Using the inference rules and proof sequence for predicates, prove the following argument: Ken is a member of the Titans. Ken can hit the ball a long way. Everyone who can hit the ball a long way can make a lot of money. Conclusion: Some member of the Titans can make a lot of money.
Introduction:
Inference rules are logical rules that can be used to derive new logical statements from existing ones. In predicate logic, also known as first-order logic, there are several inference rules that can be used to manipulate statements involving predicates and quantifiers. Some common inference rules and proof sequences for predicates are:
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Universal instantiation: If ∀x P(x) is true, then P(c) is true for any constant c. For example, if ∀x (x > 0), then 1 > 0.
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Existential instantiation: If ∃x P(x) is true, then P(c) is true for some specific constant c. For example, if ∃x (x > 0), then we can choose c to be any positive number.
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Universal generalisation: If P(c) is true for any constant c, then ∀x P(x) is true. For example, if P(c) is the statement "c is a prime number," and c can be any positive integer, then we can conclude that ∀x (x is a prime number).
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Existential generalisation: If P(c) is true for some specific constant c, then ∃x P(x) is true. For example, if P(c) is the statement "c is divisible by 5," and c = 15, then we can conclude that ∃x (x is divisible by 5).
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Modus ponens: If P → Q and P are both true, then Q is true. For example, if P is the statement "x is even," and Q is the statement "x is divisible by 2," then if we know that P is true, and we also know that P → Q is true (which it is), then we can conclude that Q is true.
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Modus tollens: If P → Q and ¬Q are both true, then ¬P is true. For example, if P is the statement "x is even," and Q is the statement "x is divisible by 2," then if we know that ¬Q is true (i.e., x is not divisible by 2), and we also know that P → Q is true (which it is), then we can conclude that ¬P is true (i.e., x is not even).
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Universal modus ponens: If ∀x P(x) → Q is true, then Q is true. For example, if ∀x (x > 0) → (2x > 0), then we can conclude that 2x > 0 for any value of x.
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Existential modus ponens: If ∃x P(x) and ∀x (P(x) → Q) are both true, then Q is true. For example, if ∃x (x > 0) and ∀x (x > 0 → 2x > 0) are both true, then we can conclude that 2x > 0 for some value of x.
These are just a few of the many inference rules and proof sequences that can be used in predicate logic. Each rule follows logically from the axioms and definitions of predicate logic and can be used to prove or disprove statements involving predicates and quantifiers.
Assumptions:
Let T(x) be the predicate "x is a member of the Titans,"
H(x) be the predicate "x can hit the ball a long way," and
M(x) be the predicate "x can earn a lot of money."
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