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Propositional and predicate logic - Computing Science
Propositional and predicate logic - Computing Science

... information. To form conclusions, inferences, or judgments [Q] How to automate reasoning? ...
Logical nihilism - University of Notre Dame
Logical nihilism - University of Notre Dame

... Thus we see a sense in which the phenomenon of structural completeness is related to a sort of semantic completeness: A structurally incomplete logic will be incomplete with respect to the most naive procedural reading of its connectives. It also happens that structural completeness bears a precise ...
Document
Document

... two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ...
propositions and connectives propositions and connectives
propositions and connectives propositions and connectives

... two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ...


... is valid iff all substitution instances of the formula are valid. However, when it comes to inference, this distinction is crucial. For example, if we did not consider substitution instances, then p F,, false would hold, where p is a primitive proposition, since p is not valid.4 However, when viewed ...
The Gödelian inferences - University of Notre Dame
The Gödelian inferences - University of Notre Dame

... according to whom the various ways of stating the consistency of S are equivalent regardless of whether S proves such equivalences. Are the conditions sufficient for G2 also sufficient for S to be able to prove all such equivalences? If not, then it is consistent with G2 to assume that S proves its ...
Label-free Modular Systems for Classical and Intuitionistic Modal
Label-free Modular Systems for Classical and Intuitionistic Modal

... for all logics in the intuitionistic modal S5-cube [18]. This concerns the modal axioms d, t, b, 4, and 5, shown in Figure 1. In classical logic only one of the two conjuncts in each axiom shown in that Figure is needed because the other follows from De Morgan duality. However, in the intuitionistic ...
A Simple Tableau System for the Logic of Elsewhere
A Simple Tableau System for the Logic of Elsewhere

... Proof. Similar to the proof of Proposition 3.1 in [HM92]. For each model M = (W, V ) we define the equivalence relation ≡M such that u ≡M v iff for all P ∈ φ0 , u ∈ V (P ) iff v ∈ V (P ). We write |u|V to denote the equivalence classes with respect to ≡M . Observe that for all A ∈ for(φ0 ), if |u|V ...
Label-free Modular Systems for Classical and Intuitionistic Modal
Label-free Modular Systems for Classical and Intuitionistic Modal

... for all logics in the intuitionistic modal S5-cube [18]. This concerns the modal axioms d, t, b, 4, and 5, shown in Figure 1. In classical logic only one of the two conjuncts in each axiom shown in that Figure is needed because the other follows from De Morgan duality. However, in the intuitionistic ...
From Syllogism to Common Sense Normal Modal Logic
From Syllogism to Common Sense Normal Modal Logic

... ‣ A U B: A is true until B becomes true ‣ G = ‘always’ , F = ‘eventually’, ‣ liveness properties state that something good keeps happening: ...
Answer Sets for Propositional Theories
Answer Sets for Propositional Theories

... Note that, even if (5) is classically equivalent to (7), they are not equivalent in the logic of here-and-there. This shows that it is generally incorrect to “move” a negation from a binary relation symbol (such as 6=) in front of the aggregate as the unary connective ¬. Some properties of aggregate ...
No Syllogisms for the Numerical Syllogistic
No Syllogisms for the Numerical Syllogistic

... an atom is said to be positive; all other literals are said to be negative. If l = p̄ is a negative literal, then we take l̄ to denote the positive literal p. An N † -formula is an expression of either of the forms (≤ C)[l, m] or (> C)[l, m], where C is a decimal string representing a non-negative i ...
Knowledge Representation and Reasoning
Knowledge Representation and Reasoning

... An inference rule characterises a pattern of valid deduction. In other words, it tells us that if we accept as true a number of propositions — called premisses — which match certain patterns, we can deduce that some further proposition is true — this is called the conclusion. Thus we saw that from t ...
Proof Search in Modal Logic
Proof Search in Modal Logic

... DL. System Z as described can prove the self-reference lemma. So, this is not an issue. However, if a modal logic is used to capture provability (i.e., has a representation of the predicate B, along with axioms that capture the derivability conditions), it is easy to represent D1, D2, and D3 (in ter ...
Propositional Logic
Propositional Logic

... users • Needs to be combined with a search algorithm. • Can be very appropriate for general problem solving (e.g., using PROLOG) CSE 415 -- (c) S. Tanimoto, 2008 Propositional Logic ...
ANNALS OF PURE AND APPLIED LOGIC I W
ANNALS OF PURE AND APPLIED LOGIC I W

... and combines ideas from both [14] and [17]. In Section 4, we propose an extension of CPL for modeling concurrent processes, called ICPL. It uses ‘11’ to denote interleaving. Interleaving is one of the simplest ways to model concurrent computation (see, e.g., [ 12]), capturing processes that run conc ...
Classical First-Order Logic Introduction
Classical First-Order Logic Introduction

... propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, functions, and relations. ...
Chapter 2, Logic
Chapter 2, Logic

... Extracting from the third diagram just the information about gardeners and pipe smokers, we have the last diagram, which corresponds to the conclusion. Does Logic just state the obvious? Perusal of examples provided by logicians often suggests that their validity is so obvious that it is hardly wort ...
Everything is Knowable - Computer Science Intranet
Everything is Knowable - Computer Science Intranet

... Indeed, Fitch does ignore it and makes no difference between known and th-knowable, between ‘knows to be true’ and ‘can know to be true’. Fitch (1963) does not distinguish two distinct modalities 䉫 and K. A standard analysis of the Fitch paradox is as follows. We base our exposition on the excellent ...
Lecture Notes
Lecture Notes

... Theorem: For finite M , and φ ∈ L{K1 ,...,Kn ,CG } there exists an algorithm that solves the problem in time linear in |M | · |φ|, where |M | and |φ| are the amount of space needed to write down M and φ, ...
Lectures on Proof Theory - Create and Use Your home.uchicago
Lectures on Proof Theory - Create and Use Your home.uchicago

... way of putting it is that R(α) is the result P α (∅) of iterating the PowerSet operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upo ...
A Critique of the Foundations of Hoare-Style
A Critique of the Foundations of Hoare-Style

... rules are sufficiently powerful to prove all true statements in the Hoare language of conditionaf-while programs. ...
Elementary Logic
Elementary Logic

... A (propositional) sequent is an expression of the form Γ ` ∆, where Γ = A1 , A2 , · · · , Am and ∆ = B1 , B2 , · · · , Bn are finite (possibly empty) sequences of (propositional) formulae. In a sequent Γ ` ∆, Γ is called the antecedent (also context) and ∆ the consequent. Note: many authors prefer t ...
A Critique of the Foundations of Hoare-Style Programming Logics
A Critique of the Foundations of Hoare-Style Programming Logics

... rules are sufficiently powerful to prove all true statements in the Hoare language of conditionaf-while programs. ...
Introduction to Artificial Intelligence
Introduction to Artificial Intelligence

... The goal is to be able, in the end, to instantly see that KB |= Q. We call this syntactic process derivation and write KB ⊢ Q. Such syntactic proof systems are called calculi. Definition of Soundness and Completeness A calculus is called sound if every derived proposition follows semantically. That ...
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Abductive reasoning

Abductive reasoning (also called abduction, abductive inference or retroduction) is a form of logical inference which goes from an observation to a theory which accounts for the observation, ideally seeking to find the simplest and most likely explanation. In abductive reasoning, unlike in deductive reasoning, the premises do not guarantee the conclusion. One can understand abductive reasoning as ""inference to the best explanation"".The fields of law, computer science, and artificial intelligence research renewed interest in the subject of abduction. Diagnostic expert systems frequently employ abduction.
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