modern versions are based on Common Lisp • Scheme is one of the major variants – We’ll use Scheme, not Lisp, in this ...
X - UOW
X - UOW

... by defining symbols and establishing ‘rules’. Roughly speaking, in arithmetic an operation is a rule for producing new numbers from a pair of given numbers, like addition (+) or multiplication (× ). In logic, we form new statements by combining short statements using connectives, like the words and, ...
Downloadable PowerPoint file
Downloadable PowerPoint file

... Quicksort is a good example of a program for which there is no direct and simple conversion to tail recursion The second recursive call can be made tail recursive, but not the first Quicksort can be implemented using iteration, but only by implementing a stack internally ...
Modal Logic for Artificial Intelligence
Modal Logic for Artificial Intelligence

... is valid, regardless of the sentences we use in the place of A and B. The only items that need to be fixed are ‘or’ and ‘not’ in this case. If we would replace ‘not’ by ‘maybe’, then the argument would not be valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . the ...
Revisiting Preferences and Argumentation
Revisiting Preferences and Argumentation

... (we assume their strict counterparts defined in the usual way, i.e., l < l0 iff l ≤ l0 and l0  l) are assumed to be used in defining an ordering  on the constructed arguments (see Section 4). Henceforth, we assume the strict counterpart ≺ of  defined in the usual way. Arguments are now defined, w ...
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Combinatory logic

Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.
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