The Ontological Argument
A Most Merry and Illustrated History,
Explanation, and Correction
(with a Short Course in Formal Logic Thrown In)
The Ontological Argument
Sometime around 1077 or 1078 a gentleman named Thomas Anselm (who later became the Archbishop of Canterbury) sat down and penned a book called the Proslogion. Although it is one of the most famous tomes ever written it is not, to say the least, a volume crafted with great journalistic clarity. In fact, you can barely tell what Tom's talking about.
He starts off saying, "Come on now, little man, get away from your worldly occupations for a while, escape from your tumultuous thoughts." Well, that's all right. Obviously Tom wants us to sit down, and can of beer in hand and bag of chips at side, watch the NFL playoffs, right? Nope. He wants to talk about whether God exists.
He continues on, ranting and raving, and ends up talking directly to God. So you might wonder why he's bothering trying to prove someone exists that he's already talking to. Then you realize he's trying to convince you - the "little man" - that God exists. And to do this he writes one of the most incomprehensible books ever.
What's all the more strange is when all is said and done, the entire book can be condensed into one paragraph. And you still can't tell what he's talking about! Anyway, we'll start off here, edited a bit for brevity and clarity (at least as much as possible), with Thomas Anselm's famous Proslogion, the "ontological" proof for the existence of God.
|For it is one thing for something to exist in a person's thought and quite another for the person to think that thing exists. And certainly that greater than which cannot be understood cannot exist only in thought, for if it exists only in thought it could also be thought of as existing in reality as well, which is greater. Therefore something than which greater cannot be thought undoubtedly exists both in thought and in reality. Thus that than which nothing can be thought so undoubtedly exists that it cannot even be thought of as not existing. And you, Lord God, are this being. You exist so undoubtedly, my Lord God, that you cannot even be thought of as not existing.|
Not surprisingly, Tom's argument didn't convince a lot of people, including many of his fellow theologians. Officially the Catholic Church never bought his argument, preferring the simpler (and later) "prime mover" argument of St. Thomas Aquinas which has the triple virtue that it's easy to understand, it's easy to believe, and it's just as easy to pooh-pooh.
Not so with Tom's argument. Although virtually all philosophers don't believe Tom's argument is correct, it will, to quote the chairman of a philosophy department at a major university, drive you crazy. For nearly a thousand years, people have spent their time trying to prove it's true, correct it so it will be true, or refute it completely. And believe it or not, people don't argue whether Tom proved God really exists or not. What they want to know is whether his argument is valid.
C. S. Lewis thought so and said it was Tom's argument that got him away from being an atheist. But even people who don't buy the argument have had trouble refuting it. Bertrand Russell - about as far from Jack as you can expect in both personal philosophy, training in logic, and fiddling around - said it was far easier to believe the argument is not valid than to point out exactly where the fallacy lies. That's a pretty heady statement coming from one of the greatest mathematical logicians of the Nineteenth and Twentieth Century. So if Tom's argument even threw Bertie for a loop, you'd think it has to have something to it. Even Kurt Godel, who had to have one of the greatest mathematical and logical minds ever, worked on the problem, and people still continue to debate it today.
Tom's argument even threw Bertie for a loop.
The biggest stumbling block to figuring out if Tom's argument is valid, though, is figuring out just what the heck he really meant. Fortunately we do have an abundance of philosophy professors whose job it is to think about stuff like that. Although it took a lot of sabbaticals and philosophy conferences at the various resorts and convention centers, we think they finally figured out what Tom meant.
In a nutshell, then here's what Tom is saying:
1. First, God is defined or identified as the being above which no greater can be conceived. You can think of this fairly accurately as saying God is the greatest being imaginable. If you happen to be thinking about something and can't picture anything greater, then you're thinking about God.
2. Next, Tom says that if something exists, then it's greater than if it didn't exist.
3. Therefore we can conclude God exists.
Before the reader splutters the expected, "Say, what?" and slams the book shut (or rather, resumes surfing on the net), consider Tom's reasoning. His proof strategy is what is known as a reductio ad absurdum or indirect proof. What you do in an absurdum proof is to first assume the contrary of what you're trying prove. Then by (correct) reasoning you arrive at some contradiction. Therefore, logicians tell us, what you wanted to prove in the first place is, in fact, correct.
So Tom reasoned this way. Suppose you say God doesn't exist. Then you are saying that the greatest being imaginable doesn't exist. But wait a minute. You can imagine a greater being, that is, a God that does exist. So a greatest being imaginable that doesn't exist is a contradiction. And so we must conclude that God exists.
Q. E. D.
At this point we'll anticipate the curmudgeons and say, yes, there are problems with what Tom is saying outside of the argument itself. First his definition is less a definition of what God is than an algorithm for determining what God is not. Look at something, if you can picture something greater, then what you're looking at isn't God. Although this is a good way for wives to realize their husband isn't God, true, they had probably figured that out already.
The Being Above Which NO Greater Can Be Conceived?
Secondly, this "definition" of God isn't fixed but is dependent upon the ability of the individual. For some people (like men with properly functioning endocrine systems), the being above which no greater can be conceived may very well be Halle Berry (and if she isn't she's not too far from it). Certainly if Halle Berry is the being above which no greater can be conceived, then the ladies shouldn't have too much trouble dragging their husbands to church since any man should be more than willing to spend an eternity of bliss with Halle.
Then there's Point 2 and we have to admit it's a bit fuzzy as to what exactly Tom means. Is he saying that anything which exists is greater than anything that doesn't? For instance, is a garden slug you see laying a slime trail on your carefully landscaped rock garden greater than an Albert Einstein who discovered the Unified Field Theory? (For those who have to have it spelled out, the real Albert Einstein never discovered the Unified Field Theory.) Certainly a physicist would say Albert - or anyone - who discovered the UFT would be the greatest physicist imaginable. Existing or not, that Albert would have to be above a lowly garden slug.
So a physicist would say Anselm is being very specific, but limited. That is, if something has a given set of properties but doesn't exist, then it's not as great as something with the same properties but does exist.
On the other hand, an invertebratologist who has a fascination with limacii maximae might consider those little beasties to be marvels far above any non-existent physicist, particularly one thought up by a philosophy professor sitting on his rear end in an overly comfortable office. An invertebratologist, then, would say Tom is being very general. Anything which exits is greater than anything which does not.
So with the latter interpretation we can say that Halle Berry that starred in "Die Another Day" (and therefore does exist) is clearly greater than an Albert Einstein who formulated the Unified Field Theory (and doesn't). Albert himself (the real one) even agreed with such a premise, although he was referring to a glamour gal from his own era. In the late 1930's the publisher Bennett Cerf was walking down the street in New York City and saw Einstein walking ahead of him. Bennett quickened his pace and when he caught up with the professor asked whether he thought it odd that he, a famous and well-recognized figure, could be walking down a busy city street with no one paying attention to him. If Lana Turner was walking down the street, Bennett added, people would certainly be stopping to look at her. "Lana Turner," Albert replied, "has a great deal more to offer than I do." So there you have it. If Lana Turner was greater than an Albert Einstein that did exist, we know Halle Berry is greater than an Albert that doesn't.
Who would Tom think is greater?
So it's settled. God is the being above which no greater can be conceived, and if you exist you're greater than anything which doesn't.
Now that we've got our premises set up, we can continue. What we need to determine next is if Tom's argument is valid. This itself requires a little excursion into what philosophers mean by validity.
In logic, a valid argument is one that in every case where the premises are all true, the conclusion is also true. Strangely enough, a valid argument does not have to be true and an argument composed of true statements is not necessarily valid. For instance, take two premises, "All theoretical physicists have dynamite abs" and "Oliver Hardy was a theoretical physicist", then a valid conclusion is "Oliver Hardy had dynamite abs." None of these statements is true, although the argument is perfectly valid.
In the same vein we can say, "Halle Berry is a drop dead Hollywood starlet" and "Some drop dead Hollywood starlets were born in Cleveland". If you deduce, "Halle Berry was born in Cleveland", yes, the statements are true, but the argument is not valid.
Normally to prove validity you have to either run through an absurdum proof, like Tom tried, or do a direct deductive proof. In a deductive proof, you start off with your premises only and then by using proper rules of deduction arrive at the conclusion. The problem is that if you don't arrive at the conclusion you want, the argument might still be valid. Maybe you just haven't found the right chain of reasoning.
Fortunately, though, disproving an argument is valid can be a bit easier. For that, all you need is a counterexample. That is, you need to find at least one specific case where the premises are true, but the conclusion is false. Then the argument can't be valid. It might be salvageable; that is, by refashioning your argument and coming up with better premises, you might craft a new argument that is valid. But your argument as originally stated is not.
At this point it will also help if we avoid vagaries of natural language. At first, this seems like an impossibility. After all, if we're debating a point, how the heck can you do that without using a language? Well, around 1840 a math professor in Ireland named George Boole, came up with a way for representing human reasoning as a form of mathematics. Now called Boolean algebra, George's invention eventually became the basis for designing electrical circuits (which includes computers), but it's original use - a precise formulation for representing thought - was developed further until by the late Nineteenth and early Twentieth Century most logical arguments could be cast into what is called formal first order predicate logic.
"Formal" doesn't mean we have to wear tuxedos while we're arguing. What it means is our logic uses formulas in place of the sentences. Then using an accepted set of transformation rules, ergo, a set of instructions for manipulating the symbols, we can arrive at a final statement which is (hopefully) equivalent to what we want to prove. So if we follow the transformation rules properly, our proof must be accepted as valid. Finding a counter example to prove an argument is invalid is possible by a number of ways. You might be able to find that there are some easily definable conditions that allow you to set the conclusions as false and then determine how to make the premises true. Or you can use transformation rules to deduce a counterexample. Then you can just use trial and error - a lengthy process and not recommended.
The first question is how do we write Tom's statement "God is the being above which no greater can be conceived in formal logic?
First we need to look at the logical symbols as in Table 1:
Table 1. A Brief Exposition of Formal Symbolism
|∃x||For some (at least one) x|
|(x)||For all (every) x (Note: sometimes ∀x is used as "for all x")|
|x > y||x is greater than y (used here in the sense of "is conceptually greater", not the mathematical "is greater than")|
|¬ x||Denial ("It is false that x" or "not x")|
Armed with this battery of symbols, maybe we can say something like:
|For some x and all y, if x is greater than y, then x is God.|
which is rendered symbolically as:
∃x (y): x > y → G(x)
... where again we point out that we use the "greater than" sign, >, to mean "is conceptually greater than".
You state this in logicese as "If there is some x which is conceptually greater than every y, the x is God."
... where it is understood by definition of > that the x cannot be equal to any y (that is, nothing can be greater than itself.
In a bit better English you can say, "If there is some x that is greater than every other y, then x is God" or "If something is greater than everything else, it is God.
Now for some this can be a fine representation of Tom's argument. But for others, this representation doesn't quite work. Remember in Tom's wording, he was saying - which is also an interpretation of the the philosophy professor at the major university (great football team) - not what God is, but what God is not.
In that case we can re-write the definition. And it's:
|For all x and all y, if it is false that x is greater than y, then x is not God.|
... which is rendered as:
(x)(y): ¬ (x > y) → ¬ G(x)
For simplicity, we'll go with this and come back to the other formalization later.
So we have the following symbolic formulation of Tom's arguments. It should be pretty easy to make the connection with the plain English version, the rather stilted English of the logic translations, and the exact rendering into the formal symbolism.
Table 2. Tom's Argument
|Tom Argument (Modernized)||Logic||Formal Representation|
|God is the being above which no greater can be conceived.||For any x and for any y, if x is not greater than y, then x is not God.||(x)(y): ¬ (x > y)→ ¬ Gx|
|Existence is greater than non-existence.||For any x and for any y, if x exists and y does not exist, then x is greater than y.||(x)(y): Ex.¬Ey → x > y|
|God exists.||For at least one x, x is God and x exists.||∃x: Gx . Ex|
At this point we should again point out that Anselm's - quote "definition of God" - unquote - is really stating what God is not. That is if you can have someone in mind and you can picture someone greater, then the first person you were thinking about isn't God. This needs to be kept in mind since there is a tendency to want to interpret the first premise, "God is the being above which no greater can be conceived" as " If Person a is Greater than Person b, then Person a is God (symbolically, a > b → Ga). However this cannot be the proper interpretation. Someone may be greater than someone else - even if only in his own mind - and that doesn't mean he is God.
Now with our argument set up in formal terms, we'll give a quick run down on basic rules on how to determine if a sentence is true or false. Mostly this is just common sense. For instance, if you use the word "and" to connect two statements, both have to be true for the whole "and" sentence (called a conjunction) to be true. If you use "or" (a disjunction) then only one does.
Table 3. Truth Values: Conjunctions (AND, ∧)
("And" Statements: ∧)
("Or" Statements: ∨)
|T ∧ T = T||T ∨ T = T|
|T ∧ F = F||T ∨ F = T|
|F ∧ T = F||F ∨ T = T|
|F ∧ F = F||F ∨ F = F|
If you're having a slow day, "T" means "True" and "F" means "False".
And some even easier rules to remember are:
Table 4. Truth Values: Negation (Denial)
|¬ T = F|
|¬ F = T|
[Note: Contrary to popular belief, "Denial" is not de river dat runs t'rough Egypt.]
Also keep in mind that an even number of negations cancel each other; odd number are the same as a single denial. That is,
¬ ¬T = ¬ F = T
¬ ¬ ¬T = ¬ ¬F = ¬ T = F
and vice versa.
Finally, we have to give the truth values for conditionals. These are nothing more than the "if-then" statements which are so essential to logic. These statements are composed of an "antecedent" - that is a condition that will or not be fulfilled - followed by a conclusion - which may or may not be true. Like disjunctions and conjunctions, conditionals are assigned truth values based on the individual truth or falsehood of the two parts.
Table 5. Truth Values: Conditionals
|T → T = T|
|T → F = F|
|F → T = T|
|F → F = T|
These truth assignments sometimes throw people, but when you think about them they make sense. The main thing to remember is these are conditionals. Both parts do not actually have to be true for the entire conditional to be true.
The first two assignments usually aren't much trouble. After all suppose you say, "Yes, if I stop by the store, then I'll pick up a loaf of bread ". Then if you do stop by the store and do pick up a loaf of bread, then the whole "if-then" statement was clearly true.
Similarly if you do stop by the store and don't pick up a loaf of bread, then the whole "if-then" statement was false (and you'll probably get your tail kicked).
It's the last two assignments that often seem confusing. But the last one (F → F = T) really isn't that much trouble to grasp. For instance suppose your kid says, "After supper I'll go to the rock concert rather than do my homework". You, the parent, then stands arms akimbo and says, "Well, if you go to the concert before you do your homework, buster, then I'm flying to the moon". By asserting the entire statement to be true (you are the parent, after all) and by making sure that the last part is false, you are maintaining that the first part is also false. Ergo, you've just shown that an "if-then" statement is true if both the antecedent and conclusion are false.
It's really the F → T = T one that doesn't make much sense to people. That's because it doesn't make much sense in natural language. Instead, the assignment comes about since formal logic requires all sentences to be true or false (rather than having an option that they might be nonsense). Consider the statement "If the submarines built in the Philadelphia shipyard are three miles long, then some of the submarines will be painted battleship gray". You might think it should be false since there are no submarines three miles long built in the Philadelphia shipyard even if they were all battleship gray. Nope, the statement - according to logicians - is true.
The reasoning seems a little iffy, but it goes something like this. No, we never build submarines three miles long in the City of Brotherly Love (aka The Town That Snowballed Santa Claus), but if we did maybe some would be battleship gray. Since we can't assign an unambiguous "False" value to the statement, it becomes "True" by default.
But the simplest way out of any confusion is just to accept that the definition of a conditional "if-then" statement is the same as a "not-or" disjunction. That is, writing A → B is the same as ¬ A V B. If you ponder it a bit, it makes sense (sort of). If you say "If I stop by the store, then I'll pick up a loaf of bread", then you're saying the same thing as "I will not stop by the store, or I will pick up a loaf of bread." Use the rules for disjunctions and you'll see all the truth assignments fall into place. If all this still seem a bit wishy-washy, rest assured these rules do work in logic. Like an American President once said, "Trust me."
Now as the psychiatrist said after a year of twice a week $500-an-hour sessions, we can begin, yes?
First we'll break the bad news, and say, yes, a counterexample to Tom's argument - at least as we've formulated it - is pretty easy to come up with. For instance, we'll take the case of the non-existent Albert Einstein who discovered the Unified Field Theory and Halle Berry who starred in "Die Another Day" and who very much exists (boy, does she ever).
Two points to note. An Albert who doesn't exist is not greater than our Halle, who does. Then since Albert's not greater than Halle, he is not the being above which no greater can be conceived. So he can't be God.
All right. Now what we do is state the specific case of the ontological argument in logical formulas. But instead of using the x and y values, we use the specific names (called "constants" in logic lingo). We'll use "a" for Albert and "h" for Halle. Then by using the rules in Tables 3, 4, and 5, we will determine whether all the premises are true and the conclusion false. If so, then we have our counterexample, and we've proven Tom's argument is not valid.
And sure enough, that's what Table 6 shows. The premises are true and the conclusion is false. Tom's argument is not valid.
Table 6. The Great Albert/Halle Counterexample
|Specific Case||Interpretation||Substituted Ontological Argument||Interpretation||Truth Value of Argument|
|¬ (a > h)||Albert is not greater than Halle.||¬ (a > h) → ¬ Ga||If Albert is not greater than Halle, then Albert is not God||T →T = True|
|¬ Ea ∧ Eh||Our non-existent Albert does not exist and Halle does.||Ea .¬Eh → a > h||If Albert exists and Halle doesn't exist, then Albert is greater than Halle.||
F ∧ ¬T →F
= F ∧ F →F
= F → F
|¬ Ga||Albert is not God.||Ga.Ea||Albert is God and he exists.||F ∧ F = False|
Of course, if an argument isn't valid, we can have more than one counterexample. Note that you could still have a counterexample if we used Albert who discovered the theory of relativity (that is, an Albert that really did exist but isn't God). And if we include this with the Halle who was born in Saskatchewan (and so doesn't exist) we will plug the values in the argument, and we still have a counterexample.
7. The Second Great Albert/Halle Counterexample
|Specific Case||Interpretation||Substituted Ontological Argument||Interpretation||Truth Value of Argument|
|a > h||Albert is greater than Halle||¬(a > h) → ¬Ga||If Albert is not greater than Halle then Albert is not God.||F → T = True|
|Ea∧¬Eh||Albert exists and Halle does not exist.||Ea∧¬Eh → a > h||If Albert exits and Halle does not exist, then Albert is greater than Halle||
T ∧ ¬F →T
= T∧T →T
|¬Ga||Albert is not God.||Ga ∧ Ea||Albert is God and Albert exists||F ∧ T = False|
At this point, you may say, fine, so we've got counterexamples. But all that means is our particular formulation of Tom's argument is not valid. But it's still not clear - as Bertie Russell pointed out - where or if Tom's original reasoning is incorrect. So we still can't be sure if Tom's basic argument is or isn't valid.
Which Albert is greater?
Actually, the counterexamples do show us where Tom went wrong. Remember Tom said that if you say that the being above which no greater can be conceived does not exist, then you can conceive of a greater one? Therefore, he said, that greater being must be a God that really exists.
That is where Tom messed up. Yes, you can conceive of a being greater than a non-existent God. But unfortunately that being is not limited to an existing powerful being above which no greater can be conceived. Why? Because in Tom's second premise existence trumps non-existence. Therefore, any being which actually exists is greater than Tom's non-existent God. It can be our real garden slug, our real Albert, or our real Halle (particularly our real Halle.) So his argument, with its ample room for counterexamples, isn't valid.
For those who have some knowledge of teaching elementary logic, you are aware of the trick of using truth trees. These are diagrams where if you follow certain rules you 1) end up with all closed (X) pathways and have a valid argument or 2) end up with at least open open branch and have an invalid argument. You can see - if you click here that we have open pathways.
[Note: If you want to see the truth tree for the first definition, that is ∃ x (y): x > y → G(x), then click here. You see this makes no difference on the argument's validity.
Some people might argue that, well, maybe Tom was not really saying simple existence is greater than non-existence (like our physicists from an earlier paragraph). Instead, they say, he meant that a being with all of God's properties that doesn't exist is not as great as a being with all of God's properties that does exist. This can be formulated in predicate logic as
(x)(y): Gx ∧ Ex → (Gy ∧ ¬ Ey → x > y)
This, though, is simply a special case of the original axiom (x)(y) Ex ∧ ¬Ey → x > y. It doesn't make the argument any more valid as the readers can now prove for themselves.
Still when all is said and done philosophers keep working on Tom's problem and no doubt will continue to do so. It seems that even though most people don't think Tom got the cigar, he was coming close. So if they twiddle a bit with Tom's reasoning, they will get the cigar.
To date the most successful adaptation is one formulated by Kurt Godel, the gentleman who proved predicate logic is complete and consistent and who discovered the famous incompleteness theorems of math and logic. However, to tackle the ontological problem, Kurt had to use a more - quote - "advanced" - unquote - type of logic known as modal logic which includes the concept of possibility and necessity. Although no one seems to dispute that Kurt's manipulation of the symbols is correct (the proof is actually quite short), he needs some extra premises, properties, axioms, concepts, symbols, and transformation rules. And for the hard nosed realist these axioms, rules, and properties are far from being clear and distinct to borrow a phrase from René Descartes. Instead the argument comes off more as a logician's game of reverse engineering to figure out which axioms we need to get the conclusion we want.
Kurt got as close as anybody?
Of course if you don't want to try Kurt's tack - that is, use a completely different type of logic - then we can use the logician's game of reverse engineering to figure out which axioms we need to get the conclusion we want - and still stick with first order predicate logic.
Briefly, we reason like this. If you assume that the universe is not democratic - that is, no one has equal greatness - then you can rank everybody in order as to how great they art. Then as long as you assume there are, were, and will be a finite number of people, places, and things (including divine beings), there is something or someone in the universe who comes out "greater" than anyone else. We express a revised premise as:
(∃ x) (y): x > y
We also, though, have to return to our earlier more positive definition of God. That is, that if at least one x exists which is conceptually greater than all other y, then x is God:
(∃ x) (y): x > y → Gx
Now it is of interest for the readers to substitute this into our first Albert/Halle counter example (Table 6). Keeping everything else the same then all the premises will be TRUE and the conclusion FALSE. That's because with Albert not being greater than Hall:
¬(a > h)
... is True and so
a > b → Ga
... is TRUE regardless of whether Albert is God or not (in our counter example he is not). All the other rows of the Table are unchanged (Albert still doesn't exist and Halle does) and Albert is not God. All our premises are TRUE and our conclusion is FALSE.
Alas, Tom's argument as formulated is not valid.
But as we said all is not lost!
Oh, yes. One slight, tiny addition. Although it is not explicitly stated in the earlier formal arguments, there is one more premise we need. That is that at least something exits. In other words we state explicitly that
∃ x: Ex.
... that is, that at least something exists.
So here we are. Although to cast all of reasoning for justifying the new premises into formal logic we would need some extra symbolism that would include concepts such as finite numbers, upper bounds, and stuff like that. It can be done. But most importantly, we have established ∃ x (y): (x > y) as a legitimate premise. So we're finally at the point where we can do one up on Tom and reason correctly, and we do so in Table 8. We just wish Tom had gotten it right the first time!
Table 8. The Amazing Simplified and Corrected Ontological Argument of Which No Reasonable Person Can Harbor the Slightest Doubt
|1.||∃ x (y): x > y → Gx||Premise 1 (Tom's Positive Definition)|
|2.||∃ x (y): (x > y)||Premise 2 (From the Finite Non-Democratic Universe)|
|3.||∃ x: Ex||Premise 3|
|4.||∃ x: Gx||Premise 1, Premise 2, and F → F = True|
|5.||∃ x: Gx . Ex||Premise 3, Premise 4, and T → T = True (Q. E. D)|
There is, though, just another wee, tiny bit of a glitch here (aside from having three steps of five step argument being premises). Although we've established that a being of maximal greatness exits, he need not be the stereotypical Big Guy of traditional Judaeo-Christian-Islamic theology. God could be some guy with a white beard sitting on a cloud surrounded by harp players, true. But depending on your ranking rules, God might very well be Albert Einstein.
Or who knows? Fiddle with your rules just right, and God could end up being Halle Berry.
Or should we say "Halle-lujah!"?
References and Bibliography
The Prayers and Meditations of Saint Anslem with the Proslogion, Saint Anselm, Penguin Books (Various Editions). You can, though, find Tom's book on-line (see below).
Formal Logic: Its Scope and Limits, Richard Jeffrey, McGraw-Hill (1981). Richard was a great popularizer of formal logic and an exponent of using truth trees to find validity or non-validity in arguments. With truth trees - which are stick type diagrams - you either show the argument valid or automatically arrive at counterexamples. Truth trees are really slick, and it was fiddling with truth trees that led to finding the counterexamples given above.
Introduction to Logic, Patrick Suppes, Originally published by Van Nordstrom but reprinted by Dover. A classic introduction and includes a lot of formal first order predicate logic with the various techniques. There's nothing about truth trees, though.
Set Theory and Logic, Robert Stoll, Freeman and Company (1963) (Later reprints by Dover). A more advanced treatment of mathematical logic which gets pretty hairy sometimes and ends up with Kurt Godel's and Alonzo Church's theorems on undecidability.
One irritation is this book is it throws in the statement that the union of a set of indexed empty sets is the empty set (which is OK), but that the intersection of the same sets is the universal set - that is, the set that contains everything. For those condemned to self-study, it seems particularly odd that the intersection of a bunch of nothings comes out to be everything, particularly when the union of the same bunch of nothings (and unions of sets contain the set composed of their intersections) is nothing! And then all the book does is say it is up to the readers to "convince themselves" this is true.
What's worse, some authors write that the intersection of indexed empty sets are undefined! But other texts say it's simply defined as being the universal set. What the hey?
I mean, if it's a theorem, prove it; if it's a weird definition or undefined, just say so, or if it's the odd ball reasoning along the lines of three mile submarines - that is, a "trivial" truth (which is ultimately what it is) then demonstrate it.
Convince themselves! Sheesh!
Theory and Problems of Logic, John Eric Nolt, Dennis A. Rohatyn, and Achille C. Varzi, Schaum's Outline Series (1998). This gives a lot of problems to work through starting with simple propositional logic and on through predicate logic and probability. Teaching yourself? Try Schaum's Outlines.
A History of Western Philosophy, Bertrand Russell, Simon and Shuster (1972) (Originally published by George Allen & Unwin, 1946). A broad survey which was drawn from Bertie's lectures when he was living in Malvern, Pennsylvania and lecturing (briefly) at the Barnes Foundation. Not much on Anselm, though, but he goes into more depth in the arguments of Aquinas and Descartes. The book is dated, but is still a good relaxed introduction to the history of western philosophy. But it's big.
Bertie, of course, was the author of the (at the time) controversial essay (also delivered as a lecture) "Why I Am Not a Christian". In the talk he pointed out that although he was not a Christian he probably agreed with Christ more than most people who did think they were Christians. So it makes you think Bertie was quite familiar with American Sunday morning television.
The Case of the Philosopher's Ring, Randall Collins, Crown Publishing (1978). As sad as it is to state, this book is capable of initiating an interest into the serious study of mathematical logic. This was one of the many Sherlock Holmes pastiches that came out in the 1970's following the success of Nicholas Meyer's The Seven Percent Solution. In this book Sherlock Holmes and his trusty sidekick, Dr. John H. Watson, get a telegram from Cambridge University that a great mind is about to be stolen.
Holmes and Watson seek out the writer of the incomprehensible message who turns out to be none other than Bertrand Russell (yes, Sherlock Holmes and Bertrand Russell). In the course of the adventure Holmes meets Bertie, Bertie's mathematical colleague Alfred North Whitehead, their fellow philosophers, George Hardy and Ludwig Wittgenstein, mathematician Srinivasa Ramanujan, all around weirdo Aleister Crowley, economist John Maynard Keynes, and a host of others. Ignoring the various liberties with the facts (the book has Ramanujan dying of constipation in his Cambridge room in 1914 rather than in India in 1920 of what was diagnosed as tuberculosis), the book is a fun read for Sherlock Holmes fans, but beyond that it's literary merit is debatable.
But during the course of the novel, Bertie makes a reference to the famous Russell Paradox. Does the set of sets that do not contain themselves contain itself? You can show that if it does, it doesn't and if it doesn't it does. Clearly a case for further investigation which inevitably can lead the curious to the study of mathematical logic and philosophy in general.
Try and Stop Me, Bennett Cerf, Simon and Schuster (1945). Bennett tells the story of his meeting with Albert on the sidewalks of New York. Bennett is known mostly to the baby boomers as the panelist of "What's My Line", but those who also read books also know him as the co-founder of Random House and the man who was responsible for getting James Joyce's Ulysses unbanned in the United States.
What isn't that well known is Bennett was one of the greatest essayists American letters has produced. His lack of fame in this area is partly his fault since his essays are scattered through his books which are largely made up of humorous anecdotes and jokes. His best essays are from the 1930's through 1950 and include memoirs of people as varied as his good friend George Gershwin and his cantankerous acquaintance, Alexander Woollcott.
Bennett, though, like most writers would not hesitate to stretch the facts a bit if it made a better story. In his memoirs, At Random, he more or less admitted (albeit obliquely) that a story that he had told as first hand was really recycled and apocryphal. He was sitting in his office when his secretary knocked on the door. She said a young man who said he was the greatest writer in the world wanted to see Mr. Cerf. Bennett (the story goes) immediately told his secretary to send William Saroyan in.
By the way, does anyone still read William Saroyan?
Proslogion, Thomas Anselm. One particularly irritating aspect of the Fount of All Knowledge (i. e., the Internet) is that the sites are often unstable and short-lived. The following three references were originally available when this essay was first written but have now vanished from the ether.
So type "Proslogion" in a search engine, dang it, and maybe you'll find an edition.
Kurt Godel's Ontological Argument, Christopher Small, Department of Statistics and Actuarial Sciences, University of Waterloo, http://www.stats.uwaterloo.ca/~cgsmall/ontology.html. This website was still up as of January 1, 2016. Very nice explanation of the problems with Tom's argument, and how Kurt did his best to fix it. Which he did logically speaking, although it can't be said Kurt convinced a lot of people, including himself (Kurt never published his findings).
For the best biography of Kurt see Who Got Einstein's Office? Eccentricity and Genius at the Institute for Advanced Study by Ed Regis, Addison Wesley, 1988).
Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/ The articles on this site sometimes come off a bit much like personal essays and reflections (but unlike some other frivolous articles on philosophy do not contain cartoons). On the other hand, they do give a good introduction to the various philosophical topics. The article on the ontological arguments (and there's more than one type of argument) is at
This URL, too, still is working at the beginning of 2016.
One instructive section discusses parodies of the ontological argument. What the reader notices is that - like much of modern art - the parodies are often indistinguishable from the real thing.
The main point of this article is you can blow hot air all day about the relative merits of the arguments but none of them are of sufficient clarity or are definite enough that they will whop the heathen (i. e., convince the non-theist). And the really sharp reader will be tempted to argue that the "corrected" argument given above can be lumped with other "correct but uninteresting" resolutions to Tom's argument.