SITUATIONS, TRUTH AND KNOWABILITY — A
... is quite implausible that it should be possible to establish Fitch's strong conclusion: For each agent who is not omniscient, there is a true proposition which the agent cannot know. ...
... is quite implausible that it should be possible to establish Fitch's strong conclusion: For each agent who is not omniscient, there is a true proposition which the agent cannot know. ...
Logic and Proof Jeremy Avigad Robert Y. Lewis Floris van Doorn
... From this perspective, logic is not so much a language for asserting truth, but a language for describing possible states of affairs. In other words, logic provides a specification language, with expressions that can be true or false depending on how we interpret the symbols that are allowed to vary. ...
... From this perspective, logic is not so much a language for asserting truth, but a language for describing possible states of affairs. In other words, logic provides a specification language, with expressions that can be true or false depending on how we interpret the symbols that are allowed to vary. ...
Elementary Logic
... A proposition is also said to be valid if it is a tautology. So, the problem of determining whether a given proposition is valid (a tautology) is also called the validity problem. Note: the notion of a tautology is restricted to propositional logic. In first-order logic, we also speak of valid formu ...
... A proposition is also said to be valid if it is a tautology. So, the problem of determining whether a given proposition is valid (a tautology) is also called the validity problem. Note: the notion of a tautology is restricted to propositional logic. In first-order logic, we also speak of valid formu ...
Let me begin by reminding you of a number of passages ranging
... maximally general science. As Frege says in the opening paragraph of “Der Gedanke”, The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean prescriptions which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature [on the other h ...
... maximally general science. As Frege says in the opening paragraph of “Der Gedanke”, The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean prescriptions which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature [on the other h ...
Chapter 0. Introduction to the Mathematical Method
... Axioms, postulates and propositions Mathematical language has to be uniform (everybody must use it in the same way) and univocal (i.e., without any kind of ambiguity). We start from some initial statements called axioms, postulates and definitions. These elements are not questioned, they are not tru ...
... Axioms, postulates and propositions Mathematical language has to be uniform (everybody must use it in the same way) and univocal (i.e., without any kind of ambiguity). We start from some initial statements called axioms, postulates and definitions. These elements are not questioned, they are not tru ...
Normal modal logics (Syntactic characterisations)
... a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Which closure conditions? See below. Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilber ...
... a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Which closure conditions? See below. Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilber ...
Propositional logic - Cheriton School of Computer Science
... The meaning of proof-theoretic negation represents a major point of departure between schools of logical thought, and the choice we make fundamentally affects the properties of the resulting logic. If we believe that ¬φ means that φ is false, then we are classicists and our proof theory becomes a pr ...
... The meaning of proof-theoretic negation represents a major point of departure between schools of logical thought, and the choice we make fundamentally affects the properties of the resulting logic. If we believe that ¬φ means that φ is false, then we are classicists and our proof theory becomes a pr ...
X - UOW
... false, otherwise it is true. We call P the hypothesis (or antecedent) of the conditional and Q the conclusion (or consequent). In determining the truth values for conditional, consider the following example. Suppose your lecturer say to you: “If you arrive for the lecture on time, then I will mark y ...
... false, otherwise it is true. We call P the hypothesis (or antecedent) of the conditional and Q the conclusion (or consequent). In determining the truth values for conditional, consider the following example. Suppose your lecturer say to you: “If you arrive for the lecture on time, then I will mark y ...
Systems of modal logic - Department of Computing
... a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Which closure conditions? See below. Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilber ...
... a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Which closure conditions? See below. Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilber ...
P,Q
... Definition: An integer n is called odd iff n=2k+1 for some integer k; n is even iff n=2k for some k. Axiom: Every integer is either odd or even. Theorem: (For all numbers n) If n is an odd integer, then n2 is an odd integer. Proof: If n is odd, then n = 2k+1 for some integer k. Thus, n2 = (2 ...
... Definition: An integer n is called odd iff n=2k+1 for some integer k; n is even iff n=2k for some k. Axiom: Every integer is either odd or even. Theorem: (For all numbers n) If n is an odd integer, then n2 is an odd integer. Proof: If n is odd, then n = 2k+1 for some integer k. Thus, n2 = (2 ...
Inference and Proofs - Dartmouth Math Home
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
A Judgmental Reconstruction of Modal Logic
... of introduction rules for logical connectives which allow us to conclude when propositions are true. They are complemented by elimination rules which allow us to obtain further knowledge from the knowledge of compound propositions. The elimination rules for a connective should be locally sound and c ...
... of introduction rules for logical connectives which allow us to conclude when propositions are true. They are complemented by elimination rules which allow us to obtain further knowledge from the knowledge of compound propositions. The elimination rules for a connective should be locally sound and c ...
Gentzen`s original consistency proof and the Bar Theorem
... the notion of a reduction procedure for deductions of sequents in the formal system of first-order number theory in natural deduction and an assignment of ordinals to these deductions such that each reduction step results in a decrease in ordinal. A reduction tree for a deduced sequent, along with a ...
... the notion of a reduction procedure for deductions of sequents in the formal system of first-order number theory in natural deduction and an assignment of ordinals to these deductions such that each reduction step results in a decrease in ordinal. A reduction tree for a deduced sequent, along with a ...
Bilattices and the Semantics of Logic Programming
... a complete semi-lattice) is behind most proposed semantics, we believe the mathematical structure is actually richer than that in practice, and bilattices are the right abstract tools. Furthermore, we will see below that one of the nice ‘extras’ that comes from the bilattice approach is that both th ...
... a complete semi-lattice) is behind most proposed semantics, we believe the mathematical structure is actually richer than that in practice, and bilattices are the right abstract tools. Furthermore, we will see below that one of the nice ‘extras’ that comes from the bilattice approach is that both th ...
logic for computer science - Institute for Computing and Information
... reasoning and of methods of attaching meaning to them. So there are strong parallels between formal computer science and logic. Both involve the study of formal systems and ways of giving them meaning (semantics). However in Logic you study a wider variety of formal systems than you do in Computer S ...
... reasoning and of methods of attaching meaning to them. So there are strong parallels between formal computer science and logic. Both involve the study of formal systems and ways of giving them meaning (semantics). However in Logic you study a wider variety of formal systems than you do in Computer S ...
Proofs by Contradiction and Contraposition
... statement and make one deduction after another until you reach the conclusion. • Indirect proofs are more roundabout. One kind of indirect proof, argument by contradiction, is based on the fact that either a statement is true or it is false but not both. • So if you can show that the assumption that ...
... statement and make one deduction after another until you reach the conclusion. • Indirect proofs are more roundabout. One kind of indirect proof, argument by contradiction, is based on the fact that either a statement is true or it is false but not both. • So if you can show that the assumption that ...
Modalities in the Realm of Questions: Axiomatizing Inquisitive
... Fact 2.5 (Persistence) If M, s |= ϕ and t ⊆ s, then M, t |= ϕ. Thus, more formulas are supported as information grows. In the limit, the empty state supports all formulas. Thus, we refer to ∅ as the absurd state. Fact 2.6 M, ∅ |= ϕ for any M and ϕ. Together, these two properties guarantee that [ϕ]M ...
... Fact 2.5 (Persistence) If M, s |= ϕ and t ⊆ s, then M, t |= ϕ. Thus, more formulas are supported as information grows. In the limit, the empty state supports all formulas. Thus, we refer to ∅ as the absurd state. Fact 2.6 M, ∅ |= ϕ for any M and ϕ. Together, these two properties guarantee that [ϕ]M ...
CH8B
... – Low true signal names are followed by ‘(L)’ to indicate low true – High true signal names are followed by ‘(H)’ to indicate low true ...
... – Low true signal names are followed by ‘(L)’ to indicate low true – High true signal names are followed by ‘(H)’ to indicate low true ...
On the Complexity of the Equational Theory of Relational Action
... for all a, b, c ∈ A. Operations /, \ are called the left and right residual, respectively, with respect to product. Pratt writes a → b for a\b and a ← b for a/b; we use the slash notation of Lambek [18]. Pratt [22] proves that the class of action algebras is a finitely based variety. Furthermore, i ...
... for all a, b, c ∈ A. Operations /, \ are called the left and right residual, respectively, with respect to product. Pratt writes a → b for a\b and a ← b for a/b; we use the slash notation of Lambek [18]. Pratt [22] proves that the class of action algebras is a finitely based variety. Furthermore, i ...
PPT
... In Classical Logic, which is what we’ve been discussing, the goal is to formalize theories. In Intuitionistic Logic, theorems are viewed as programs. They give explicit evidence that a claim is true. ...
... In Classical Logic, which is what we’ve been discussing, the goal is to formalize theories. In Intuitionistic Logic, theorems are viewed as programs. They give explicit evidence that a claim is true. ...
Linear Contextual Modal Type Theory
... from no other resources. They can be used as often as desired, and thus, constructively speaking, they are true. Finally, we introduce the judgment of contextual validity, which will ultimately serve as the logical justification of the existence of logic variables. Usually we say a proposition is va ...
... from no other resources. They can be used as often as desired, and thus, constructively speaking, they are true. Finally, we introduce the judgment of contextual validity, which will ultimately serve as the logical justification of the existence of logic variables. Usually we say a proposition is va ...
Introduction to mathematical arguments
... evidence” or “beyond any reasonable doubt” isn’t good enough. In principle we try to prove things beyond any doubt at all — although in real life people make mistakes, and total rigor can be impractical for large projects. (There are also some subtleties in the foundations of mathematics, such as Go ...
... evidence” or “beyond any reasonable doubt” isn’t good enough. In principle we try to prove things beyond any doubt at all — although in real life people make mistakes, and total rigor can be impractical for large projects. (There are also some subtleties in the foundations of mathematics, such as Go ...
term rewriting.
... • These two interpretations are closely related. Usually we talk about syntactic 'terms' and semantic 'expressions'. • A good specification should be written in such a way that it can be used to predict how the system will behave. ...
... • These two interpretations are closely related. Usually we talk about syntactic 'terms' and semantic 'expressions'. • A good specification should be written in such a way that it can be used to predict how the system will behave. ...