HU Logic

You are now in a position to understand a more formal presentation of the argument from reason. This version will also be presented a bit too quickly, but you should be able to grasp its point. Consider the following argument: All humans are mortal Socrates is a human Hence, Socrates is mortal. The argument is valid: the truth of the premises guarantees the truth of the conclusion. If the premises are the true the conclusion must be true. It is not possible for the premises to be true and the conclusion false. The point with all of these different ways of saying the same thing is that there is a logical connection between the premises of the argument and its conclusion. We can grasp that connection via reason. Now suppose that materialism is true. If materialism is true, then humans are wholly material beings--there are no non-material features of humans (no immaterial minds, souls, spirits, etc). Materialism is the dominant view of most scientists, especially biologists, cog...

1.      P & ~P 2.      P          1, simp 3.      P v Q    2, add 4.      ~P         1, simp 5.      Q          3, 4, DS The addition rule is useful for a number of reasons. First, it simply follows from the truth-table for ‘or’ and the rules of validity. But as you can see here it is really helpful in showing why contradictions cannot be tolerated. If we allow contradictions into our beliefs or premises, then we can prove literally any proposition. This is what logicians call explosion. From one single contradiction, we can show that every statement, no matter what it is, is true.   Of course, it is absurd that every statement is true. So contradictions cannot be tolerated, and the addition rule nicely shows us why. Applications : one of t...

Here is another extra credit opportunity. Attempt to re-present the following argument using the tools of logic you have learned thus far. It is one of the most famous arguments in the history of the world.  Please email your attempts to me. Do not post them on the blog. From Anselm's Proslogion Ch 3 God cannot be conceived not to exist. --God is that, than which nothing greater can be conceived. --That which can be conceived not to exist is not God. AND it assuredly exists so truly, that it cannot be conceived not to exist. For, it is possible to conceive of a being which cannot be conceived not to exist; and this is greater than one which can be conceived not to exist. Hence, if that, than which nothing greater can be conceived, can be conceived not to exist, it is not that, than which nothing greater can be conceived. But this is an irreconcilable contradiction. There is, then, so truly a being than which nothing greater can be conceived to exist, that it cannot even be c...

Here are two very famous arguments that are easily put in the form of a constructive dilemma. I will give extra credit to those who write up substantive comments on either or both of these. Pascal's Wager 1. If God exists, I have everything to gain by believing in Him and if God does not exist, I have nothing to lose by believing in Him.  2. Either God exists or God does not exist.  3. Therefore, I have everything to gain or nothing to lose by believing in Him.  The Euthyphro Dilemma 1. If God willed the moral law arbitrarily, then he is not essentially good and if God willed the moral law according to an ultimate standard beyond himself, then He is not God (because there is something beyond Him). 2. But God willed the moral law either arbitrarily or according to a standard beyond him. 3. Therefore, either God is not good, or He is not God.

On the first exam, the portion that seemed to give a lot of you the most difficulty was the counter-example section. That is not very surprising. To construct a good counter-example can be quite difficult, and often takes quite a bit of time and ingenuity.  The truth table method of determining validity and invalidity eliminates the need for ingenuity to determine validity or invalidity. The process is algorithmic. All you have to do is plug in the truth-values consistently and exhaustively, determine the main connectives, and check the rows accurately. If there is a row where each premise is true and the conclusion is false, then the argument is invalid; otherwise, it is valid.  The indirect truth-table method does require a bit more ingenuity, but not much. All you really have to do is get very familiar with the nature of the logical connectives. Once you have mastered them, the indirect method is usually fairly simple. In fact, the indirect method will require at m...

Truth Functional Logic and the Weird Conditionals (I will use an arrow instead of a horseshoe) The logic we are studying in our course is truth-functional.    This is a very useful thing and it applies to a huge number of actual arguments and situations we encounter every day. Of course, there are some limitations but those need not disturb us because the types of statements that are not truth-functional can still be analyzed using the tools you will acquire plus some other ones.  As I said in class, the most controversial truth-functional analysis of a logical connective is the analysis of conditionals. Conditionals are if…, then… statements. In ordinary language, we assume that when someone states a conditional he intends the antecedent to be related to the consequent in some way. However, our analysis of conditionals allows some of them to be true even when there is no relationship between the antecedent and the consequent. Here’s the truth-functional a...

According to the grammar of ‘only if’ ‘p only if q’ is equivalent to ‘if p then q’.  I think we can see this from a syntactical point of view and from a semantical point of view.  I’ll start with the semantic claim first. Semantic The following seem to be semantically equivalent: a. p only if q b. p occurs only if q occurs c. p is true only if q is true d. if q fails to happen, then p fails to happen. What seems to be the common element in b-d is q appears to function as a necessary condition for the occurrence of p.  Perhaps d. captures the necessary condition idea the clearest, but it seems to be there in b. and c. as well.  If that is right, then in a. q is a necessary condition for p.  But ‘if p, then q’ states that q is a necessary condition for p. Furthermore, ‘if p, then q’ is logically equivalent to ‘if ~q, then ~p.’  But ‘if ~q, then ~p’ is d. above.  Hence, ‘p only if q’ is equivale...