Philosophy of Math 012Translate the following arguments into symbolic form then use the eighteen rules of inference to derive the conclusion. Problem#1 If a superconducting particle collider is built, then the data yielded will benefit scientists of all nations and it deserves international funding. Either a superconducting particle collider will be built, or the ultimate nature of matter will remain hidden and the data yielded will benefit scientists of all nations. Therefore, the data yielded by a superconducting particle collider will benefit scientists of all nations. (S,D,I,U) S=a superconducting particle collider is built D=the data yielded will benefit scientists of all nations I=it deserves international funding U=the ultimate nature of matter will remain hidden This is how I've started the argument in symbolic form. Please check then use the 18 forms of inference to derive the conclusion. Examples of argument format are in screen shot next page. S ( D I ) Premise1 S v (UD) /D Premise 2/Conclusion Problem #2- If there is a direct correlation between what a nations spends on healthcare and the health of its citizens then America has the lowest incidence of disease and the lowest mortality rates of any nation on earth. But America does not have the lowest mortality rates of any nation on earth. Therefore, there is not a direct correlation between what a nation spends on healthcare and the health of its citizens. (C,D,M) C= there is a direct correlation between what a nations spends on healthcare and health of its citizens D= America has the lowest incidence of disease M= America has the lowest mortality rates of any nation on earth. Here's how I've converted the argument to symbolic form. Please check then use the 18 forms of inference to derive the conclusion. C D M Premise 1 ~M / ~C Premise 2/Conclusion DeMorgan's Rule DM ~(p v q) :: (~p . ~q) ~( p . q) :: (~p v ~q) 1. ~M premise 2. ~M v ~G___ 1, ad 3. ~(M . G)___ 2, dm \"neither...nor...\" is the same as \"not the one and not the other\" \"not both...\" is the same as \"either not this one or not that one.\" 1. ~(H v K) premise 2. ~ H . ~ K 1. dm 3. ~K ___ 2 cm, sm Transposition TR (p > q) :: (~q > ~p) Contraposition, but in the context of propositional logic All Popes are Catholics so All non-Catholics are non-Popes. If he's the Pope, he's Catholic, so if he's not Catholic, he's not the Pope. Material Implication IMP (p v q) :: (~p > q) \"or\" means the same thing as \"if not\" 1. ~A premise 2. (M > L) v A premise 3. ~(M > L) > A_2_IMP_ 4. ~A > (M > L) 5. M > L A>B ~A v B IMP _3_TR_ _1,4_mp_ When you change a \"v\" to a \">\" or vice versa, add a tilde to the expression on the left ~A v B ~~A > B IMP A > B DN Distribution DIST [ p v (q . r)] : : [(p v q) . (p v r)] [p . (q v r)] : : [(p . q) v (p . r)] \"p\" is being distributed through a disjunction or a conjunction Material Equivalence EQ (p q) : : [(p > q) . (q > p)] Biconditional: p and q are necessary and sufficient conditions for each other: p implies q and q implies p. (p q) : : [(p . q) v (~p . ~q) They have the same truth values: either both are true or both are false. Either both or neither. Exportation EXP [p > (q > r)] : : [(p . q) > r] If p is true, then if q is, so is r if p and q are both true, then so is r Tautology (p v p) : : p (p . p) : : p TAUT Eliminates redundancy Rules of inference (8) MP p > q / p // q MT p > q / ~q // ~p HS p > q / q > r // p > r DS p v q / ~p // q SM p . q // p CN p / q // p. q AD p // p v q CD (p > q) . (r > s) / p v r // q v s Rules of Equivalence/ Replacement (10) DN p :: ~~p CM (p . q) :: (q . p) AS ((p . q) . r) :: (p . ( q . r)) DM ~(p v q) :: (~p . ~q) (p v q) :: (q v p) ((p v q) v r) :: (p v (q v r)) ~(p .q) :: (~p v ~q) DIST (p v (q . r)) :: ((p v q) . (p v r)) (p . (q v r)) :: ((p . q) v (p . r)) TRAN (p > q) :: (~q > ~p) IMP (p v q) :: (~p > q) EQ (p q) :: ((p > q) . (q > p)) (p q) :: ((p . q) v (~p . ~q)) EXP (p > (q > r)) :: ((p . q) > r) TAUT (p v p) :: p (p . p) :: p Philosophy of Math 012Translate the following arguments into symbolic form and the use the nineteen rules of inference to derive the conclusion of each. Use the letter abbreviations in the order in which they are listed. [50 points] I've started the 2 problems below and defined the variables. Please check them before you start to derive the conclusion. Please verify that I have translated to symbolic form correctly before you work on deriving (proving )the conclusion using the 18 rules of inference 1. If birthcontrol devices are made available in high school clinics, then the incidence of teenage pregnancy will decrease. Therefore, if both birthcontrol information and birthcontrol devices are made available in high school clinics, then the incidence of teenage pregnancy will decrease. (D, P, I) D= birthcontrol devices are made available in high school clinics P= the incidence of teenage pregnancy will decrease I= birthcontrol information DP (I D) P PR PR/C 2. Vitamin E is an antioxidant and a useless food supplement if and only if it does not reduce heart disease. It is not the case either that vitamin E does not reduce heart disease or is not an antioxidant. Therefore, vitamin E is not a useless food supplement. (A, U, R) A= Vitamin E is an antioxidant U=Vitamin E is useless food supplement R=Vitamin E does not reduce heart disease 1.A U R 2. ~(Rv~A)/ ~U PR PR/C 3. Either human choices are determined by antecedent factors or they are free. If human choices are determined by antecedent factors, then we cannot act in any way different from the way we do act. If we cannot act in any way different from the way we do act, then we are not morally responsible for our actions. But we are morally responsible for our actions, so our choices must be free. A = Human choices are determined by antecedent factors; F = Human choices are free; D = We can act different from the way we do; M = We are morally responsible for our actions. Examples below of deriving conclusion using the 18 Inference Rules of Equivilence Rules - See list of 18 Rules and definitions uploaded as separate document. Philosophy of Math 012Translate the following arguments into symbolic form and the use the nineteen rules of inference to derive the conclusion of each. Use the letter abbreviations in the order in which they are listed. [50 points] I've started the 2 problems below and defined the variables. Please check them before you start to derive the conclusion. Please verify that I have translated to symbolic form correctly before you work on deriving (proving )the conclusion using the 18 rules of inference 1. If birthcontrol devices are made available in high school clinics, then the incidence of teenage pregnancy will decrease. Therefore, if both birthcontrol information and birthcontrol devices are made available in high school clinics, then the incidence of teenage pregnancy will decrease. (D, P, I) D= birthcontrol devices are made available in high school clinics P= the incidence of teenage pregnancy will decrease I= birthcontrol information DP (I D) P PR PR/C 2. Vitamin E is an antioxidant and a useless food supplement if and only if it does not reduce heart disease. It is not the case either that vitamin E does not reduce heart disease or is not an antioxidant. Therefore, vitamin E is not a useless food supplement. (A, U, R) A= Vitamin E is an antioxidant U=Vitamin E is useless food supplement R=Vitamin E does not reduce heart disease 1.A U R 2. ~(Rv~A)/ ~U PR PR/C 3. Either human choices are determined by antecedent factors or they are free. If human choices are determined by antecedent factors, then we cannot act in any way different from the way we do act. If we cannot act in any way different from the way we do act, then we are not morally responsible for our actions. But we are morally responsible for our actions, so our choices must be free. A = Human choices are determined by antecedent factors; F = Human choices are free; D = We can act different from the way we do; M = We are morally responsible for our actions. Examples below of deriving conclusion using the 18 Inference Rules of Equivilence Rules - See list of 18 Rules and definitions uploaded as separate document