Lecture Notes on the Lambda Calculus
... information than classical ones, and in particular, they allow one to compute solutions to problems (as opposed to merely knowing the existence of a solution). The resulting algorithms can be useful in computational mathematics, for instance in ...
... information than classical ones, and in particular, they allow one to compute solutions to problems (as opposed to merely knowing the existence of a solution). The resulting algorithms can be useful in computational mathematics, for instance in ...
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The Logic of Compound Statements
... called proposition forms or formulas built from propositional variables (atoms), which represent simple propositions and symbols representing logical connectives Proposition or propositional variables: p, q,… each can be true or false Examples: p=“Socrates is mortal” q=“Plato is mortal” ...
... called proposition forms or formulas built from propositional variables (atoms), which represent simple propositions and symbols representing logical connectives Proposition or propositional variables: p, q,… each can be true or false Examples: p=“Socrates is mortal” q=“Plato is mortal” ...
LPF and MPLω — A Logical Comparison of VDM SL and COLD-K
... to be expressed as formulae of MPLω . This was first sketched in [KR89, Section 4] and later worked out in detail by Renardel de Lavalette in [Ren89]. If A is a formula, then the term ιx : S (A) can be formed which is called a description. Its intended meaning is the unique value x of sort S that sa ...
... to be expressed as formulae of MPLω . This was first sketched in [KR89, Section 4] and later worked out in detail by Renardel de Lavalette in [Ren89]. If A is a formula, then the term ιx : S (A) can be formed which is called a description. Its intended meaning is the unique value x of sort S that sa ...
Rich Chapter 5 Predicate Logic - Computer Science
... But a major motivation for choosing to use logic at all is that if we use logical statements as a way of representing knowledge, then we have available a good way of reasoning with that knowledge. Determining the validity of a proposition in propositional logic is straightforward, although it may be ...
... But a major motivation for choosing to use logic at all is that if we use logical statements as a way of representing knowledge, then we have available a good way of reasoning with that knowledge. Determining the validity of a proposition in propositional logic is straightforward, although it may be ...
Problems on Discrete Mathematics1
... We use Dx , Dy to denote the domains of x and y, respectively. Note that Dx and Dy do not have to be the same. In the above example, P (3, 2) is the proposition 3 ≥ 22 with truth value F . Similarly, Q(Boo, dog) is a proposition with truth value T if there is a dog named Boo. Note: Any proposition i ...
... We use Dx , Dy to denote the domains of x and y, respectively. Note that Dx and Dy do not have to be the same. In the above example, P (3, 2) is the proposition 3 ≥ 22 with truth value F . Similarly, Q(Boo, dog) is a proposition with truth value T if there is a dog named Boo. Note: Any proposition i ...
Essentials Of Symbolic Logic
... discoveries was not realized at the time they were made. The general belief that all the important logical discoveries have been made by Aristotle naturally tended to prevent philosophers from assessing any new discovery at it’s true value. The undeveloped state of the mathematical sciences prior to ...
... discoveries was not realized at the time they were made. The general belief that all the important logical discoveries have been made by Aristotle naturally tended to prevent philosophers from assessing any new discovery at it’s true value. The undeveloped state of the mathematical sciences prior to ...
Notes on Modal Logic - Stanford University
... • Alethic Reading: 2ϕ means ‘ϕ is necessary’ and 3ϕ means ‘ϕ is possible’. • Deontic Reading: 2ϕ means ‘ϕ is obligatory’ and 3ϕ means ‘ϕ is permitted’. In this literature, typically ‘O’ is used instead of ‘2’ and ‘P ’ instead of ‘3’. • Epistemic Reading: 2ϕ means ‘ϕ is known’ and 3ϕ means ‘ϕ is cons ...
... • Alethic Reading: 2ϕ means ‘ϕ is necessary’ and 3ϕ means ‘ϕ is possible’. • Deontic Reading: 2ϕ means ‘ϕ is obligatory’ and 3ϕ means ‘ϕ is permitted’. In this literature, typically ‘O’ is used instead of ‘2’ and ‘P ’ instead of ‘3’. • Epistemic Reading: 2ϕ means ‘ϕ is known’ and 3ϕ means ‘ϕ is cons ...
Formal Foundations of Computer Security
... P1 , P2 , P3 , and the events - all the actions taken, say e1 , e2 , e3 , ... Each action has a location apparent from its definition, say loc(e). Some of the events are comparable ei < ej and others aren’t, e.g. imagine two processes that never communicate e1 , e2 , ... at i and e0i , e02 , ... at ...
... P1 , P2 , P3 , and the events - all the actions taken, say e1 , e2 , e3 , ... Each action has a location apparent from its definition, say loc(e). Some of the events are comparable ei < ej and others aren’t, e.g. imagine two processes that never communicate e1 , e2 , ... at i and e0i , e02 , ... at ...
notes
... Cook’s proof of relative completeness depends on the notion of weakest liberal preconditions. Given a command c and a postcondition Q the weakest liberal precondition is the weakest assertion P such that {P } c {Q} is a valid triple. Here, “weakest” means that any other valid precondition implies P ...
... Cook’s proof of relative completeness depends on the notion of weakest liberal preconditions. Given a command c and a postcondition Q the weakest liberal precondition is the weakest assertion P such that {P } c {Q} is a valid triple. Here, “weakest” means that any other valid precondition implies P ...
article - British Academy
... prove that I cannot prove that 0 = 1. Then the F can prove that I cannot prove that 0 = 1. It doesn’t follow that the F can prove that the F cannot prove that 0 = 1. There are two separable issues here. First, the existence of deviant provability operators, with which the theorem does not go through ...
... prove that I cannot prove that 0 = 1. Then the F can prove that I cannot prove that 0 = 1. It doesn’t follow that the F can prove that the F cannot prove that 0 = 1. There are two separable issues here. First, the existence of deviant provability operators, with which the theorem does not go through ...
Knowledge of Logical Truth Knowledge of Logical Truth
... The question is then whether those truths are derivable from any set of premises. All other possible premises imagined will be additions to E, so the question is whether adding any set of those could ruin the implication. That is, we need: For all S and p, if E├ p then E,S ├ p. So, it looks like the ...
... The question is then whether those truths are derivable from any set of premises. All other possible premises imagined will be additions to E, so the question is whether adding any set of those could ruin the implication. That is, we need: For all S and p, if E├ p then E,S ├ p. So, it looks like the ...
Default Logic (Reiter) - Department of Computing
... • α follows from (D, S W ) by ‘brave’/‘credulous’ reasoning when α in any extension of (D, W ): α ∈ ext(D, W ); • α follows from (D, T W ) by ‘cautious’/‘sceptical’ reasoning when α in all extensions of (D, W ): α ∈ ext(D, W ). ...
... • α follows from (D, S W ) by ‘brave’/‘credulous’ reasoning when α in any extension of (D, W ): α ∈ ext(D, W ); • α follows from (D, T W ) by ‘cautious’/‘sceptical’ reasoning when α in all extensions of (D, W ): α ∈ ext(D, W ). ...
Introduction to Modal and Temporal Logic
... Lemma 1 For any Kripke model hW, R, ϑi, any w ∈ W and any formula ϕ, either ϑ(w, ϕ) = t or else ϑ(w, ϕ) = f . Proof: Pick any Kripke model hW, R, ϑi, any w ∈ W , and any formula ϕ. Proceed by induction on the length l of ϕ. Base Case l = 1: If ϕ is an atomic formula p, either ϑ(w, p) = t or ϑ(w, p) ...
... Lemma 1 For any Kripke model hW, R, ϑi, any w ∈ W and any formula ϕ, either ϑ(w, ϕ) = t or else ϑ(w, ϕ) = f . Proof: Pick any Kripke model hW, R, ϑi, any w ∈ W , and any formula ϕ. Proceed by induction on the length l of ϕ. Base Case l = 1: If ϕ is an atomic formula p, either ϑ(w, p) = t or ϑ(w, p) ...
full text (.pdf)
... In all these examples, the proofs we give are quite short and involve establishing a coinductive step analogous to the inductive step in proofs by induction. What is missing is the final argument that the proof is a valid application of the coinduction principle; but it is not necessary to include th ...
... In all these examples, the proofs we give are quite short and involve establishing a coinductive step analogous to the inductive step in proofs by induction. What is missing is the final argument that the proof is a valid application of the coinduction principle; but it is not necessary to include th ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... as least fixpoint of a logic program, it has been due to an excess of information in the program (possibly erroneous information). As a result, rejecting noncoherent interpretations seems convenient as well. An important remark is that coherence can be interpreted with an empirical sense and that th ...
... as least fixpoint of a logic program, it has been due to an excess of information in the program (possibly erroneous information). As a result, rejecting noncoherent interpretations seems convenient as well. An important remark is that coherence can be interpreted with an empirical sense and that th ...