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What is a theorem? (50) A theorem is the conclusion of a provable sequent in which the number of assumptions is zero. How is '|-' read in sequents provable with no assumptions? (50) It is read as 'it is a theorem that...' Read: ' |- -(P & -P) (50) It is a theorem that it is not the case that P and not P. The importance of theorems is? (52) The importance of theorems is, since they are provable as conclusions from no assumptions, they are propositions which are true simply on logical grounds. What are two other names for or ways to refer to theorems? (52) Logical truths or logical laws The Law of Non-Contradiction (52) It is a theorem that it is not the case that P and not P. The Law of Identity (52) It is a theorem that if P then P. The Law of Double Negation It is a theorem that if P then not not P. Why or how is 'P -> P' different from 'P |- P'? 'P -> P' is a (logically) true proposition 'P |- P is an argument frames/patterns of valid argument. To confuse arguments with propositions is analogous to confusing ______ with _______. (52) validity, truth The Law of the Excluded Middle (53) It is a theorem that, for any proposition, either it or its negation is true. In proving a theorem, we are implicitly proving a wide variety of other theorems closely related to the proved theorem by substitutions. (T/F) True Define a substitution-instance of a given wff? (53) A substitution-instance of a given wff is a wff which results from the the given wff by replacing one or more of the variables occurring in the wff throughout by some other wffs, it being understood that each variable so replaced is replaced by the same wff. (Q -> R) & -(Q -> R) is a substitution-instance of 'P v -P'. (T/F) False What are the two features of substitution which are easily forgotten? (54) First, a substitution must be made uniformly - i.e. throughout - for each substituted variable: the same wff must be substituted for every occurrence of a given variable. Second, it is only on propositional variables that this substitution can be performed, and not, for example, on negated variables. A substitution instance of a wff will, at times, be shorter than the given wff. (T/F) (54) False Connectives can disappear in an instance of substitution. (T/F) (54) False What does a proof of a theorem constitute? (54) It constitutes implicit proof of all the (indefinitely many) possible substitution-instances of that theorem. (S1) A proof can be found for any substitution-instance of a proved theorem. (S2) A proof can be found for any substitution-instance of a proved sequent. The Rule of Theorem Introduction (TI) (56) This rule permits us to introduce, at any stage of a proof, a theorem already proved or a substitution-instance of such a theorem. Explain the uses of TI or TI(S) respectively. (56) TI is the citation used on the right of a line of proof for theorem introduction. TI(S) is the citation used on the right of a line of proof for a substitution-instance of a theorem. Numbers appear on the left of a line of proof that uses Theorem Introduction or a substitution-instance of Theorem Introduction. (T/F) (56) False. Since theorems depend on no assumptions, no numbers appears. TI(S) 44 (56)(52) The Law of the Excluded Middle - (Q v -Q) TI 41 (51) (P & Q -> P) The Rule of Sequent Introduction (SI) (58) ... The Rule of Sequent Introduction is a primitive rule. (T/F) (58) False. It is a derived rule. Derived rules increase our derivational power. (T/F) False. They expedite our proof-techniques.