Al-Kindi

. Expand any quantifiers and calculate the truth values of the following argument in the possible world provided. Is the given world a counter-example to the argument? (In other words, are the premises true and the conclusion false in the given world?) (Show your working. 3 points each). (a) Argument: (∀x)(~Fx  ~Gx), (∀x)(Fx  Hx), (x)Gx ∴ (x)Hx Possible world: --------------------------- | | | F | G | H | | a | | 1 | 0 | 1 | ---------------------------- Domain = {a}

In contemplating the argument put forth, we must first analyze the quantifiers involved and their implications in the given possible world. Let us begin by examining the first premise, which states that for all x, if x is not F, then x is not G. In our world, we have a single individual a, who is indeed F (F(a) = 1), but not G (G(a) = 0). This means that the premise holds true for this individual. Moving on to the second premise, it asserts that for all x, if x is F, then x is H. In our world, individual a is indeed F and H (H(a) = 1). Therefore, this premise is also satisfied. The third premise states that there exists an x such that x is G. In this world, we have individual a, who is not G. Consequently, the conclusion that there exists an x such that x is H cannot be inferred from the premises in this given world. It is evident that the premises are true in the possible world provided, but the conclusion is false, making this world a counter-example to the argument. Through meticulous analysis and logical reasoning, we have uncovered the truth values and ultimately determined the validity of the argument in the context of this specific world.