The Fundamental Theorem of World Theory
... Note that, given Coherence and some basic modal and propositional 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 equivalen ...
... Note that, given Coherence and some basic modal and propositional 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 equivalen ...
Translating the Hypergame Paradox - UvA-DARE
... Note that here the argument which leads to the paradox has an asymmetric pattern which consists of two parts: (1) ‘p is true’ (hypergame is a founded game); (2) ‘if p is true then lp is true’ (if hypergame is founded, then it is not founded). In other words, we arrive to a contradiction by showing ‘ ...
... Note that here the argument which leads to the paradox has an asymmetric pattern which consists of two parts: (1) ‘p is true’ (hypergame is a founded game); (2) ‘if p is true then lp is true’ (if hypergame is founded, then it is not founded). In other words, we arrive to a contradiction by showing ‘ ...
The Development of Mathematical Logic from Russell to Tarski
... Padoa lists a number of notions that he considers as belonging to general logic such as class (“which corresponds to the words: terminus of the scholastics, set of the mathematicians, common noun of ordinary language”). The notion of class is not defined but assumed with its informal meaning. Extens ...
... Padoa lists a number of notions that he considers as belonging to general logic such as class (“which corresponds to the words: terminus of the scholastics, set of the mathematicians, common noun of ordinary language”). The notion of class is not defined but assumed with its informal meaning. Extens ...
Document
... it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication To show that ∀x (P(x) → Q(x)) is false by finding an x in D such ...
... it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication To show that ∀x (P(x) → Q(x)) is false by finding an x in D such ...
Lecture01 - Mathematics
... a) Propositional logic is the study of propositions (true or false statements) and ways of combining them (logical operators) to get new propositions. It is effectively an algebra of propositions. In this algebra, the variables stand for unknown propositions (instead of unknown real numbers) and the ...
... a) Propositional logic is the study of propositions (true or false statements) and ways of combining them (logical operators) to get new propositions. It is effectively an algebra of propositions. In this algebra, the variables stand for unknown propositions (instead of unknown real numbers) and the ...
AppA - txstateprojects
... inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms and that characterizes the standard behavior of the ...
... inference rules R such that, given any set of axioms A and a sentence c, there is a proof of c, starting with A and applying the rules in R, iff c is entailed by A. • Incompleteness Theorem: any theory that is derived from a decidable set of axioms and that characterizes the standard behavior of the ...
Notes
... Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last cent ...
... Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last cent ...
PPT
... • Program – S-expression and consists of Sexpressions, e.g. (A (B C …) (S (G H …) K)) – The basic elements of s-expressions are lists and atoms. – S-expression may be interpreted as list (data structure) – Or function with arguments, e.g. (ADD (SUB 4 3) 6) return 7. Sexpression may be evaluated and ...
... • Program – S-expression and consists of Sexpressions, e.g. (A (B C …) (S (G H …) K)) – The basic elements of s-expressions are lists and atoms. – S-expression may be interpreted as list (data structure) – Or function with arguments, e.g. (ADD (SUB 4 3) 6) return 7. Sexpression may be evaluated and ...
Logic and Existential Commitment
... This second way of changing a sentence’s truth value leads to what might be termed the possible meaning (PM) account of logical consequence: the conclusion of an argument is a logical consequence of its premises iff there is no possible use or meaning of its constituent nonlogical elements under whi ...
... This second way of changing a sentence’s truth value leads to what might be termed the possible meaning (PM) account of logical consequence: the conclusion of an argument is a logical consequence of its premises iff there is no possible use or meaning of its constituent nonlogical elements under whi ...
Tools-Slides-3 - Michael Johnson`s Homepage
... For any two sets A and B, A = B iff (for all x)(x ∈ A iff x ∈ B) Therefore, {1, 1} = {1} iff (for all x)(x ∈ {1, 1} iff x ∈ {1}) ...
... For any two sets A and B, A = B iff (for all x)(x ∈ A iff x ∈ B) Therefore, {1, 1} = {1} iff (for all x)(x ∈ {1, 1} iff x ∈ {1}) ...
Factoring out the impossibility of logical aggregation
... either or ¬ belongs to B. This maximal consistency property implies the weaker one that B is deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequ ...
... either or ¬ belongs to B. This maximal consistency property implies the weaker one that B is deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequ ...
completeness theorem for a first order linear
... for the rst order temporal logics with since and until over linear time and rationals were given in [16]. In the case of FOLTL (and similarly when the ow of time is isomorphic to reals or integers) the set of valid formulas is not recursively enumerable, and there is no recursive axiomatization of ...
... for the rst order temporal logics with since and until over linear time and rationals were given in [16]. In the case of FOLTL (and similarly when the ow of time is isomorphic to reals or integers) the set of valid formulas is not recursively enumerable, and there is no recursive axiomatization of ...
Hilbert Calculus
... The calculus defines a syntactic consequence relation ⊢ (notation: F1 , . . . , Fn ⊢ G), intended to “mirror” semantic consequence. We will have: F1 , . . . , Fn ⊢ G iff F1 , . . . , Fn |= G (syntactic consequence and semantic consequence will coincide). ...
... The calculus defines a syntactic consequence relation ⊢ (notation: F1 , . . . , Fn ⊢ G), intended to “mirror” semantic consequence. We will have: F1 , . . . , Fn ⊢ G iff F1 , . . . , Fn |= G (syntactic consequence and semantic consequence will coincide). ...
Overview of proposition and predicate logic Introduction
... In these examples brackets indicate in what order the proposition is constructed by the rules from the definition. In order to avoid too many brackets, connectives can be ordered according to their binding strength as follows (from strong to weak): ¬, ∧, ∨, →, ↔. Thus, the final example above might ...
... In these examples brackets indicate in what order the proposition is constructed by the rules from the definition. In order to avoid too many brackets, connectives can be ordered according to their binding strength as follows (from strong to weak): ¬, ∧, ∨, →, ↔. Thus, the final example above might ...
A. Formal systems, Proof calculi
... did not know which formulas are axioms). It means that there is an algorithm that for any WFF given as its input answers in a finite number of steps an output Yes or NO on the question whether is an axiom or not. A finite set is trivially decidable. The set of axioms can be infinite. In such a c ...
... did not know which formulas are axioms). It means that there is an algorithm that for any WFF given as its input answers in a finite number of steps an output Yes or NO on the question whether is an axiom or not. A finite set is trivially decidable. The set of axioms can be infinite. In such a c ...
Scharp on Replacing Truth
... By now we should all be familiar with problems that beset this naı̈ve conception. If L is the sentence ‘L is not true’, then by substituting L into T and applying Leibniz’s law we can infer that L is true if and only if it isn’t. If we furthermore assume the classical laws of logic we can derive fro ...
... By now we should all be familiar with problems that beset this naı̈ve conception. If L is the sentence ‘L is not true’, then by substituting L into T and applying Leibniz’s law we can infer that L is true if and only if it isn’t. If we furthermore assume the classical laws of logic we can derive fro ...
The origin of the technical use of "sound argument": a postscript
... original). (By a deductive argument he means one "in which the truth of the premises guarantees (or is intended to guarantee) the truth of the conclusion without appeal to other reasons" (35-36).) The idea that a deductively valid argument with true premisses is a good argument appears in the textbo ...
... original). (By a deductive argument he means one "in which the truth of the premises guarantees (or is intended to guarantee) the truth of the conclusion without appeal to other reasons" (35-36).) The idea that a deductively valid argument with true premisses is a good argument appears in the textbo ...
lec5 - Indian Institute of Technology Kharagpur
... • If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal. • If the unicorn is either immortal or a mammal, then it is horned. • The unicorn is magical if it is horned Can we prove that the unicorn is mythical? Magical? Horned? ...
... • If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal. • If the unicorn is either immortal or a mammal, then it is horned. • The unicorn is magical if it is horned Can we prove that the unicorn is mythical? Magical? Horned? ...
XR3a
... • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it resulted in the impossible conclusion that a number could be rational and irrational at the same time. (It can be put in a form that implies n n is true, which ...
... • In the previous proof we showed that assuming that the sum of a rational number and an irrational number is rational and showed that it resulted in the impossible conclusion that a number could be rational and irrational at the same time. (It can be put in a form that implies n n is true, which ...
full text (.pdf)
... The scheme p must also verify that oracle responses are correct. Without loss of generality, we can assume that M uses the following mechanism to query the oracle. We assume that M has an integer counter initially set to 0. In each step, M may add one to the counter or not, depending on its current ...
... The scheme p must also verify that oracle responses are correct. Without loss of generality, we can assume that M uses the following mechanism to query the oracle. We assume that M has an integer counter initially set to 0. In each step, M may add one to the counter or not, depending on its current ...
Beyond first order logic: From number of structures to structure of
... in logic. For the last 50 years most research in model theory has focused on first order logic. Motivated both by intrinsic interest and the ability to better describe certain key mathematical structures (e.g. the complex numbers with exponentiation), there has recently been a revival of ‘nonelement ...
... in logic. For the last 50 years most research in model theory has focused on first order logic. Motivated both by intrinsic interest and the ability to better describe certain key mathematical structures (e.g. the complex numbers with exponentiation), there has recently been a revival of ‘nonelement ...
Aristotle`s particularisation
... language—in Cohen’s sense—for which we can insist that B(x) is true under an interpretation without inviting inconsistency? We note that Query 1 essentially questions the introduction of Cantor’s first transfinite ordinal ω into the formal language of Set Theory, purely on the debatable basis that w ...
... language—in Cohen’s sense—for which we can insist that B(x) is true under an interpretation without inviting inconsistency? We note that Query 1 essentially questions the introduction of Cantor’s first transfinite ordinal ω into the formal language of Set Theory, purely on the debatable basis that w ...
slides1
... A proof of A ⇒ B is a function f that maps each proof p of A to the proof f (p) of B. ¬A is treated as A ⇒ ⊥ where ⊥ is a sentence without proof. A proof of ∀ξ.A is a function f that maps each point a in the domain of definition to a proof f (a) of A[a/ξ]. A proof of ∃ξ.A is a pair (a, p) where a is ...
... A proof of A ⇒ B is a function f that maps each proof p of A to the proof f (p) of B. ¬A is treated as A ⇒ ⊥ where ⊥ is a sentence without proof. A proof of ∀ξ.A is a function f that maps each point a in the domain of definition to a proof f (a) of A[a/ξ]. A proof of ∃ξ.A is a pair (a, p) where a is ...