A Judgmental Reconstruction of Modal Logic

... We call this pattern a local expansion since we obtain more complex evidence for the original judgment. An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verific ...

... We call this pattern a local expansion since we obtain more complex evidence for the original judgment. An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verific ...

AN EARLY HISTORY OF MATHEMATICAL LOGIC AND

... or Richard Dedekind. I believe this is because Styazhkin believes first, that logic from Leibniz to Peano was largely separate from set theory; and second, he believed that logic after Peano changed radically in its relationship with set theory. This is just the view that I want to combat. The reaso ...

... or Richard Dedekind. I believe this is because Styazhkin believes first, that logic from Leibniz to Peano was largely separate from set theory; and second, he believed that logic after Peano changed radically in its relationship with set theory. This is just the view that I want to combat. The reaso ...

On the Meaning of the Logical Constants and the

... and Frege. And, through Frege’s influence, the whole of modern logic has come to be based on the single form of judgement, or assertion, A is true. Once this step was taken, the question arose, What sort of thing is it that is affirmed in an affirmation and denied in a denial? that is, What sort of ...

... and Frege. And, through Frege’s influence, the whole of modern logic has come to be based on the single form of judgement, or assertion, A is true. Once this step was taken, the question arose, What sort of thing is it that is affirmed in an affirmation and denied in a denial? that is, What sort of ...

On the meanings of the logical constants and the justifications of the

... That is, what are you allowed to insert into the places indicated by these variables? The standard answer to this question, by someone who has received the now current logical education, would be to say that A and B range over arbitrary formulas of the language that you are considering. Thus, if the ...

... That is, what are you allowed to insert into the places indicated by these variables? The standard answer to this question, by someone who has received the now current logical education, would be to say that A and B range over arbitrary formulas of the language that you are considering. Thus, if the ...

PDF - University of Kent

... The basic question is, does a proposition such as “all s are m” imply that there are actually any s’s? One might think yes, but consider the propositions: “all unicorns are white” and “all shoplifters will be prosecuted”. As far as we know, there are no unicorns, and there may be no shoplifters, and ...

... The basic question is, does a proposition such as “all s are m” imply that there are actually any s’s? One might think yes, but consider the propositions: “all unicorns are white” and “all shoplifters will be prosecuted”. As far as we know, there are no unicorns, and there may be no shoplifters, and ...

Propositional Logic

... which is even more serious, is that the meaning of an English sentence can be ambiguous, subject to different interpretations depending on the context and implicit assumptions. If the object of our study is to carry out precise rigorous arguments about assertions and proofs, a precise language whose ...

... which is even more serious, is that the meaning of an English sentence can be ambiguous, subject to different interpretations depending on the context and implicit assumptions. If the object of our study is to carry out precise rigorous arguments about assertions and proofs, a precise language whose ...

The Foundations

... 3 is a constant, min is a function symbol with arity 2 “min(3,2)” behaves more like x, 3 than “x >y”. So if let P(x,y) “x > y”, then s1 can be represented as P(y, min(x,3)) we call any expression that can be put on the argument position of an atomic proposition a term Obviously, cons ...

... 3 is a constant, min is a function symbol with arity 2 “min(3,2)” behaves more like x, 3 than “x >y”. So if let P(x,y) “x > y”, then s1 can be represented as P(y, min(x,3)) we call any expression that can be put on the argument position of an atomic proposition a term Obviously, cons ...

Q - GROU.PS

... Idea: Assume that the hypothesis of this implication is true (n is odd). Then use rules of inference and known theorems of math to show that q must also be true (n2 is odd). Spring 2003 ...

... Idea: Assume that the hypothesis of this implication is true (n is odd). Then use rules of inference and known theorems of math to show that q must also be true (n2 is odd). Spring 2003 ...

FC §1.1, §1.2 - Mypage at Indiana University

... mulas themselves. But it is the applications that bring the subject to life for most people. We will, of course, cover some applications as we go along. In a sense, though, the real applications of logic include much of computer science and of mathematics itself. Among the fundamental elements of th ...

... mulas themselves. But it is the applications that bring the subject to life for most people. We will, of course, cover some applications as we go along. In a sense, though, the real applications of logic include much of computer science and of mathematics itself. Among the fundamental elements of th ...

Algebraic foundations for the semantic treatment of inquisitive content

... matter whether such an account is cast within the framework of inquisitive semantics or not. The second question that arises is how the propositions expressed by complex sentences should be defined in a compositional way. In particular, if we limit ourselves to a first-order language, what is the ro ...

... matter whether such an account is cast within the framework of inquisitive semantics or not. The second question that arises is how the propositions expressed by complex sentences should be defined in a compositional way. In particular, if we limit ourselves to a first-order language, what is the ro ...

SITUATIONS, TRUTH AND KNOWABILITY — A

... Aϕ → NK(ϕ, @), where '@' is a constant that rigidly refers to the actual situation and K(ϕ, @) means that it is known of @ that ϕ. By the actual situation we here simply mean a designated situation supplied by context. These ideas, having been indicated here, will be developed in detail below. ...

... Aϕ → NK(ϕ, @), where '@' is a constant that rigidly refers to the actual situation and K(ϕ, @) means that it is known of @ that ϕ. By the actual situation we here simply mean a designated situation supplied by context. These ideas, having been indicated here, will be developed in detail below. ...

Argumentations and logic

... can think that we have deduced a conclusion thought to be false from a hypothesis augmented by premises thought to be true and then discover that the hypothesis itself played no role in the reasoning. This means that we have arrived at an argumentation that seems to deduce a conclusion thought to be ...

... can think that we have deduced a conclusion thought to be false from a hypothesis augmented by premises thought to be true and then discover that the hypothesis itself played no role in the reasoning. This means that we have arrived at an argumentation that seems to deduce a conclusion thought to be ...

Chapter 11: Other Logical Tools Syllogisms and Quantification

... true that if the accused was not in the neighborhood on the night of the crime, then he had no knowledge that the crime took place," ~(~N ~K), it follows only that it is possible that the accused was not in the neighborhood but still knew about the crime. Not that it is necessary that the accused ...

... true that if the accused was not in the neighborhood on the night of the crime, then he had no knowledge that the crime took place," ~(~N ~K), it follows only that it is possible that the accused was not in the neighborhood but still knew about the crime. Not that it is necessary that the accused ...

Lecture 2

... • b c “b differs c” (AKA exclusive or, XOR) • b c “b equivales c” or b and c are equivalent. The Boolean expression b c is evaluate exactly as b=c, except that can be used only when b and c are Boolean expressions (= can be used for arithmetic equality e.g. 2 = 1 + 1). • b c “the inequival ...

... • b c “b differs c” (AKA exclusive or, XOR) • b c “b equivales c” or b and c are equivalent. The Boolean expression b c is evaluate exactly as b=c, except that can be used only when b and c are Boolean expressions (= can be used for arithmetic equality e.g. 2 = 1 + 1). • b c “the inequival ...

Chapter 2 Propositional Logic

... The world logic refers to the use and study of valid reasoning. Logic contains rules and techniques to formalize statements, to make them precise. Logic is studied by philosophers, mathematicians and computer scientists. Logic appears in different areas of computer science, such as programming, circ ...

... The world logic refers to the use and study of valid reasoning. Logic contains rules and techniques to formalize statements, to make them precise. Logic is studied by philosophers, mathematicians and computer scientists. Logic appears in different areas of computer science, such as programming, circ ...

INTERMEDIATE LOGIC – Glossary of key terms

... A combination of logic gates, in which the outputs of some gates are joined to the input of other gates, used to perform complex operations. Logic gate Lesson 32, page 265 A logical operator represented by a symbol in digital logic, with one or two inputs and a single output, which can be joined tog ...

... A combination of logic gates, in which the outputs of some gates are joined to the input of other gates, used to perform complex operations. Logic gate Lesson 32, page 265 A logical operator represented by a symbol in digital logic, with one or two inputs and a single output, which can be joined tog ...

Chapter 2, Logic

... Formal logic studies patterns of argument such that any argument conforming to the pattern is valid. A particular argument can usually be fitted into several different patterns, but to establish it’s validity we need only point to one valid pattern, so when we formalise the propositions of an argume ...

... Formal logic studies patterns of argument such that any argument conforming to the pattern is valid. A particular argument can usually be fitted into several different patterns, but to establish it’s validity we need only point to one valid pattern, so when we formalise the propositions of an argume ...

The Fundamental Theorem of World Theory

... logic, the Equivalence Principle is equivalent to: The Leibniz Principle It is necessary that p if and only if p is true at every possible world. More formally, in terms of the language at hand: LP p ↔ ∀w(w |= p) Given this equivalence between the two principles,3 one can take either principle as a ...

... logic, the Equivalence Principle is equivalent to: The Leibniz Principle It is necessary that p if and only if p is true at every possible world. More formally, in terms of the language at hand: LP p ↔ ∀w(w |= p) Given this equivalence between the two principles,3 one can take either principle as a ...

Knowledge of Logical Truth Knowledge of Logical Truth

... That is, we need: For all S and p, if E├ p then E,S ├ p. So, it looks like the account will be available in monotonic logic, where adding premises does not change the implication, but not in nonmonotonic, where adding premises can ruin the implication. (Think of induction.) However, what we need is ...

... That is, we need: For all S and p, if E├ p then E,S ├ p. So, it looks like the account will be available in monotonic logic, where adding premises does not change the implication, but not in nonmonotonic, where adding premises can ruin the implication. (Think of induction.) However, what we need is ...

On Linear Inference

... there is no premiss, we can apply this inference for any t in place of x, but this yields infinitely many different conclusions. One way to resolve this to allow parametric truths to be asserted, rather than just ground truths. This leads to what is traditionally called resolution, where any clause ...

... there is no premiss, we can apply this inference for any t in place of x, but this yields infinitely many different conclusions. One way to resolve this to allow parametric truths to be asserted, rather than just ground truths. This leads to what is traditionally called resolution, where any clause ...

POSSIBLE WORLDS SEMANTICS AND THE LIAR Reflections on a

... Now, Kaplan’s argument shows that the principle of plenitude is incompatible with assumptions commonly made in possible worlds semantics. Here is how the argument goes: (i) There is a set W of possible worlds and a set P rop of propositions. (ii) There is, for every subset X of W , a corresponding p ...

... Now, Kaplan’s argument shows that the principle of plenitude is incompatible with assumptions commonly made in possible worlds semantics. Here is how the argument goes: (i) There is a set W of possible worlds and a set P rop of propositions. (ii) There is, for every subset X of W , a corresponding p ...

An Introduction to SOFL

... 2.1 Propositional logic Definition 2.1 A proposition is a statement that must be either true or false. For example, the following statements are propositions: (1) A tiger is an animal (true) (2) An apple is a fruit (true) (3) 3 + 5 > 10 (false) ...

... 2.1 Propositional logic Definition 2.1 A proposition is a statement that must be either true or false. For example, the following statements are propositions: (1) A tiger is an animal (true) (2) An apple is a fruit (true) (3) 3 + 5 > 10 (false) ...

A Textbook of Discrete Mathematics

... or It is not the case that Kolkata is a city. Although the two statements ‘Kolkata is not a city’ and ‘It is not the case that Kolkata is a city’ are not identical, we have translated both of them by G p. The reason is that both these statements have the same meaning. Notes: (i) A given statement (p ...

... or It is not the case that Kolkata is a city. Although the two statements ‘Kolkata is not a city’ and ‘It is not the case that Kolkata is a city’ are not identical, we have translated both of them by G p. The reason is that both these statements have the same meaning. Notes: (i) A given statement (p ...

Propositional Logic - faculty.cs.tamu.edu

... You should very carefully inspect this table! It is critical that you memorize and fully understand the meaning of each connective. The semantics of the language Prop is given by assigning truth values to each proposition in Prop. Clearly, an arbitrary assignment of truth values is not interesting, ...

... You should very carefully inspect this table! It is critical that you memorize and fully understand the meaning of each connective. The semantics of the language Prop is given by assigning truth values to each proposition in Prop. Clearly, an arbitrary assignment of truth values is not interesting, ...

Logic is a discipline that studies the principles and methods used in

... Predicate Logic: Existential Quantifier Suppose P(x) is a predicate on some universe of discourse. The existential quantification of P(x) is the proposition: “There exists at least one x in the universe of discourse such that P(x) is true.” ∃ x P(x) reads “for some x, P(x)” or “There exists x, P(x) ...

... Predicate Logic: Existential Quantifier Suppose P(x) is a predicate on some universe of discourse. The existential quantification of P(x) is the proposition: “There exists at least one x in the universe of discourse such that P(x) is true.” ∃ x P(x) reads “for some x, P(x)” or “There exists x, P(x) ...

# Bernard Bolzano

Bernhard Placidus Johann Nepomuk Bolzano (Bernard Bolzano in English; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views. Bolzano wrote in German, his mother tongue. For the most part, his work came to prominence posthumously.