Exam 4: Monadic Predicate Logic

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof. -1. (∃x)[Hx • (Ix $\lor$ Jx)] 2. (∀x)(Kx ⊃ ∼Ix) 3. (∀x)(Hx ⊃ Kx) / (∃x)Jx

1. (∃x)∼Ix ⊃ (∀x)(Jx $\lor$ Kx) 2. ∼(∀x)Ix • ∼Jb -Which of the following propositions is derivable from the given premises in M?

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain. -[(∀x)(Fx ⊃ Gx) • (∀x)(Gx ⊃ Hx)] ⊃ (∃x)(Fx • Hx)

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $\lor$ Dx)] -Are there any free variables? If so, which are they?

refer to the following formula: (∀x)[(Ex $\lor$ Fx) ⊃ (Gx • Hd)] -Which of the following variables in the formula are free?

refer to the following formula: (∃x)[Mx • (∼Nc $\lor$ ∼Ox)] ≡ (Py • Pb) -Is the given formula open or closed?

derive the conclusions of each of the following arguments using the rules of inference for M, including conditional or indirect proof. -1. (∀x)[Ax ⊃ (Bx ⊃ Cx)] 2. ∼(∀x)(Bx ⊃ Dx) / (∀x)Ax ⊃ (∃x)(Cx • ∼Dx)

1. (∀x)(Px ⊃ Qx) 2. ∼(∃x)[(Px • Rx) • Qx] 3. (∃x)Rx / ∼(∀x)Px -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

derive the conclusions of each of the following arguments using the rules of inference for M, including conditional or indirect proof. -1. (∀x)[(Px $\lor$ Qx) ⊃ (Rx • ∼Sx)] 2. (∀x)[Rx ⊃ (Tx ⊃ ∼Sx)] / (∀x)[Px ⊃ (Sx ⊃ ∼Tx)]

If some journalists are not respectable, then all journalists on the web have work to do. Some journalists on the web lack assignments. It is not the case that something without an assignment has work to do. So all journalists are respectable. -Which of the following claims can also be derived from the premises of this argument?

Everything is material, or ideal or transcendental. All atoms are not ideal and not transcendental. And nothing is material. So there are no atoms. -Which of the following is the best translation into M of this argument?

If anything is stuck in traffic, Bruno will be mad if it is late. Something is neither annoyed nor not late. If Bruno is mad, then something is annoyed. So something is not stuck in traffic. -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

1. (∀x)(Ax ⊃ Bx) 2. (∃x)Ax 3. (∃x)Bx ⊃ (∃x)Dx -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

provide a conterexample in a finite domain to each given invalid argument. -1. (∀x)[Tx ⊃ (∼Wx $\lor$ Fx)] 2. (∀x)[(∼Wx • Tx) ⊃ ∼Fx] 3. (∃x)(Tx • Fx) / (∃x)(Tx • ∼Wx)

Translate each sentence into predicate logic, using the given translation keys. Ax: x is an athlete Dx: x has determination Px: x plays professional sports Sx: x receives a scholarship Tx: x is tall Wx: x works hard -Some tall athletes don't receive scholarships.

1. (∃x)(Hx • ∼Ix) 2. (∀x)(Hx ⊃ Jx) 3. ∼(∃x)(Jx • Kx) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

use: t: Tortuga Bx: x creates bricks Cx: x is a city Nx: x is nicely placed Px: x is productive Sx: x is a settlement Tx: x has a trading port Wx: x is on the water -There are no settlements on the water with trading ports.

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (?x)(Gx ? Hx) 2) (?x)(Gx • Ix) 3) (?x)(Gx • Jx) 4) (?x)(Jx ? Hx) / (?x)(Ix • Jx)

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $\lor$ Dx)] -Which wffs below are not in the scope of '(∃x)'?

refer to the following formula: (∀x)[(Ex $\lor$ Fx) ⊃ (Gx • Hd)] -Is the given formula open or closed?