God, the Devil, and Gödel
... would have to look to another aspect of His infinite bounty for proof of His existence. These are heady matters, dark doings, and I do not propose to discourse on the present state of mathematical theology. I only raise them to give an indication of how far-reaching the philosophical consequences of ...
... would have to look to another aspect of His infinite bounty for proof of His existence. These are heady matters, dark doings, and I do not propose to discourse on the present state of mathematical theology. I only raise them to give an indication of how far-reaching the philosophical consequences of ...
Remarks on Second-Order Consequence
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
many-valued logics - University of Sydney
... propositions Γ (written Γ |= α) if, on every model on which every proposition in Γ has the value 1, α has the value 1 (e.g. {p, p → q} |= q, {p} |= p ∨ q). Classical logic is then the language just introduced together with either the set of tautologies, or the consequence relation, just defined. The ...
... propositions Γ (written Γ |= α) if, on every model on which every proposition in Γ has the value 1, α has the value 1 (e.g. {p, p → q} |= q, {p} |= p ∨ q). Classical logic is then the language just introduced together with either the set of tautologies, or the consequence relation, just defined. The ...
Structural Multi-type Sequent Calculus for Inquisitive Logic
... An easy inductive proof shows that InqL-formulas have the downward closure property and the empty team property: (Downward Closure Property) If S |= φ and S ′ ⊆ S , then S ′ |= φ. (Empty Team Property) ∅ |= φ. CPL extended with the dependence atoms =(p1 , . . . , pn , q) is called propositional depe ...
... An easy inductive proof shows that InqL-formulas have the downward closure property and the empty team property: (Downward Closure Property) If S |= φ and S ′ ⊆ S , then S ′ |= φ. (Empty Team Property) ∅ |= φ. CPL extended with the dependence atoms =(p1 , . . . , pn , q) is called propositional depe ...
full text (.pdf)
... We formulate a noncommutative sequent calculus for partial correctness that subsumes propositional Hoare Logic. Partial correctness assertions are represented by intuitionistic linear implication. We prove soundness and completeness over relational and trace models. As a corollary we obtain a comple ...
... We formulate a noncommutative sequent calculus for partial correctness that subsumes propositional Hoare Logic. Partial correctness assertions are represented by intuitionistic linear implication. We prove soundness and completeness over relational and trace models. As a corollary we obtain a comple ...
Discrete Mathematics: Chapter 2, Predicate Logic
... set of premises, then it is a logical consequence of those premises: If P − Q, then P = Q. It is complete because if a sentence is a logical consequence of a set of premises, then it can be proved from them using rules of inference from our system: If P = Q, then P − Q. We explained why SL’s Deducti ...
... set of premises, then it is a logical consequence of those premises: If P − Q, then P = Q. It is complete because if a sentence is a logical consequence of a set of premises, then it can be proved from them using rules of inference from our system: If P = Q, then P − Q. We explained why SL’s Deducti ...
Section 1: Propositional Logic
... • A function can be written in many ways. For example, xy + x, x + yx, x(y + 1) and (x + z)y + x − yz are all ways of writing the same function. Logicians refer to the particular way a function is written as a statement form. You may wonder why we’re concerned with statement forms since we’re not co ...
... • A function can be written in many ways. For example, xy + x, x + yx, x(y + 1) and (x + z)y + x − yz are all ways of writing the same function. Logicians refer to the particular way a function is written as a statement form. You may wonder why we’re concerned with statement forms since we’re not co ...
Introduction to Artificial Intelligence
... To prove that from a knowledge base KB, a query Q follows, we carry out a proof by contradiction. We must show that a contradiction can be derived from KB ∧ ¬Q. In CNF, a contradiction appears in the form of two clauses (A) and (¬A), which lead to the empty clause as their resolvent. To ensure this ...
... To prove that from a knowledge base KB, a query Q follows, we carry out a proof by contradiction. We must show that a contradiction can be derived from KB ∧ ¬Q. In CNF, a contradiction appears in the form of two clauses (A) and (¬A), which lead to the empty clause as their resolvent. To ensure this ...
What is "formal logic"?
... Other striking results are about the axiomatization of simple notions such as identity, it has been shown that the identity relation (the “diagonal”) is not axiomatizable in first-order logic in the same sense that e.g. the notion of wellordering is not first-order axiomatizable (see e.g. Hodges 198 ...
... Other striking results are about the axiomatization of simple notions such as identity, it has been shown that the identity relation (the “diagonal”) is not axiomatizable in first-order logic in the same sense that e.g. the notion of wellordering is not first-order axiomatizable (see e.g. Hodges 198 ...
John Nolt – Logics, chp 11-12
... The operators '•' and ' 0 ' are thus akin, respectively, to universal and existential quantifiers over a domain of possible worlds. So, for example, to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroye ...
... The operators '•' and ' 0 ' are thus akin, respectively, to universal and existential quantifiers over a domain of possible worlds. So, for example, to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroye ...
Consequence relations and admissible rules
... become theorems of L, then so does the conclusion. In talks and informal expositions on admissibility the first definition is often used, while the second one seems to be preferred in formal settings. Informally, it is quite easy to argue that the two definitions are equivalent, but if one wishes to ...
... become theorems of L, then so does the conclusion. In talks and informal expositions on admissibility the first definition is often used, while the second one seems to be preferred in formal settings. Informally, it is quite easy to argue that the two definitions are equivalent, but if one wishes to ...
Knowledge Representation: Logic
... A structural implementation of the map would be done by creating a graph of streets etc. Adding a new component to the map requires some modifications to be introduced to the data structure, so the code must be rewritten. The elements of the map could be associated with object classes divided into p ...
... A structural implementation of the map would be done by creating a graph of streets etc. Adding a new component to the map requires some modifications to be introduced to the data structure, so the code must be rewritten. The elements of the map could be associated with object classes divided into p ...
Logic
... The truth of Q(x), however, depends on the value of x. This is called a propositional function or an open sentence. More than one variable may be present, as in R(x, y ). The truth of this open sentence can only be determined when both x and y are known. ...
... The truth of Q(x), however, depends on the value of x. This is called a propositional function or an open sentence. More than one variable may be present, as in R(x, y ). The truth of this open sentence can only be determined when both x and y are known. ...
Chapter 2 - Princeton University Press
... (b) for each set x there exists a set y with the property that the members of y are the same as the members of members of x. The issue is not whether empty sets and unions “really exist,” but rather, what consequences can be proved about some abstract objects from an abstract system of axioms, consi ...
... (b) for each set x there exists a set y with the property that the members of y are the same as the members of members of x. The issue is not whether empty sets and unions “really exist,” but rather, what consequences can be proved about some abstract objects from an abstract system of axioms, consi ...
Building explicit induction schemas for cyclic induction reasoning
... of induction hypotheses representing ‘not yet proved’ formulas. The induction hypotheses can be defined before their use, by explicit induction schemas that can be directly embedded in inference systems using explicit induction rules. On the other hand, the induction hypotheses can also be defined b ...
... of induction hypotheses representing ‘not yet proved’ formulas. The induction hypotheses can be defined before their use, by explicit induction schemas that can be directly embedded in inference systems using explicit induction rules. On the other hand, the induction hypotheses can also be defined b ...
Modal Logic for Artificial Intelligence
... Logic is concerned with the study of reasoning or, more specifically, the study of arguments. An argument is an act or process of drawing a conclusion from premises. We call an argument sound if the premises are all true, and valid if the truth of the premises guarantees the truth of the conclusion. ...
... Logic is concerned with the study of reasoning or, more specifically, the study of arguments. An argument is an act or process of drawing a conclusion from premises. We call an argument sound if the premises are all true, and valid if the truth of the premises guarantees the truth of the conclusion. ...
Implication - Abstractmath.org
... false would result in even more peculiar situations. For example, if you made P⇒ Q false when P and Q are both false, you would then have to say that the statement discussed previously, "For any integers m and n, if m>5 and 5>n then m>n," is not always true (substitute 3 for m and 4 for n and you ge ...
... false would result in even more peculiar situations. For example, if you made P⇒ Q false when P and Q are both false, you would then have to say that the statement discussed previously, "For any integers m and n, if m>5 and 5>n then m>n," is not always true (substitute 3 for m and 4 for n and you ge ...
Everything is Knowable - Computer Science Intranet
... 3. Successful – the Dynamic Turn The further development of the Moore-sentence firstly gives a multi-agent perspective of announcements of the form “(I tell you that:) p is true and you don’t know that”, and, secondly, gives a dynamic perspective namely that such announcements cannot be believed aft ...
... 3. Successful – the Dynamic Turn The further development of the Moore-sentence firstly gives a multi-agent perspective of announcements of the form “(I tell you that:) p is true and you don’t know that”, and, secondly, gives a dynamic perspective namely that such announcements cannot be believed aft ...
In order to define the notion of proof rigorously, we would have to
... discharging mechanism but they all involve some form of tagging (with “new” variable). For example, the rule formalizing the process that we have just described to prove an implication, A ⇒ B, known as ⇒-introduction, uses a tagging mechanism described precisely in Definition 1.2.1. Now, the rule th ...
... discharging mechanism but they all involve some form of tagging (with “new” variable). For example, the rule formalizing the process that we have just described to prove an implication, A ⇒ B, known as ⇒-introduction, uses a tagging mechanism described precisely in Definition 1.2.1. Now, the rule th ...
BASIC COUNTING - Mathematical sciences
... – Arithmetic operators (operations) such as addition, subtraction, multiplication, division, and negation act on numbers to give new numbers. Logical operators such conjunction (and), disjunction (or), and negation (not) act on propositions to give new (compound) propositions. Logical operators shou ...
... – Arithmetic operators (operations) such as addition, subtraction, multiplication, division, and negation act on numbers to give new numbers. Logical operators such conjunction (and), disjunction (or), and negation (not) act on propositions to give new (compound) propositions. Logical operators shou ...
Complete Sequent Calculi for Induction and Infinite Descent
... which case Γω ⊢ ∆ω is the “limit union” of the sequents along this branch. • Either way, we show Γω ⊢ ∆ω is not provable (this uses the trace condition). • Thus we can use Γω ⊢ ∆ω to construct a syntactic counter-model (the inductive predicate case also uses the ...
... which case Γω ⊢ ∆ω is the “limit union” of the sequents along this branch. • Either way, we show Γω ⊢ ∆ω is not provable (this uses the trace condition). • Thus we can use Γω ⊢ ∆ω to construct a syntactic counter-model (the inductive predicate case also uses the ...
7 LOGICAL AGENTS
... must obey the requirement that when one A SK s a question of the knowledge base, the answer should follow from what has been told (or T ELL ed) to the knowledge base previously. Later in this chapter, we will be more precise about the crucial word “follow.” For now, take it to mean that the inferenc ...
... must obey the requirement that when one A SK s a question of the knowledge base, the answer should follow from what has been told (or T ELL ed) to the knowledge base previously. Later in this chapter, we will be more precise about the crucial word “follow.” For now, take it to mean that the inferenc ...
8.3 Conditional Statements and Material Implication
... p, then q.” But it is not the same kind of implication as any of those mentioned earlier. It is called material implication by logicians. In giving it a special name, we admit that it is a special notion, not to be confused with other, more usual, types of implication. Not all conditional statements ...
... p, then q.” But it is not the same kind of implication as any of those mentioned earlier. It is called material implication by logicians. In giving it a special name, we admit that it is a special notion, not to be confused with other, more usual, types of implication. Not all conditional statements ...
1 LOGICAL CONSEQUENCE: A TURN IN STYLE KOSTA DO SEN
... the sentences below ‘conclusions’; i.e., in a deduction, we pass from premises to a conclusion. There may be several premises from which we deduce a conclusion. Syntactically, a deduction is correct when it is done in conformity to some prescribed rules; semantically, a correct deduction, often call ...
... the sentences below ‘conclusions’; i.e., in a deduction, we pass from premises to a conclusion. There may be several premises from which we deduce a conclusion. Syntactically, a deduction is correct when it is done in conformity to some prescribed rules; semantically, a correct deduction, often call ...
The Omnitude Determiner and Emplacement for the Square of
... TOM: "Subject false" sounds more like an insult than comfort. I don't think Lord Strawson would have agreed that all sentences with an S- P+ profile are false, because you've made your point only with singular subjects, not with universally quantified statements, such as ...
... TOM: "Subject false" sounds more like an insult than comfort. I don't think Lord Strawson would have agreed that all sentences with an S- P+ profile are false, because you've made your point only with singular subjects, not with universally quantified statements, such as