Introduction to Mathematical Logic lecture notes
... the convention that ¬ binds more strongly than the binary connectives). When wanting to prove that all formulae have a certain property, we usually use “proof by induction on the construction of the formula”: Theorem 1.1.1 (Proof by induction on the structure). Let X is a property that a formula may ...
... the convention that ¬ binds more strongly than the binary connectives). When wanting to prove that all formulae have a certain property, we usually use “proof by induction on the construction of the formula”: Theorem 1.1.1 (Proof by induction on the structure). Let X is a property that a formula may ...
CATEGORICAL MODELS OF FIRST
... setting, it is often a useful short-cut to consider only one half of any De Morgan dual pair, and/or use a one-sided presentation of the logic; this will, for example, half the length of any presentation of the system. However, when considering the system for its own merits, one may appreciate the b ...
... setting, it is often a useful short-cut to consider only one half of any De Morgan dual pair, and/or use a one-sided presentation of the logic; this will, for example, half the length of any presentation of the system. However, when considering the system for its own merits, one may appreciate the b ...
Formal systems of fuzzy logic and their fragments∗
... T, ϕ → ψ `L χ T, ψ → ϕ `L χ T `L χ Sometimes we just say that the logic is fuzzy. As we do not want to go into details about general semantics of (weakly) implicative logics (logical matrices) we gave just a syntactic definition. The original definition in [8] involves both syntax and semantics and ...
... T, ϕ → ψ `L χ T, ψ → ϕ `L χ T `L χ Sometimes we just say that the logic is fuzzy. As we do not want to go into details about general semantics of (weakly) implicative logics (logical matrices) we gave just a syntactic definition. The original definition in [8] involves both syntax and semantics and ...
A Proof Theory for Generic Judgments
... cut-free proof (that is, it can be computed) and we know that A ⊃ B can be proved (possibly with cuts), cut-elimination allows us to conclude that B has a cut-free proof (that is, it can be computed). As we mentioned above, such direct reasoning on logic specification involves instantiations of eige ...
... cut-free proof (that is, it can be computed) and we know that A ⊃ B can be proved (possibly with cuts), cut-elimination allows us to conclude that B has a cut-free proof (that is, it can be computed). As we mentioned above, such direct reasoning on logic specification involves instantiations of eige ...
Godel`s Proof
... Starting from scratch, AM discovered many concepts of number theory. Rather than logically proving theorems, AM wandered around the world of numbers, following its primitive esthetic nose, sniffing out patterns, and making guesses about them. As with a bright human, most of AM’s guesses were right, s ...
... Starting from scratch, AM discovered many concepts of number theory. Rather than logically proving theorems, AM wandered around the world of numbers, following its primitive esthetic nose, sniffing out patterns, and making guesses about them. As with a bright human, most of AM’s guesses were right, s ...
Intermediate Logic
... will introduce you to the concepts, results, and methods of formal logic necessary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic (no ...
... will introduce you to the concepts, results, and methods of formal logic necessary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic (no ...
On the specification of sequent systems
... to specify and reason about a variety of proof systems. Since the encodings of such logical systems are natural and direct, the meta-theory of linear logic can be used to draw conclusions about the object-level proof systems. More specifically, in [MP02], the authors present a decision procedure for ...
... to specify and reason about a variety of proof systems. Since the encodings of such logical systems are natural and direct, the meta-theory of linear logic can be used to draw conclusions about the object-level proof systems. More specifically, in [MP02], the authors present a decision procedure for ...
slides
... added to ASP input languages but is not covered by the original semantics. Example: The expression #count{X:p(X)} = 0 intuitively says that the cardinality of the set of all X such that p(X) holds is 0. If there are infinitely many possible values for X the meaning of this expression cannot be repre ...
... added to ASP input languages but is not covered by the original semantics. Example: The expression #count{X:p(X)} = 0 intuitively says that the cardinality of the set of all X such that p(X) holds is 0. If there are infinitely many possible values for X the meaning of this expression cannot be repre ...
Kripke Semantics for Basic Sequent Systems
... {π0 }. In LK-legal frames, R consists of one relation Rπ0 which is the identity relation. π0 imposes a trivial condition, v(a, ψ) = v(b, ψ) whenever a = b. The basic rules of LK impose the usual truth-tables in each world, e.g. v(a, ψ ⊃ ϕ) = t iff either v(a, ψ) = f or v(a, ϕ) = t. Example 6. Assume ...
... {π0 }. In LK-legal frames, R consists of one relation Rπ0 which is the identity relation. π0 imposes a trivial condition, v(a, ψ) = v(b, ψ) whenever a = b. The basic rules of LK impose the usual truth-tables in each world, e.g. v(a, ψ ⊃ ϕ) = t iff either v(a, ψ) = f or v(a, ϕ) = t. Example 6. Assume ...
Complete Sequent Calculi for Induction and Infinite Descent
... • We work in first-order logic with inductive definitions. • We formulate and compare proof-theoretic foundations of ...
... • We work in first-order logic with inductive definitions. • We formulate and compare proof-theoretic foundations of ...
Group knowledge is not always distributed (neither is it always implicit)
... Theorem 3.3. Let X and Y range over hK1 ,K2 , . . . ,Km ,Gj. Then: £Xw ⇔ £Yw. Theorem 3.3 has, for both the reading as group knowledge as well as that of a receiving agent for G, some remarkable consequences. It implies that the knowledge in the group is nothing else than the knowledge of any partic ...
... Theorem 3.3. Let X and Y range over hK1 ,K2 , . . . ,Km ,Gj. Then: £Xw ⇔ £Yw. Theorem 3.3 has, for both the reading as group knowledge as well as that of a receiving agent for G, some remarkable consequences. It implies that the knowledge in the group is nothing else than the knowledge of any partic ...
Conjunctive normal form - Computer Science and Engineering
... functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables (Enderton, 2001). In particular, truth tables can be used to tell whether a propositional expre ...
... functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables (Enderton, 2001). In particular, truth tables can be used to tell whether a propositional expre ...
Definable maximal cofinitary groups.
... VERA FISCHER, SY DAVID FRIEDMAN, AND ASGER TÖRNQUIST Abstract. Using countable support iteration of S-proper posets, for some appropriate stationary set S, we obtain a generic extension of the constructible universe, in which b = c = ℵ2 and there is a maximal cofinitary group with a Π12 -definable ...
... VERA FISCHER, SY DAVID FRIEDMAN, AND ASGER TÖRNQUIST Abstract. Using countable support iteration of S-proper posets, for some appropriate stationary set S, we obtain a generic extension of the constructible universe, in which b = c = ℵ2 and there is a maximal cofinitary group with a Π12 -definable ...
Systems of modal logic - Department of Computing
... In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Whi ...
... In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Whi ...
Constructive Mathematics in Theory and Programming Practice
... There are model theories for this logic—Kripke models and Beth models. These models are often useful for showing that classical results, such as LPO, cannot be derived within Heyting arithmetic; see Dummett [1977] and Chapter 7 of Bridges and Richman [1987]. To carry out the development of mathemati ...
... There are model theories for this logic—Kripke models and Beth models. These models are often useful for showing that classical results, such as LPO, cannot be derived within Heyting arithmetic; see Dummett [1977] and Chapter 7 of Bridges and Richman [1987]. To carry out the development of mathemati ...
The logic and mathematics of occasion sentences
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
... ABSTRACT. The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of ...
duality of quantifiers ¬8xA(x) 9x¬A(x) ¬9xA(x) 8x¬A(x)
... some of each of them. Also there are some grains, and grains are plants. Every animal either likes to eat all plants or all animals much smaller than itself that like to eat some plants. Caterpillars and snails are much smaller than birds, which are much smaller than foxes, which in turn are much sm ...
... some of each of them. Also there are some grains, and grains are plants. Every animal either likes to eat all plants or all animals much smaller than itself that like to eat some plants. Caterpillars and snails are much smaller than birds, which are much smaller than foxes, which in turn are much sm ...
The Relative Efficiency of Propositional Proof
... We close this section by establishing some notation and terminology specific for propositional proof systems which will be used in the rest of this paper. The letter n will always stand for an adequate set of propositional connectives which are binary, unary, or nullary (have two, one, or zero argum ...
... We close this section by establishing some notation and terminology specific for propositional proof systems which will be used in the rest of this paper. The letter n will always stand for an adequate set of propositional connectives which are binary, unary, or nullary (have two, one, or zero argum ...
Refinement Modal Logic
... Such a relation always holds: a refinement of a given structure can always be seen as the model restriction of a bisimilar copy of the given structure. This work deals with the semantic operation of refinement, as in this example, in full generality, and also applied to the multi-agent case. Previo ...
... Such a relation always holds: a refinement of a given structure can always be seen as the model restriction of a bisimilar copy of the given structure. This work deals with the semantic operation of refinement, as in this example, in full generality, and also applied to the multi-agent case. Previo ...
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction
... search for proofs, and we were able to do so in a blind, fully automatic way. We were able to conduct an argument of the type: if this formula has a proof the only way to construct it is from such and such formulas by the means of one of the inference rules, and that formula can be found automatical ...
... search for proofs, and we were able to do so in a blind, fully automatic way. We were able to conduct an argument of the type: if this formula has a proof the only way to construct it is from such and such formulas by the means of one of the inference rules, and that formula can be found automatical ...
the partition property for certain extendible
... §1 introduces notation, defines the various combinatorial properties to be studied, and recalls some elementary facts. §2 contains a theorem which gives an alternate characterization of a combinatorial property used by Menas in his proofs of Theorems A and C. §3 depends on §2 and contains the statem ...
... §1 introduces notation, defines the various combinatorial properties to be studied, and recalls some elementary facts. §2 contains a theorem which gives an alternate characterization of a combinatorial property used by Menas in his proofs of Theorems A and C. §3 depends on §2 and contains the statem ...
Normal modal logics (Syntactic characterisations)
... In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Whi ...
... In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ simply when A ∈ Σ. Whi ...
On presenting monotonicity and on EA=>AE (pdf file)
... of [1], Dijkstra and Scholten discuss the monotonic properties of negation and implication. But they don’t state the general theorem (5) and they don’t give a convention for indicating its use. On page 93 of [1], a hint explicitly states the use of monotonicity of ∧ and ∃ in a weakening step, but on ...
... of [1], Dijkstra and Scholten discuss the monotonic properties of negation and implication. But they don’t state the general theorem (5) and they don’t give a convention for indicating its use. On page 93 of [1], a hint explicitly states the use of monotonicity of ∧ and ∃ in a weakening step, but on ...
Document
... 2. Mapping fJ: (TUF,V)n TUF,V assigned to every f F(n) with fJ(t1, ... , tn) : f(t1, ... , tn) A term interpretation I for F and consists of: 1. A term algebra for F 2. I TB,,F,V (set of atoms that are true; equivalently it can be seen as an assignment of a relation pI (TUF,V)n to every ...
... 2. Mapping fJ: (TUF,V)n TUF,V assigned to every f F(n) with fJ(t1, ... , tn) : f(t1, ... , tn) A term interpretation I for F and consists of: 1. A term algebra for F 2. I TB,,F,V (set of atoms that are true; equivalently it can be seen as an assignment of a relation pI (TUF,V)n to every ...
pdf
... correct writing, those sentences are written following all the standard conventions of the language in which they are written. Because of the precision of the thoughts that a proof must convey, it is especially important that the prose be clear and correct. In order to help the reader follow the arg ...
... correct writing, those sentences are written following all the standard conventions of the language in which they are written. Because of the precision of the thoughts that a proof must convey, it is especially important that the prose be clear and correct. In order to help the reader follow the arg ...