The Science of Proof - University of Arizona Math
... The thesis of this book is that there is a science of proof. Mathematics prides itself on making its assumptions explicit, but most mathematicians learn to construct proofs in an unsystematic way, by example. This is in spite of the known fact that there is an organized way of creating proofs using ...
... The thesis of this book is that there is a science of proof. Mathematics prides itself on making its assumptions explicit, but most mathematicians learn to construct proofs in an unsystematic way, by example. This is in spite of the known fact that there is an organized way of creating proofs using ...
Logic for Computer Science. Lecture Notes
... Finally we have to state clearly what kind of opinions (sentences) can be formulated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Now we can investigate the subject of reasoning via the validity of expressed opinions. Such an ab ...
... Finally we have to state clearly what kind of opinions (sentences) can be formulated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Now we can investigate the subject of reasoning via the validity of expressed opinions. Such an ab ...
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF
... Ehrenfeucht Game with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free v ...
... Ehrenfeucht Game with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free v ...
PROVING THE CORRECTNESS OF REGULA DETERMINISTIC
... standard way. The copstruct x + e in ( 1) is called a (simple) assignment. The progralm constructs in (2) can be expressed as regular expressions over assignments and tests (see, e.g., [§, 47]), hence the adjective ‘regular’ in the title of this paper. The semantic; of a program in WL is based on th ...
... standard way. The copstruct x + e in ( 1) is called a (simple) assignment. The progralm constructs in (2) can be expressed as regular expressions over assignments and tests (see, e.g., [§, 47]), hence the adjective ‘regular’ in the title of this paper. The semantic; of a program in WL is based on th ...
Aristotle, Boole, and Categories
... aeio reverse to become the string oiea) but is not itself part of the language. Nor is any other Boolean operation, the complete absence of which is a feature of syllogistic, not a bug. It follows from all this that a syllogism contains six occurrences of terms, two in each of the three sentences. A ...
... aeio reverse to become the string oiea) but is not itself part of the language. Nor is any other Boolean operation, the complete absence of which is a feature of syllogistic, not a bug. It follows from all this that a syllogism contains six occurrences of terms, two in each of the three sentences. A ...
The Complete Proof Theory of Hybrid Systems
... systems [PC07] that it is often inadvertently forsaken. In logic, we can simply ensure soundness locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s i ...
... systems [PC07] that it is often inadvertently forsaken. In logic, we can simply ensure soundness locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s i ...
PPT
... We reason about the “truth” of wffs using the concept of assignments. An assignment gives a truth value to every propositional variable in the wff. is true if and only if ...
... We reason about the “truth” of wffs using the concept of assignments. An assignment gives a truth value to every propositional variable in the wff. is true if and only if ...
Separation Logic with One Quantified Variable
... first-order quantifiers can be found in [11, 4]. However, these known results crucially rely on the memory model addressing cells with two record fields (undecidability of 2SL in [6] is by reduction to the first-order theory of a finite binary relation). In order to study decidability or complexity ...
... first-order quantifiers can be found in [11, 4]. However, these known results crucially rely on the memory model addressing cells with two record fields (undecidability of 2SL in [6] is by reduction to the first-order theory of a finite binary relation). In order to study decidability or complexity ...
Sequent-Systems for Modal Logic
... will be used for set terms of any level F,Fza,... The schemata F, a, 0, EH > 1. Substitution for these schemata is subject to the same proviso as above. The allowable substitutions for a schema like "F"can be inferredfrom the schema of the formula in which it occurs. In particular,they will be restr ...
... will be used for set terms of any level F,Fza,... The schemata F, a, 0, EH > 1. Substitution for these schemata is subject to the same proviso as above. The allowable substitutions for a schema like "F"can be inferredfrom the schema of the formula in which it occurs. In particular,they will be restr ...
A Nonstandard Approach to the. Logical Omniscience Problem
... What about logical omniscience? Notice that notions like "validity" and "logical consequence" (which played a prominent part in our informal description of logical omniscience) are not absolute notions; their formal definitions depend on how truth is defined and on the class of worlds being consider ...
... What about logical omniscience? Notice that notions like "validity" and "logical consequence" (which played a prominent part in our informal description of logical omniscience) are not absolute notions; their formal definitions depend on how truth is defined and on the class of worlds being consider ...
Propositional Logic - Department of Computer Science
... • Only one person can come in first, etc: represent this using Q, where Q = (¬(L1 ∧ R1) ∧ ¬(L2 ∧ R2) ∧ ¬(L3 ∧ R3) ∧ (R1 ∧ J 1) · · · ) Any interpretation I with I(J ∧ A ∧ P1 ∧ P2 ∧ Q) = 1 corresponds to a possible placing of the three contestants. Logic in Computer Science ...
... • Only one person can come in first, etc: represent this using Q, where Q = (¬(L1 ∧ R1) ∧ ¬(L2 ∧ R2) ∧ ¬(L3 ∧ R3) ∧ (R1 ∧ J 1) · · · ) Any interpretation I with I(J ∧ A ∧ P1 ∧ P2 ∧ Q) = 1 corresponds to a possible placing of the three contestants. Logic in Computer Science ...
Sample pages 2 PDF
... that formalize computations. In both cases, we need to define the syntax and the semantics. The syntax defines what strings of symbols constitute legal formulas (legal programs, in the case of languages), while the semantics defines what legal formulas mean (what legal programs compute). Once the sy ...
... that formalize computations. In both cases, we need to define the syntax and the semantics. The syntax defines what strings of symbols constitute legal formulas (legal programs, in the case of languages), while the semantics defines what legal formulas mean (what legal programs compute). Once the sy ...
A Note on the Relation between Inflationary Fixpoints and Least
... Being a regular logic comes with a range of restrictions, in particular the inability to count. Hence, specifications such as a particular event occurs on all execution traces at the same time or every request is acknowledged cannot be expressed in Lµ . To overcome the restriction to regular logics, ...
... Being a regular logic comes with a range of restrictions, in particular the inability to count. Hence, specifications such as a particular event occurs on all execution traces at the same time or every request is acknowledged cannot be expressed in Lµ . To overcome the restriction to regular logics, ...
4 The semantics of full first
... (ii) (a) v(pi ) = T if and only if pi ∈ Γ∗ . (b) v((Pin , [c1 ]R , . . . , [cn ]R )) = T if and only if Pin c1 . . . cn ∈ Γ∗ . (iii) χ(c) = [c]R for each c ∈ C∗ . Consider how (ii)(b) defines v for (P11 , [c0 ]R ). (ii)(b) says: take a representative c0 from [c0 ]R , and with it form the sentence P ...
... (ii) (a) v(pi ) = T if and only if pi ∈ Γ∗ . (b) v((Pin , [c1 ]R , . . . , [cn ]R )) = T if and only if Pin c1 . . . cn ∈ Γ∗ . (iii) χ(c) = [c]R for each c ∈ C∗ . Consider how (ii)(b) defines v for (P11 , [c0 ]R ). (ii)(b) says: take a representative c0 from [c0 ]R , and with it form the sentence P ...
PROBLEM SOLVING THROUGH FIRST-ORDER LOGIC
... There are many people, who had a significant effect to my private and professional life during the pursuit of my study and the thesis. It‘s my best pleasure to express my thanks to everybody, who have had any influence to me. I would like to start out by thanking my advisor Doc. RNDr. Alena Lukasová ...
... There are many people, who had a significant effect to my private and professional life during the pursuit of my study and the thesis. It‘s my best pleasure to express my thanks to everybody, who have had any influence to me. I would like to start out by thanking my advisor Doc. RNDr. Alena Lukasová ...
Propositional Logic
... • An inference method called “resolution” is probably the most important one for logic programming and automatic theorem proving. • Resolution can be considered as a generalization of modus ponens. • It becomes very powerful in the predicate calculus when combined with a substitution technique known ...
... • An inference method called “resolution” is probably the most important one for logic programming and automatic theorem proving. • Resolution can be considered as a generalization of modus ponens. • It becomes very powerful in the predicate calculus when combined with a substitution technique known ...
Nonmonotonic Logic II: Nonmonotonic Modal Theories
... ABSTRACT Tradmonal logics suffer from the "monotomclty problem"' new axioms never mvahdate old theorems One way to get nd of this problem ts to extend traditional modal logic in the following way The operator M (usually read "possible") is extended so that Mp is true whenever p is consistent with th ...
... ABSTRACT Tradmonal logics suffer from the "monotomclty problem"' new axioms never mvahdate old theorems One way to get nd of this problem ts to extend traditional modal logic in the following way The operator M (usually read "possible") is extended so that Mp is true whenever p is consistent with th ...
Intuitionistic Logic
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
Deciding Intuitionistic Propositional Logic via Translation into
... path of the subformula Fp which wp is meant to be associated with. Thus, we will have to deal with the set W(F ) = {wp |Fp is special}, where we call a subformula Fp special , iff op(Fp ) ∈ {⇒, ¬} and pol(Fp ) = 0. In the above example we obtain W(F ) = {w, w111 , w121 }. The respective subformulas ...
... path of the subformula Fp which wp is meant to be associated with. Thus, we will have to deal with the set W(F ) = {wp |Fp is special}, where we call a subformula Fp special , iff op(Fp ) ∈ {⇒, ¬} and pol(Fp ) = 0. In the above example we obtain W(F ) = {w, w111 , w121 }. The respective subformulas ...
An Automata Theoretic Decision Procedure for the Propositional Mu
... DEFINITION 2.2. A Kripke structure is a triple ( U, k, -+ ), where U is a universe of states, b is a satisfaction relation between states and propositional letters, and -+ gives, for each program letter A, a binary relation +A on states. DEFINITION 2.3. A model is a Kripke structure with the satisfa ...
... DEFINITION 2.2. A Kripke structure is a triple ( U, k, -+ ), where U is a universe of states, b is a satisfaction relation between states and propositional letters, and -+ gives, for each program letter A, a binary relation +A on states. DEFINITION 2.3. A model is a Kripke structure with the satisfa ...
gödel`s completeness theorem with natural language formulas
... We give a formalization of the mathematical language used in the 2example. We shall work with natural language constructs like “for all x” instead of introducing formal quantifiers “∀x”. In this way the meaning or the semantics of formulas becomes self-explanatory. For the purposes of this paper we ...
... We give a formalization of the mathematical language used in the 2example. We shall work with natural language constructs like “for all x” instead of introducing formal quantifiers “∀x”. In this way the meaning or the semantics of formulas becomes self-explanatory. For the purposes of this paper we ...
11. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand
... The Grammar of First-Order Predicate Logic (1) Terms (represent objects): 1. Every variable is a term. 2. If t1 , t2 , . . . , tn are terms and f is an n-ary function, then f (t1 , t2 , . . . , tn ) is also a term. Terms without variables: ground terms. Atomic Formulae (represent statements about o ...
... The Grammar of First-Order Predicate Logic (1) Terms (represent objects): 1. Every variable is a term. 2. If t1 , t2 , . . . , tn are terms and f is an n-ary function, then f (t1 , t2 , . . . , tn ) is also a term. Terms without variables: ground terms. Atomic Formulae (represent statements about o ...
full text (.pdf)
... Correctness assertions, on the other hand, are statements about the global behavior of a program, such as partial correctness or halting. They are typically much richer in expressive power than tests and undecidable in general. DL does not distinguish between these two categories of assertions. The ...
... Correctness assertions, on the other hand, are statements about the global behavior of a program, such as partial correctness or halting. They are typically much richer in expressive power than tests and undecidable in general. DL does not distinguish between these two categories of assertions. The ...
slides
... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
Coordinate-free logic - Utrecht University Repository
... (ii) if ϕ, ψ are formulas, then (ϕ ∧ ψ), ¬ϕ are formulas, (iii) if ϕ is a formula and x is a simple term, then ∀x ϕ is a formula. We will assume that ∨, →, ↔, ∃ are defined in an obvious way. For example, ∃x ϕ denotes ¬∀x ¬ϕ. As the definitions show, we have no terms with more than one argument-pla ...
... (ii) if ϕ, ψ are formulas, then (ϕ ∧ ψ), ¬ϕ are formulas, (iii) if ϕ is a formula and x is a simple term, then ∀x ϕ is a formula. We will assume that ∨, →, ↔, ∃ are defined in an obvious way. For example, ∃x ϕ denotes ¬∀x ¬ϕ. As the definitions show, we have no terms with more than one argument-pla ...