Please read Definitions 3.1, 3.2 and 3.3 and a) prove p (~ q p) is an...

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Please read Definitions 3.1, 3.2 and 3.3 and a) prove p→ (~ q→ p) is an axiom of our deductive system. b) prove {p} Hq V p. DEFINITION 3.1. The three axiom schema of Lp are: A1: (a (3→ a)) - -> -> A2: ((a (3)) ((a3) → (a →y))) A3: (((-3) (-a)) (((-3) → a) → 3)). Replacing a, 3, and y by particular formulas of Lp in any one of the schemas A1, A2, or A3 gives an axiom of Lp. -> -> -> → For example, (A₁ → (A₁ – A₁)) is an axiom, being an instance of axiom schema A1, but (Ag (Ao)) is not an axiom as it is not the instance of any of the schema. As had better be the case, every axiom is always true: DEFINITION 3.2 (Modus Ponens). Given the formulas and (→ ), one may infer. We will usually refer to Modus Ponens by its initials, MP. Like any rule of inference worth its salt, MP preserves truth. DEFINITION 3.3. Let Σ be a set of formulas. A deduction or proof from in Lp is a finite sequence 192...n of formulas such that for each kn, (1) k is an axiom, or (2) φκ Ε Σ, or (3) there are i, j <k such that pk follows from p; and p; by MP. A formula of Σ appearing in the deduction is called a premiss. Σ proves a formula a, written as Σa, if a is the last formula of a deduction from E. We'll usually write a fora, and take Σ A to mean that Σ Ε δ for every formula δεΔ. In order to make it easier to verify that an alleged deduction really is one, we will number the formulas in a deduction, write them out in order on separate lines, and give a justification for each formula. Like the additional connectives and conventions for dropping parentheses in Chapter 1, this is not officially a part of the definition of a deduction. Please read Definitions 3.1, 3.2 and 3.3 and a) prove p→ (~ q→ p) is an axiom of our deductive system. b) prove {p} Hq V p. DEFINITION 3.1. The three axiom schema of Lp are: A1: (a (3→ a)) - -> -> A2: ((a (3)) ((a3) → (a →y))) A3: (((-3) (-a)) (((-3) → a) → 3)). Replacing a, 3, and y by particular formulas of Lp in any one of the schemas A1, A2, or A3 gives an axiom of Lp. -> -> -> → For example, (A₁ → (A₁ – A₁)) is an axiom, being an instance of axiom schema A1, but (Ag (Ao)) is not an axiom as it is not the instance of any of the schema. As had better be the case, every axiom is always true: DEFINITION 3.2 (Modus Ponens). Given the formulas and (→ ), one may infer. We will usually refer to Modus Ponens by its initials, MP. Like any rule of inference worth its salt, MP preserves truth. DEFINITION 3.3. Let Σ be a set of formulas. A deduction or proof from in Lp is a finite sequence 192...n of formulas such that for each kn, (1) k is an axiom, or (2) φκ Ε Σ, or (3) there are i, j <k such that pk follows from p; and p; by MP. A formula of Σ appearing in the deduction is called a premiss. Σ proves a formula a, written as Σa, if a is the last formula of a deduction from E. We'll usually write a fora, and take Σ A to mean that Σ Ε δ for every formula δεΔ. In order to make it easier to verify that an alleged deduction really is one, we will number the formulas in a deduction, write them out in order on separate lines, and give a justification for each formula. Like the additional connectives and conventions for dropping parentheses in Chapter 1, this is not officially a part of the definition of a deduction.