Question

Phil-Formal Logic:

Translate the premises and conclusion into the symbols of propositional logic. Construct a truth table in which you analyze the argument for validity. You can construct a truth a table by inserting a table into a Microsoft Word document (from the INSERT option in Word, choose “table.” You will then have an opportunity to choose how many rows and columns you would like your table to be.) Is your argument valid or invalid? If valid, say why it is valid; identify the rows in the truth table that make the argument valid. If the argument is invalid, identify a counterexample; point to a row in your truth table that makes the argument invalid

2. Augustine achieves heaven if Augustine is virtuous. But Augustine is happy provided that he is not virtuous. Augustine does not achieve heaven only if he is not happy. Therefore, Augustine achieves heaven. (A: Augustine achieves heaven; V: Augustine is virtuous; H: Augustine is happy)

4. American foreign policy is bankrupt unless it is based on clear moral principles. American foreign policy is not based on clear moral principles just in case it is based primarily on the national interest. Unfortunately American foreign policy is based primarily on the national interest. Therefore, American foreign policy is bankrupt. (B: American foreign policy is bankrupt; M: American foreign policy is based on clear moral principles; N: American foreign policy is based primarily on national interest.)

5. On the condition that landmines are designed to inflict horrible suffering, they ought to be banned unless inflicting horrible suffering is sometimes justified. It is not true that inflicting horrible suffering is sometimes justified, but it is true that landmines are designed to inflict horrible suffering. Therefore, landmines ought to be banned. (L: Landmines are designed to inflict horrible suffering; B: Landmines ought to be banned; S: Inflicting horrible suffering is sometimes justified)

Then, construct abbreviated truth tables of your argument as described in section 7.4 of the textbook (rules for this are summarized on page 329).

Answer 1

Hi!

So, this is one of the basic things you will learn in philosophy: how to assess if whole arguments are valid or not by breaking it down in premises and evaluating them. What I am going to do is use your question #1 as an example and you can finish with question #2. If you have any further doubts message me and we can schedule a lesson!

Let's begin!

Argument #1 is: Humans evolved from lower life forms given that either human life evolved from inanimate matter apart from divine causes or God created human life via evolution. God created life via of evolution. It follows that humans evolved from lower life forms.

The assignment first asks you to translate the premisses into the symbols of first order predicate logic. Do you know those symbols? They are:
The quantifiers - ∀ (for all, for all that is given) and ∃ (there exists, there is at least one)
The logical connectives - ∧ for conjunction, ∨ for disjunction, → for implication, ↔ for biconditional, ¬ for negation.
Equality symbol - = (equals, is identical)

To translate the first argument, we need to break it in premises, which are the *statements that constitute the argument*. That is, in fact, given to you in:
H: Human life evolved from lower life forms;
M: Human life evolved from inanimate matter apart from divine causes;
G: God created human life via evolution
Thus we have premises H, M and G that form argument #1.

To translate it, we take a look what is said: Humans evolved from lower life forms "given that" human life evolved from causes M or G.
Translating it according to the symbols and the premises:
(M∨ G)→H
G → H

One thing we notice here is that H is our conclusion. Through M and G, we want to prove H.

We are done for the first part! Now we come to a truth table. We will analyze our premises according to their validity, and then come to terms with the argument. For this part, it is important to remember what are our premises and what is our conclusion. As we have seen in the first part, H is our conclusion. Therefore, it is dependent on G and M, and it is the *last row of our truth table*. This is always valid for conclusions.

We have G, M and, lastly, H in our rows. For our columns, we have true and false:

Truthiness Falsehood
G
M
H

Now lets assess the value of each of these premises, and see if we have a valid argument. To have a valid argument means to *get to a conclusion that implies no contradictions*.
We start with premise G (God created human life via evolution). Such assertion is stated for us. In here, it is not a matter of if we believe it or not, but it is presented as a truth in the logical argument. Therefore, let's mark it as a Truth (T).

Truthiness Falsehood
G T
M
H

We follow to premise M (Human life evolved from inanimate matter apart from divine causes). The argument does not state to us if this is true or false, right? However, there is a disjunction between G and M. One *or* the other. Therefore, it follows that if G, as we have seen in the argument, is true, M can only be false (F).

Truthiness Falsehood
G T
M F
H

Following, considering that as we have seen, H can only be true if G or M are true ((M∨ G)→H)), what is left is to evaluate H. As we concluded, G is true, and M, by disjunction, is false. Therefore, H can only be true, because G is true.

Truthiness Falsehood
G T (M∨ G)
M F (M∨ G)→H
H T

Here we finalize assessing our argument. H is true because G is true, and H depends on G or M to be true.

Concluding, we have a valid argument, which we proved trough a truth table and logic signs. Was this argument invalid, We would have found in our truth table a contradiction, that is *a premise would both be true and false*, which is logically impossible, and therefore invalidates the argument. Pay attention to those occasions when you are doing the second one!