CSI 2101 / Rules of Inference (§1.5)
... OK so what is a proof? Formal proof • sequence of statements, ending in conclusion • statements preceding the conclusion are called premises • each statement is either an axiom, or is derived from previous premises using a rule of inference Informal proof • formal proofs are too tedious to read • h ...
... OK so what is a proof? Formal proof • sequence of statements, ending in conclusion • statements preceding the conclusion are called premises • each statement is either an axiom, or is derived from previous premises using a rule of inference Informal proof • formal proofs are too tedious to read • h ...
PPT
... such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by listing all the hypotheses (marked as Hyp), then the sequence of propositional forms follo ...
... such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by listing all the hypotheses (marked as Hyp), then the sequence of propositional forms follo ...
Artificial Intelligence Applications by
... fact is not known precisely, e.g.,(make car ford .9) or make(car,ford,.9). In practice, we will often use a representation which is not strictly a triple. ...
... fact is not known precisely, e.g.,(make car ford .9) or make(car,ford,.9). In practice, we will often use a representation which is not strictly a triple. ...
slides
... A formula G is said to be a logical consequence of formulas F1 , F2 , . . . , Fn , notation F1 , . . . , Fn |= G , iff, for all interpretations I, if I |= F1 and . . . and I |= Fn then I |= G . Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first ...
... A formula G is said to be a logical consequence of formulas F1 , F2 , . . . , Fn , notation F1 , . . . , Fn |= G , iff, for all interpretations I, if I |= F1 and . . . and I |= Fn then I |= G . Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first ...
Chapter 4. Logical Notions This chapter introduces various logical
... It is sometimes difficult to say what the implicit premisses are in a particular case and hence to identify the argument under consideration. It might even be denied that there are any legitimate inductive arguments on the grounds that, once all of the implicit premisses are made explicit, such argu ...
... It is sometimes difficult to say what the implicit premisses are in a particular case and hence to identify the argument under consideration. It might even be denied that there are any legitimate inductive arguments on the grounds that, once all of the implicit premisses are made explicit, such argu ...
Logical nihilism - University of Notre Dame
... admissible rules is a deficiency of systems like IPC. The very term “structural incompleteness” suggests that something is missing from IPC because correct inferences about provability in this logic are not represented in IPC. Rybakov (1997), for example, suggests that “there is a sense in which a d ...
... admissible rules is a deficiency of systems like IPC. The very term “structural incompleteness” suggests that something is missing from IPC because correct inferences about provability in this logic are not represented in IPC. Rybakov (1997), for example, suggests that “there is a sense in which a d ...
On Probability of First Order Formulas in a Given Model
... Finally, we accept the agreement that, for simplicity, we shall speak only about probabilities of formulas instead of probabilities of satisfaction cylinders of formulas, and write only P(cp) instead of P[C(
... Finally, we accept the agreement that, for simplicity, we shall speak only about probabilities of formulas instead of probabilities of satisfaction cylinders of formulas, and write only P(cp) instead of P[C(
3.1.3 Subformulas
... speaking, a subformula of a propositional formula H is a string occurring in H which itself is a propositional formula. Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ...
... speaking, a subformula of a propositional formula H is a string occurring in H which itself is a propositional formula. Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ...
predicate
... • Let be a set of formulas (possibly infinite) and be a formula from predicate calculus • ⊨ holds iff for all models M and lookup tables l, whenever M ⊨l holds for all then M ⊨l holds as well • is satisfiable iff there is some model M and lookup table l such that M ⊨l holds • i ...
... • Let be a set of formulas (possibly infinite) and be a formula from predicate calculus • ⊨ holds iff for all models M and lookup tables l, whenever M ⊨l holds for all then M ⊨l holds as well • is satisfiable iff there is some model M and lookup table l such that M ⊨l holds • i ...
Identity in modal logic theorem proving
... construct proofs within one of these proof theories - - by which I mean both that the result generated would be recognized as a proof in [say] Whitehead Russell's axiom system and also that the "machine internal" strategies and methods are applications of what it is legal to do within the proof theo ...
... construct proofs within one of these proof theories - - by which I mean both that the result generated would be recognized as a proof in [say] Whitehead Russell's axiom system and also that the "machine internal" strategies and methods are applications of what it is legal to do within the proof theo ...
Subintuitionistic Logics with Kripke Semantics
... In 1981, A. Visser [7] had already introduced Basic logic (BPC), an extension of F with truth preservation, in the natural deduction form, and proved completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BP ...
... In 1981, A. Visser [7] had already introduced Basic logic (BPC), an extension of F with truth preservation, in the natural deduction form, and proved completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BP ...
On Sets of Premises - Matematički Institut SANU
... omit the outermost parentheses of formulae, as in pp(p ∧ q) ⊢ rr.) So Gentzen’s sequents may be conceived as expressions of the form Γ ⊢ ∆ where Γ and ∆ are finite, possibly empty, sequences of formulae. (Capital Greek letters as schemata in sequents originate from Gentzen; see op. cit.) To economize ...
... omit the outermost parentheses of formulae, as in pp(p ∧ q) ⊢ rr.) So Gentzen’s sequents may be conceived as expressions of the form Γ ⊢ ∆ where Γ and ∆ are finite, possibly empty, sequences of formulae. (Capital Greek letters as schemata in sequents originate from Gentzen; see op. cit.) To economize ...
Representing Synonymity in Causal Logic and in
... What are the counterparts of this formula in the language of logic programming under the answer set semantics (Gelfond and Lifschitz 1991) and in nonmonotonic causal logic (McCain and Turner 1997)? This question is closely related to current research on modular action description languages. In STRIP ...
... What are the counterparts of this formula in the language of logic programming under the answer set semantics (Gelfond and Lifschitz 1991) and in nonmonotonic causal logic (McCain and Turner 1997)? This question is closely related to current research on modular action description languages. In STRIP ...
Decomposing Specifications of Concurrent Systems
... Πa . In a more complicated example, Ea will be significantly simpler than Mbl , the full specification of Πlb . Verifying these hypotheses will therefore be easier than proving Mal ∧ Mbl ⇒ Ma ∧Mb directly, since this proof requires reasoning about the specification Mal ∧Mbl of the complete low-level pr ...
... Πa . In a more complicated example, Ea will be significantly simpler than Mbl , the full specification of Πlb . Verifying these hypotheses will therefore be easier than proving Mal ∧ Mbl ⇒ Ma ∧Mb directly, since this proof requires reasoning about the specification Mal ∧Mbl of the complete low-level pr ...
Deciding Intuitionistic Propositional Logic via Translation into
... path of the subformula Fp which wp is meant to be associated with. Thus, we will have to deal with the set W(F ) = {wp |Fp is special}, where we call a subformula Fp special , iff op(Fp ) ∈ {⇒, ¬} and pol(Fp ) = 0. In the above example we obtain W(F ) = {w, w111 , w121 }. The respective subformulas ...
... path of the subformula Fp which wp is meant to be associated with. Thus, we will have to deal with the set W(F ) = {wp |Fp is special}, where we call a subformula Fp special , iff op(Fp ) ∈ {⇒, ¬} and pol(Fp ) = 0. In the above example we obtain W(F ) = {w, w111 , w121 }. The respective subformulas ...
Propositional Calculus
... Sorting out equivalence and implication One should always be aware of the difference between equivalence and implication. In English, it is not always clear which connective is intended, as seen in the example below. Eating hamburgers at a fast-food bar is equivalent to aiding the destruction of th ...
... Sorting out equivalence and implication One should always be aware of the difference between equivalence and implication. In English, it is not always clear which connective is intended, as seen in the example below. Eating hamburgers at a fast-food bar is equivalent to aiding the destruction of th ...
Propositional Logic
... algorithm in which resolution is embedded. • Can be made understandable to non-technical users • Needs to be combined with a search algorithm. • Can be very appropriate for general problem solving (e.g., using PROLOG) CSE 415 -- (c) S. Tanimoto, 2008 Propositional Logic ...
... algorithm in which resolution is embedded. • Can be made understandable to non-technical users • Needs to be combined with a search algorithm. • Can be very appropriate for general problem solving (e.g., using PROLOG) CSE 415 -- (c) S. Tanimoto, 2008 Propositional Logic ...
ON PRESERVING 1. Introduction The
... transmuted into single formula truth. It may be more correct to say that, but it makes the whole paradigm somewhat less forceful or even less appealing. What we need in order to rescue the very idea of preservationism is to talk entirely about sets. So we shall have to replace the arbitrary conclusi ...
... transmuted into single formula truth. It may be more correct to say that, but it makes the whole paradigm somewhat less forceful or even less appealing. What we need in order to rescue the very idea of preservationism is to talk entirely about sets. So we shall have to replace the arbitrary conclusi ...
Logic as a Tool 3mm Chapter 2: Deductive Reasoning in
... Propositional resolution as a deductive system The underlying idea of Propositional Resolution: in order to prove the validity of a logical consequence A1 , . . . , An |= B, show that there is no truth assignment which falsifies it, i.e., show that the formulae A1 , . . . , An and ¬B cannot be sati ...
... Propositional resolution as a deductive system The underlying idea of Propositional Resolution: in order to prove the validity of a logical consequence A1 , . . . , An |= B, show that there is no truth assignment which falsifies it, i.e., show that the formulae A1 , . . . , An and ¬B cannot be sati ...
SECOND-ORDER LOGIC, OR - University of Chicago Math
... even fairly basic proofs fully would be unacceptably time-consuming—but generally one considers his proof valid if any competent person could easily translate it into a valid argument in some first-order language, had he only the time and motivation. (Of course, the level of dogmatism on this point ...
... even fairly basic proofs fully would be unacceptably time-consuming—but generally one considers his proof valid if any competent person could easily translate it into a valid argument in some first-order language, had he only the time and motivation. (Of course, the level of dogmatism on this point ...
A Logical Characterisation of Ordered Disjunction
... However, in the general case, we cannot remove the leftmost occurrence of a repeated formula. As a counterexample, the formulas a × b × a and b × a are ...
... However, in the general case, we cannot remove the leftmost occurrence of a repeated formula. As a counterexample, the formulas a × b × a and b × a are ...
The Taming of the (X)OR
... The suggested calculus in this paper can also be understood as an attempt to “lift” these properties to the case of a combined inclusive/exclusive-or logic. The underlying inference rules can roughly be divided in three classes: Resolution-type inferences to implement GE (however, one parent clause ...
... The suggested calculus in this paper can also be understood as an attempt to “lift” these properties to the case of a combined inclusive/exclusive-or logic. The underlying inference rules can roughly be divided in three classes: Resolution-type inferences to implement GE (however, one parent clause ...
presentation - Washington University in St. Louis
... M. Jamnik, M. Kerber, M. Pollet, C. Benzmüller. Automatic learning of proof methods in proof planning. L. J. of the IGPL, Vol. 11 No. 6, pages 647-673. 2003. E. Melis and J. Whittle. Analogy in inductive theorem proving. Journal of ...
... M. Jamnik, M. Kerber, M. Pollet, C. Benzmüller. Automatic learning of proof methods in proof planning. L. J. of the IGPL, Vol. 11 No. 6, pages 647-673. 2003. E. Melis and J. Whittle. Analogy in inductive theorem proving. Journal of ...
PPT
... This is going to be a contradiction proof. Assume for the sake of contradiction that is both consistent and complete. Suppose Q were provable. Then, P(G(Q)) would be provable, because a proof definitely exists. But Q is true iff G(Q) is not provable. This is a contradiction. Now suppose Q were not p ...
... This is going to be a contradiction proof. Assume for the sake of contradiction that is both consistent and complete. Suppose Q were provable. Then, P(G(Q)) would be provable, because a proof definitely exists. But Q is true iff G(Q) is not provable. This is a contradiction. Now suppose Q were not p ...