Notes
... Proofs are a “proven way” to find programs in a specification type. Proof rules organize a systematic search for programs and data given a typing specification, e.g. a computing task. The idea of a proof rule in “top down” style. solve x : α → β(x) ` x : α → β(x) ...
... Proofs are a “proven way” to find programs in a specification type. Proof rules organize a systematic search for programs and data given a typing specification, e.g. a computing task. The idea of a proof rule in “top down” style. solve x : α → β(x) ` x : α → β(x) ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... There are two primary rules of inference. Modus ponens (MP) “if A and A → B, then infer B”, and generalization (Gen) “if A, then infer (∀x)A, for any object variable x”. We shall always work within a mathematical theory, generically denoting its set of nonlogical axioms by T . Examples of T are ZFC, ...
... There are two primary rules of inference. Modus ponens (MP) “if A and A → B, then infer B”, and generalization (Gen) “if A, then infer (∀x)A, for any object variable x”. We shall always work within a mathematical theory, generically denoting its set of nonlogical axioms by T . Examples of T are ZFC, ...
Predicate Logic - Teaching-WIKI
... • We'd like to be able to conclude that Pat will get wet, but nothing we have stated so far will help us do this • The problem is that we aren't able to represent any of the details of these propositions – It's the internal structure of these propositions that make the reasoning valid. – But in prop ...
... • We'd like to be able to conclude that Pat will get wet, but nothing we have stated so far will help us do this • The problem is that we aren't able to represent any of the details of these propositions – It's the internal structure of these propositions that make the reasoning valid. – But in prop ...
Cocktail
... (sub)goal into smaller goals until the goal is trivial. Rewriting: Either by using a single rule containing an equation Or by exhaustively using a set of rules in a specified order and direction (computing normal-forms) ...
... (sub)goal into smaller goals until the goal is trivial. Rewriting: Either by using a single rule containing an equation Or by exhaustively using a set of rules in a specified order and direction (computing normal-forms) ...
slides
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
logical axiom
... 2. (a → (b → c)) → ((a → b) → (a → c)) 3. (¬a → ¬b) → (b → a) where → is a binary logical connective and ¬ is a unary logical connective, and a, b, c are any (well-formed) formulas. Let us take these formulas as axioms. Next, we pick a rule of inference. The popular choice is the rule “modus ponens ...
... 2. (a → (b → c)) → ((a → b) → (a → c)) 3. (¬a → ¬b) → (b → a) where → is a binary logical connective and ¬ is a unary logical connective, and a, b, c are any (well-formed) formulas. Let us take these formulas as axioms. Next, we pick a rule of inference. The popular choice is the rule “modus ponens ...
Logical Fallacies Chart APLAC TERM DEFINITION EXAMPLE 1
... Someone tries to win support for their argument or idea by exploiting her or his opponent's feelings of pity or guilt. In false analogies, though A and B may be similar in one respect (such as color) they may not both share property X (e.g. size). This fallacy consists of offering a poorly supported ...
... Someone tries to win support for their argument or idea by exploiting her or his opponent's feelings of pity or guilt. In false analogies, though A and B may be similar in one respect (such as color) they may not both share property X (e.g. size). This fallacy consists of offering a poorly supported ...
An Introduction to Lower Bounds on Formula
... properties of Kripke frames and models. To put things in perspective, I am going to start by giving an informal overview of some techniques used for proving lower bounds on the size of Boolean formulae and then I am going to show how to extend and apply them in the modal case where we have obtained ...
... properties of Kripke frames and models. To put things in perspective, I am going to start by giving an informal overview of some techniques used for proving lower bounds on the size of Boolean formulae and then I am going to show how to extend and apply them in the modal case where we have obtained ...
Is the principle of contradiction a consequence of ? Jean
... thought. They generally don’t interpret as a fundamental law of thought and as the principle of contradiction. 4. Boolean algebra from the point of view of model theory Boolean algebra can be considered today from the point of view of classical first-order logic. First-order logic can itself be cons ...
... thought. They generally don’t interpret as a fundamental law of thought and as the principle of contradiction. 4. Boolean algebra from the point of view of model theory Boolean algebra can be considered today from the point of view of classical first-order logic. First-order logic can itself be cons ...
Modal Logic
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
Modal_Logics_Eyal_Ariel_151107
... Let be a set of propositions. These propositions describe facts about the system as “the system is deadlocked” or “the value of variable x is 5”. An interpreted system is a tuple (S,V), where ...
... Let be a set of propositions. These propositions describe facts about the system as “the system is deadlocked” or “the value of variable x is 5”. An interpreted system is a tuple (S,V), where ...
Lecture 10 Notes
... 1. Reflecting on Evidence Semantics We see both philosophical and technical reasons for exploring this new semantics. On the philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important ...
... 1. Reflecting on Evidence Semantics We see both philosophical and technical reasons for exploring this new semantics. On the philosophical side we hear phrases such as “mental constructions” and intuition used to account for human knowledge. On the technical side we see that computers are important ...
Lecture #3
... and four transistors to make a two-input nand or nor gate (two transistors per input). A two input and gate is then built using a nand gate and an inverter. It makes sense to learn how to build logic gates using only nand or nor gates (as well as inverters). The sequence of figures 3, 4, and 5. ...
... and four transistors to make a two-input nand or nor gate (two transistors per input). A two input and gate is then built using a nand gate and an inverter. It makes sense to learn how to build logic gates using only nand or nor gates (as well as inverters). The sequence of figures 3, 4, and 5. ...
Syntax of first order logic.
... be using the logical symbols ∀, ∃, ∧, ∨, →, ¬, ↔, equality =, and a set of variables Var. Definition of an L-term. Every variable is an L-term. If σ(f˙i ) = n, and t1 , ..., tn are L-terms, then f˙i (t1 , ..., tn ) is an L-term. Nothing else is an L-term. Definition of an L-formula. If t and t∗ are ...
... be using the logical symbols ∀, ∃, ∧, ∨, →, ¬, ↔, equality =, and a set of variables Var. Definition of an L-term. Every variable is an L-term. If σ(f˙i ) = n, and t1 , ..., tn are L-terms, then f˙i (t1 , ..., tn ) is an L-term. Nothing else is an L-term. Definition of an L-formula. If t and t∗ are ...
Semantics of intuitionistic propositional logic
... Example 2.13 Here are some examples of distributive lattices. The first, second and fourth lattices on the top row are boolean algebras, while the other lattices are not. (Exercise: in each such case find the elements which lack complements.) ...
... Example 2.13 Here are some examples of distributive lattices. The first, second and fourth lattices on the top row are boolean algebras, while the other lattices are not. (Exercise: in each such case find the elements which lack complements.) ...
Propositional Logic .
... Close stations are not assigned the same frequency. For each (i,j) 2 E, ...
... Close stations are not assigned the same frequency. For each (i,j) 2 E, ...
323-670 ปัญญาประดิษฐ์ (Artificial Intelligence)
... • It is equivalent to a single long sentence: the conjunction of all sentences (JerryGivingLecture (TodayIsTuesday TodayIsThursday)) JerryGivingLecture ...
... • It is equivalent to a single long sentence: the conjunction of all sentences (JerryGivingLecture (TodayIsTuesday TodayIsThursday)) JerryGivingLecture ...
Chapter 1 Section 2
... Poole and Alan Mackworth, 2010) for details on this problem and how the method of consistency based diagnosis can determine possible diagnoses for the electrical system. The approach yields 7 possible faults in the system. At least one of these must hold: Circuit Breaker 1 is not ok. Both Swit ...
... Poole and Alan Mackworth, 2010) for details on this problem and how the method of consistency based diagnosis can determine possible diagnoses for the electrical system. The approach yields 7 possible faults in the system. At least one of these must hold: Circuit Breaker 1 is not ok. Both Swit ...
Probabilistic Propositional Logic
... FOPC, it is computationally semi-decidable, which is a far cry from polynomial property of GMP inferences. • So, most common uses of FOPC involve doing GMP-style reasoning rather than the full theorem-proving.. • There is a controversy in the community as to whether the right way to handle the compu ...
... FOPC, it is computationally semi-decidable, which is a far cry from polynomial property of GMP inferences. • So, most common uses of FOPC involve doing GMP-style reasoning rather than the full theorem-proving.. • There is a controversy in the community as to whether the right way to handle the compu ...