Wednesday, May 27, 2009

Logic Primer 3: Thinking Logically

Logical thinking is a process. As long as the rules are not broken, the thought process will bring good conclusions. Now we will look at logical syllogisms.

The most basic logical structure is the syllogism. The syllogism is a deductive argument consisting of premises and a conclusion.1

It should be noted from the outset that for each of the following syllogisms presented, pages and pages could be (and have been) written with much more detail, explanation, and exceptions. What has been presented here is only a cursory glance at each one and should be treated as such. The reader is encouraged to delve into a more systematic textbook to explore fully.

A categorical syllogism is composed of two unconditional statements that lead deductively to an unconditional conclusion. An example of a categorical syllogism is as follows:
1. All cats are mammals.
2. Fuzzy is a cat.
3. Therefore, Fuzzy is a mammal.
The categorical syllogism has various forms and moods, which will not be detailed here, but the basic form simply entails two statements leading to a conclusion.

Hypothetical syllogisms take the form of a hypothetical statement. This syllogism has the word “IF” at its core. The hypothetical proposition uses the word if to make a conditional statement: if one state of affairs is true, then another state of affairs will follow. The first hypothetical syllogism is the Modus Ponens, structured like this:
If P, then Q.
P.
Therefore, Q.
Modus ponens means “way of affirmation” in Latin because it affirms the antecedent of the first proposition. One form of the cosmological argument takes the form of modus ponens:
If a contingent being exists, then a necessary being must exist as its cause.
A contingent being exists.
Therefore, a necessary being must exist as its cause.2
The other hypothetical syllogism is called Modus Tollens, which means “the way of denial.” This form of syllogism denies the consequent (the “then Q” part of the first statement). It is structured like this:
If P, then Q.
Not Q.
Therefore, Not P.
Disjunctive Syllogisms are either/or sentences. One statement is made with two alternatives, of which only one can be true.3 The disjunctive syllogism looks like this:
Either P or Q.
Not Q.
Therefore, P.
The way the disjunctive syllogism works requires for one alternate to be denied for the other one to be true. It is a fallacy to affirm one alternate to eliminate the other, because it is possible for them both to be true. Geisler and Brooks offer an excellent example of this fallacy found in Bertrand Russell’s book Why I am not a Christian:
Life was caused either by evolution or by design.
Life was caused by evolution.
Therefore, it was not caused by design (so there is no reason to posit God).
Geisler and Brooks explain: “This approach commits the formal fallacy of affirming one alternate. Even if the minor premise were true, the conclusion would not follow. For it is possible that both are true; that is, that evolution is designed.”4

The conjunctive syllogisms take the form of “both…and” statements. Here is the form:
Both P and Q are true.
Therefore, P.
Therefore, Q.
The conjunctive syllogism is fairly straightforward. Both terms in the first statement are separated and can be affirmed individually.

The Dilemma form of syllogism takes two hypothetical syllogisms and weds them with a disjunction. Here is what the dilemma looks like:
(If P, then Q) and (If R, then S).
P or R.
Therefore, Q or S.
The mathematician Pascal presented a dilemma with this syllogism:
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.
Either God does exist or he does not exist.
Therefore, I have everything to gain or nothing to lose by believing in God.5
The final syllogism presented here is the Sorites. This comes from a Greek word meaning “heap.” The premises are stacked together in a heap to come to a final conclusion. An example:
All A are B...............or...............If A then B
All B are C...............or...............If B then C
All C are D...............or...............If C then D
Therefore, all A are D......or.....Therefore, if A then D.
That is a basic look at basic logical syllogisms.

Here are some resources that will get you started:

Audio resource:
- Critical Thinking Audio Course

Helpful book:
- Introduction to Logic by Harry Gensler

Some web Sites on logic:
- Philosophy Pages logic index
- Atheism Analyzed looks at atheism from a logical perspective.

Tomorrow we will take a look at language.

1 Geisler & Brooks, p. 194.
2 Ibid., p. 61.
3 In a weak disjunction both may be true.
4 Ibid., p. 66.
5 Ibid., p. 69.

11 comments :

Aaron said...

1. Either Apologetics 315 is the coolest blog site around or the moon is made of green cheese.

2. The moon is not made of green cheese.

3. Therefore, Apologetics 315 is the coolest blog site around.

What do you think? I know it's valid...but is it sound??

Martin Cothran said...

A slight quibble. You said:

The Dilemma form of syllogism takes two hypothetical syllogisms and weds them with a disjunction. Here is what the dilemma looks like:

(If P, then Q) or (If R, then S).
P or R.
Therefore, Q or S.

That 'or' in the major premise needs to be an 'and'. But it is correct in your example.

In my opinion, the dilemma is the pinnacle of traditional logic, partly because it incorporates so many other logical forms. If you have mastered the dilemma, then you have arrived as a logician. It is also a highly effective weapon in argumentation if you know how to use it.

Brian said...

Martin,

Thanks for catching that error and correcting it... I have now made the appropriate edit to the content.

I appreciate your input as a logician.

paulcorrado said...

Brian, thanks so much for this post on logic. I am learning a lot from it! You are doing a great job of spelling out some important ideas.

Two question:

Question 1: from your quote "The way the disjunctive syllogism works requires for one alternate to be denied for the other one to be true. It is a fallacy to affirm one alternate to eliminate the other, because it is possible for them both to be true"

in this are you referring to the fallacy Bertrand Russell has in his book or are you referring to disjunctive syllogism's. I am guessing you are referring to Bertrand Russell quote but it is not clear (to me) in the sentence. (and i agree that his quote is a fallacy)

Question two:
Pascal wager in the Dilemma syllogism is also a fallacy, correct? the second proposition is not correct (we have lots to lose if we believe something that is not true) so this would be a fallacy. Only reason i comment is because Bertrand Russell comment is noted as a fallacy and Pascal was not.

thanks

paulcorrado said...

Brian, thanks so much for this post on logic. I am learning a lot from it! You are doing a great job of spelling out some important ideas.

Two question:

Question 1: from your quote "The way the disjunctive syllogism works requires for one alternate to be denied for the other one to be true. It is a fallacy to affirm one alternate to eliminate the other, because it is possible for them both to be true"

in this are you referring to the fallacy Bertrand Russell has in his book or are you referring to disjunctive syllogism's. I am guessing you are referring to Bertrand Russell quote but it is not clear (to me) in the sentence. (and i agree that his quote is a fallacy)

Question two:
Pascal wager in the Dilemma syllogism is also a fallacy, correct? the second proposition is not correct (we have lots to lose if we believe something that is not true) so this would be a fallacy. Only reason i comment is because Bertrand Russell comment is noted as a fallacy and Pascal was not.

thanks

Brian Auten said...

Thanks, Paul.
As for question 1: in a disjunctive syllogism, you take the form:
Either P or Q
Not Q.
Therefore P.

You eliminate one, and then the other must be true. The problem comes when you try to affirm one in order to eliminate the other.

However, the fallacy takes this form:
Either P or Q
P.
Therefore not Q.

As for question 2: The FORM of the wager as presented here is not a fallacy.
So it is formally valid. However, the truth of the premise may be in question -- and that would be the point that Pascal would need to argue. (He might point out, for instance, that you don't ultimately have anything to lose by believing in God if he does not exist, for example).

I hope that helps clarify.

paulcorrado said...

Brian, thanks, i understand the Pascal wager question i had. i misused the word fallacy because it refers to the form of the argument. I should have figured that out.

still not clear on the first question i had but i think i was using the wrong terminology.

It appears to me that:

Either P or Q
P.
Therefore not Q.

is a formally valid argument. it appears to me that the truth of the premise may be in question (Life was caused either by evolution or by design) but the argument is valid, just not sound.

the statement:
penny is heads or tails
this penny is tails
the penny is not heads

this would be a valid and sound argument

this penny is heads or made in 2010
this penny is made in 2010
this penny is not heads

this would be a valid but not sound argument.

so using this Bertrand Russel argument is valid but not sound.

I may be missing something and am having a really hard time using the terminology correctly and i understand that terminology is really important. Do not feel you have to respond if doing so takes up to much time as you did not sign up to be my logic tutor and you already have done a lot to educate me on the topic. thanks for that!! Paul

Brian Auten said...

No worries, Paul.

I would recommend checking out a couple of wiki articles on this:

http://en.wikipedia.org/wiki/Disjunctive_syllogism

In particular, it should be noted that disjunctive syllogisms could be inclusive or exclusive. That is the difference between something being "and/or" (inclusive) or "only one must be true, but not both" (exclusive).

This link makes it a bit clearer, I think.

paulcorrado said...

Brian, great, when writing the last post i was on that link, i love Wikipedia, and use it daily. (the way your write is much more clear to me than Wikipedia often is but it is still such a great tool)

the inclusive or exclusive answers my question on this.

Wouldn't that would make the Bertrand Russel argument valid but not sound? he is using an exclusive form of logical disjunction and that is ok to do.

am i correct on this?

using the terminology correctly is really hard for me but i am trying.

Brian Auten said...

Paul,
I think that is still fallacious because those are not necessarily exclusive options. It seems to me that as long as both could be true (to some extend) then it is not an exclusive disjunction. So if you affirm one to negate the other it is a formal fallacy.

paulcorrado said...

Brian, ok, i am getting to understand it a bit more. a fallacy can come into play in both the argument and the proposition. on first glance and without doing the research necessary to fully understand fallacious arguments i would think the Bertrand Russel argument was an informal fallacies because it has to do with the thinking that happen apart from the structure of an argument. I guess if you changed the structure of the argument it could be correct or if you changed the proposition in the argument it could be correct. that is where i am caught up. Intuitively i would think because the argument is sound but not valid it would be an informal fallacy but i am getting confused because we have two variables that could change to make it a sound argument. The inclusive/exclusive aspect of the argument or the proposition and this would change how we determine if it is a formal or informal fallacy. (in my last post i was only thinking of Formal fallacies)

I am getting confused because a formal fallacy is is referring to the structure of the argument and that can be valid or invalid. two people with different opinions that both think logically will still agree weather or not an argument is valid or invalid, but may not agree that it apples or is the correct argument to use with a group of proposition.

I would say that pascals wager is a fallacy the way it is stated but could be fixed if changed the words "nothing to lose" to "almost nothing to loose in the big picture of all eternity" or if he did not use the "(If R, then S)" aspect you used above it.

i know we can discuss definitions used in the propositions all day long. I think both (Russell and Pascal arguments) are fallacies but can imagine if I change what I perceive the definitions of "nothing" or "evolution and design" to be they could both be valid.

ok, my brain is starting to hurt with all of these terms i am not used to using on a daily basis.

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