x - Koc Lab
... A theorem is a statement that can be shown to be true using: definitions other theorems axioms (statements which are given as true) rules of inference A lemma is a ‘helping theorem’ or a result which is needed to prove a theorem. A corollary is a result which follows directly from a th ...
... A theorem is a statement that can be shown to be true using: definitions other theorems axioms (statements which are given as true) rules of inference A lemma is a ‘helping theorem’ or a result which is needed to prove a theorem. A corollary is a result which follows directly from a th ...
Thursday Feb 9, at 1:00
... r for each row where the final evaluation is T, and then take their disjunction: (p ∧ q ∧ r) ∨ (p ∧ ¬q ∧ r) ∨ (p ∧ ¬q ∧ ¬r) ∨ (¬p ∧ q ∧ ¬r) ∨ (¬p ∧ ¬q ∧ ¬r) To get conjunctive normal form, we take the conjunction of the values for p, q, and r for each row where the final evaluation is F, and then ta ...
... r for each row where the final evaluation is T, and then take their disjunction: (p ∧ q ∧ r) ∨ (p ∧ ¬q ∧ r) ∨ (p ∧ ¬q ∧ ¬r) ∨ (¬p ∧ q ∧ ¬r) ∨ (¬p ∧ ¬q ∧ ¬r) To get conjunctive normal form, we take the conjunction of the values for p, q, and r for each row where the final evaluation is F, and then ta ...
Chapter 1 Logic and Set Theory
... appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical rules of inference used in the derivation. Rather, they focus on the mathematical ju ...
... appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical rules of inference used in the derivation. Rather, they focus on the mathematical ju ...
Chapter 1 Logic and Set Theory
... appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical rules of inference used in the derivation. Rather, they focus on the mathematical ju ...
... appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical rules of inference used in the derivation. Rather, they focus on the mathematical ju ...
CptS 440 / 540 Artificial Intelligence
... about a particular patient because all the necessary tests have not or cannot be run ...
... about a particular patient because all the necessary tests have not or cannot be run ...
A(x)
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
In order to define the notion of proof rigorously, we would have to
... 3. A proof is a deduction tree whose leaves are all discharged (Γ is empty). This corresponds to the philosophy that if a proposition has been proved, then the validity of the proof should not depend on any assumptions that are still active. We may think of a deduction tree as an unfinished proof tr ...
... 3. A proof is a deduction tree whose leaves are all discharged (Γ is empty). This corresponds to the philosophy that if a proposition has been proved, then the validity of the proof should not depend on any assumptions that are still active. We may think of a deduction tree as an unfinished proof tr ...
Q 0 - SSDI
... Base case (i=1) : If the derivation has only one step, than Q0 has a single atom and any rule must select it. Induction case (i > i+1) : - Assume that the first i steps of the successful derivation ξ are similar to the first i steps obtained by using rule R.. - Assume further that in the i+1th st ...
... Base case (i=1) : If the derivation has only one step, than Q0 has a single atom and any rule must select it. Induction case (i > i+1) : - Assume that the first i steps of the successful derivation ξ are similar to the first i steps obtained by using rule R.. - Assume further that in the i+1th st ...
Notes on Classical Propositional Logic
... Classical propositional logic is the simplest and most nicely behaved of any logic (whatever that means). In a course discussing a wide variety of logics, this is a natural place to start. Many different proof procedures have been developed for it: axiom systems, tree (tableau) systems, sequent calc ...
... Classical propositional logic is the simplest and most nicely behaved of any logic (whatever that means). In a course discussing a wide variety of logics, this is a natural place to start. Many different proof procedures have been developed for it: axiom systems, tree (tableau) systems, sequent calc ...
A(x)
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
Digital Logic and the Control Unit
... Chapter 4 – Digital Logic and the Control Unit This chapter will cover digital logic and its use to build a control unit for a computer. The function of the control unit is to interpret the binary machine language and cause the computer to do what each instruction directs, even if it is not what the ...
... Chapter 4 – Digital Logic and the Control Unit This chapter will cover digital logic and its use to build a control unit for a computer. The function of the control unit is to interpret the binary machine language and cause the computer to do what each instruction directs, even if it is not what the ...
A Contraction-free and Cut-free Sequent Calculus for
... is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form: [α](A → B) → ([α]A → [α]B), for each program α; (ii) modus ponens and ...
... is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form: [α](A → B) → ([α]A → [α]B), for each program α; (ii) modus ponens and ...
Modal Consequence Relations
... ` is structural if from ∆ ` ϕ follows ∆s ` ϕs , where s is a substitution. ` is finitary (or compact) if from ∆ ` ϕ follows that there is a finite ∆0 ⊆ ∆ such that ∆0 ` ϕ. A tautology of ` is a formula ϕ such that ` ϕ. Taut(`) is the set of tautologies of `. There is an alternative approach via dedu ...
... ` is structural if from ∆ ` ϕ follows ∆s ` ϕs , where s is a substitution. ` is finitary (or compact) if from ∆ ` ϕ follows that there is a finite ∆0 ⊆ ∆ such that ∆0 ` ϕ. A tautology of ` is a formula ϕ such that ` ϕ. Taut(`) is the set of tautologies of `. There is an alternative approach via dedu ...
Syllogistic Logic with Complements
... the form Some A are B 0 .) But let S be Some X 0 are Y 0 so that ¬S is All X 0 are Y . M |= S but N |= ¬S. We conclude from this example is that a logical system for the language with complements cannot simply be a translation into the smaller language. The work in Section 2 is not new, though after ...
... the form Some A are B 0 .) But let S be Some X 0 are Y 0 so that ¬S is All X 0 are Y . M |= S but N |= ¬S. We conclude from this example is that a logical system for the language with complements cannot simply be a translation into the smaller language. The work in Section 2 is not new, though after ...
Integrating Logical Reasoning and Probabilistic Chain Graphs
... Clearly, the function PT obeys the axioms of probability theory, as each weight is larger than or equal to 0 and, given that there is a set of consistent assumables consistent with T , there is at least one possible world for T , thus, it follows that w∈W PT (w) = 1. Therefore, it is a joint proba ...
... Clearly, the function PT obeys the axioms of probability theory, as each weight is larger than or equal to 0 and, given that there is a set of consistent assumables consistent with T , there is at least one possible world for T , thus, it follows that w∈W PT (w) = 1. Therefore, it is a joint proba ...
A course in Mathematical Logic
... To know the truth value of the propositional formula (A ∧ B) → C, we need to know the truth values of the propositional variables A, B, C. For example, if A, C are true and B is false, then (A ∧ B) → C is true. The formula A ∧ B is false because B is false, and any implication, whose premises is fa ...
... To know the truth value of the propositional formula (A ∧ B) → C, we need to know the truth values of the propositional variables A, B, C. For example, if A, C are true and B is false, then (A ∧ B) → C is true. The formula A ∧ B is false because B is false, and any implication, whose premises is fa ...
Decidability for some justification logics with negative introspection
... F , then t : F is satisfied in the model. Justification logics without negative introspection are also sound with respect to models that do not fulfill this strong evidence property. To solve the first problem, we develop a novel model construction that is based on non-monotone inductive definitions ...
... F , then t : F is satisfied in the model. Justification logics without negative introspection are also sound with respect to models that do not fulfill this strong evidence property. To solve the first problem, we develop a novel model construction that is based on non-monotone inductive definitions ...
Conditional Statements and Logic
... pq : If two angles are vertical, then they are congruent. ~q~p: If two angles are not congruent, then they are not ...
... pq : If two angles are vertical, then they are congruent. ~q~p: If two angles are not congruent, then they are not ...
Intuitionistic completeness part I
... procedure Prf that converts uniform evidence into a formal first-order proof. We have implemented Prf . Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuiti ...
... procedure Prf that converts uniform evidence into a formal first-order proof. We have implemented Prf . Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuiti ...