An introduction to the battle of logics.

Metaphysics and Epistemology
1

Having spent the day reading and studying, in response to many of LauricAcid’s comments on the Perfecting Logic thread, I do not feel that I have a full enough context to completely invalidate predicate logic. My approach to the subject is to understand its history and the result, but I am not yet strong enough in many of the key areas necessary, especially early 20th century philosophy (during which predicate logic arose as a systematic logic). I also have found almost no good information on the problems, true or imagined, with Aristotelian logic that led to the development of alternative logic systems. However, my reading does indicate that the primary reason for predicate logic’s development was an attempt to make all argument mathematical. Leinbiz, for instance, thought “that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica [language] and then simply calculated.” (http://www.rbjones.com/rbjpub/logic/jrh0103.htm). Other research indicates that theirs was a division between linguistic and mathematical logic during the early 20th century.

Since I cannot give a full argument about the nature and results of predicate logic, I have confined myself to attempting to point out the essential differences. I shall include a “good faith” account of what I learned in my graduate logic class in college, and then contrast it with the primary topics that I focus on in logic (Aristotelian and Ayn Rand). My source for predicate logic is Logics by John Nolt.

The most basic non-Aristotelian formal logic is known as Boolean logic. Boolean logic is limited in what it can analyze to the operators “and,” “or,” “not,” and “if.” Its only form of analysis consists of converting the terms between these operators into symbols, and then using the rules of logic to check the validity of the form. (The rules for Boolean analysis are in almost all cases identical to those of Aristotelian logic). An example of a Boolean argument would look like this:

If Socrates is a man, then he is mortal. Socrates is a man. Therefore, Socrates is mortal.

If a, then b. a; therefore b.

Predicate logic is the next stage of Boolean logic, introducing the concepts of universal and particular propositions. It is therefore, just as in Aristotelian logic, possible to talk about some men, or all men. At this point, predicate logic has achieved a relative parity with the Aristotelian notion of syllogisms. By parity, I mean that both systems achieve relatively the same results from the same argument. However predicate logic has not achieved its goal of being a calculus. Thus, it matures into what it currently is a series of logical systems deducted from mathematical theories: abstract algebra and set theory.

Among the problems of predicate logic is its inability to maintain any content from the original argument. (It can only use “and,” “or,” “not,” “if,” “all,” and “some.”) To deal with this problem, new symbols were introducted, such as Leibniz’s Alethic and Deontic operators, which represented ‘possibility’ and ‘necessity’ respectively (Nolt p308). Another example is Kripke, who added operators for “knowing”, “has always been the case that”, and “obligation” (Nolt p335). In order to handle these new operators, and to later fully integrate with set theory, predicate (now known as “formal”) logicians created a new field called “metalogic”. This field was highly axiomatized in place of induction and allowed for the resolution of the problems caused by the new operators with the creation of a series of logical systems. Each system was designed to handle a specific category of problems with a specific set of operators and rules. Examples of such systems are free logics, multivalue logics, supervaluations, infinite values logics, fuzzy logics, and intuitionistic logics (Nolt, chapters 15 and 16).

My conclusion from these is that they are all rationalistic and floating. Their intention is to create rules such that their arguments are correct, and they keep the student focused so much on the form that the context is ignored. To date, I have never seen, nor could I imagine any use in, anyone using boolean or predicate logic to analyze Ayn Rand. I offer this: the difference between Rand and Kant is not simply that Kant failed to use logic in the correct formal way.

What is the alternative?

Aristotelian logic is focused on grammar, not mathematics. Its syllogisms handle all the forms of predicate logic, but can also do things like using different verbs without needing any special symbols. It focuses on the relationship of words to concepts and most importantly, Aristotelian logic integrates all logical fallacies. I mentioned in my original statement that the rules of syllogisms should be automatized. No one can analyze the syllogisms of an argument while at the same time taking in the content. So it is important to learn the rules of distribution so that you can automate your ability to validate the syllogistic aspects of an argument.

The best logic book that i have ever seen is An Introduction to Logic by H.W.B. Joseph (http://www.papertig.com/Logic.htm).

I will quote a section of the table of contents starting with chapter 2.

Of terms, and their principal distinctions

Of the categories

Of the predicables

Of the Rules of Definition and Division: Classification and Dichotomy

Of the Intension and Extension of Terms and of their Denotation and Connotation

Of the Proposition or Judgment

Of the Various Forms of the Judgment

Of the Distribution of Term in the Judgment: And of the Opposition of Judgments

Of Immediate Inferences

Of Syllogisms in General

In this dense 600-page book, the topic of syllogisms is not even full discussed until chapter 11 (page 249). Furthermore, the material up to that point directly relates to a student’s ability to understand a logical argument and to create one himself. However, the Joseph book is very advanced. Loinel Ruby’s Logic: An Introduction (http://www.papertig.com/Logic.htm) is much easier, with lots of good Aristotelian content suitable for more casual students of logic. It, too, helps the reader in learning to apply logic to arguments.

In addition to the huge repository of material on logic in the Aristotelian tradition, there are also contributions to logic by Ayn Rand. Concept-formation allows for a more detailed understanding of propositions. Also see the “Objectivity” chapter of OPAR – for instance, under the heading, “Objectivity as volitional adherence to reality by means of logic”. The Ayn Rand Bookstore has a number of further topics, including Peikoff’s Introduction to Logic (http://www.aynrandbookstore2.com/store/pro…tem=19&mitem=46).

Most exciting to the field of logic are the advancements in induction by Objectivism (http://www.aynrandbookstore2.com/store/pro…tem=34&mitem=46 and http://www.aynrandbookstore2.com/store/pro…tem=44&mitem=46). Induction and deduction are the two methods of integrating your ideas.

As one last comment, I want to point out that there is nothing you can study in Boolean logic or predicate logic that does not exist in Aristotelian logic. In most cases, such as verbs, these statements are fully integrated already.

I hope that this is helpful in explaining my objections to formal logic as such.

2

Its syllogisms handle all the forms of predicate logic,

No they dont. Express the following as valid syllogisms:

  1. All Objectivists are humans, therefore anything written by an Objectivist was written by a human

  2. 2+2 = 4

  3. All pink horses are horses, all pink horses are pink, therefore some horses are pink

(The first 2 show the limitations of the syllogism - not all valid arguments are syllogistic. The last one shows the defects in the Aristotlean way of handling existential statements)

There are also deeper problems such as the way he treats general terms as being proper names (a proposition like “All Objectivists are philosophers” is treated as having the same logical form as “Ayn Rand is a philosopher”, as if “all Objectivists” was an entity in the same way that Ayn Rand was). This is tied to his strong focus on the subject-predicate form.

All of the above problems are fixed in the predicate calculus, which includes the Aristotlean syllogisms as a subset, as well as the tools to formulate vastly more arguments.

Among the problems of predicate logic is its inability to maintain any content from the original argument

This isnt a ‘problem’ with predicate logic, its one of its virtues. The focus is on the logical form of the argument, not any particular interpretation of the symbols involved.

3

My conclusion from these is that they are all rationalistic and floating. Their intention is to create rules such that their arguments are correct, and they keep the student focused so much on the form that the context is ignored. To date, I have never seen, nor could I imagine any use in, anyone using boolean or predicate logic to analyze Ayn Rand. I offer this: the difference between Rand and Kant is not simply that Kant failed to use logic in the correct formal way.

How would you use a syllogism to analyse Ayn Rand (or Kant, or anyone?). You could probably formalise Ayn Rand’s detailed arguments in predicate calculus if you really wanted to, although I’m not sure what the point would be. You almost certainly couldnt put them in syllogism form though (What would be the syllogism for “Existence is identity therefore instantaneous travel is impossible”?).

I’m not really sure what your statement about Kant’s logic means. Kant wrote before the development of modern logic, and was highly influenced by Aristotle (he even had a table of categories!). The syllogism form played a role in Kant’s work, as did subject-predicate analysis.

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  1. Objectivist’s are humans, so an Objectivist’s head is human head

This is known as a enthymeme, a missing premise. There is no syllogistic problem with this statement. You can easily find the premises missing, such as “humans have heads”.

  1. 2+2 = 4 

This is not an argument, but if you used it as a proposition to an argument, then I do not see what the problem is.

Two plus two is four. Four is two times two. So Two plus two is two times two.

I am curious how you handle it in predicate logic?

  1. All pink horses are horses, all pink horses are pink, therefore some horses are pink

This is another example of a enthymeme – some horses are pink. There is no problem in syllogistic logic with moving from all to some.

(The first 2 show the limitations of the syllogism - not all valid arguments are syllogistic. The last one shows the defects in the Aristotlean way of handling existential statements)

I do not understand how this is case. For the dogmatic Catholics who had to have each term absolutely exact, there was some funny ways around problems, but to my knowledge all such problems are linguistic and not logical.

There are also deeper problems such as the way he treats general terms as being proper names (a proposition like “All Objectivists are philosophers” is treated as having the same logical form as “Ayn Rand is a philosopher”, as if “all Objectivists” was an entity in the same way that Ayn Rand was). This is tied to his strong focus on the subject-predicate form.

All sentences are subject-predicate. To be strongly focused on a subject-predicate is to be focused on language – the tool of reasoning. I want to make it clear that it is for the reason that Aristotle is focused on tool of reasoning that I say his is the only valid system of logic.

But your statement confuses me. Assuming that it is true that all objectivists are philosophers (a premise I would reject if you mean philosopher as an occupation), how is that different? In both cases you are talking about an attribute of an existent or set of existents.

All of the above problems are fixed in the predicate calculus, which includes the Aristotlean syllogisms as a subset, as well as the tools to formulate vastly more arguments.

This isnt a ‘problem’ with predicate logic, its one of its virtues. The focus is on the logical form of the argument, not any particular interpretation of the symbols involved.

The “particular interpretation of the symbols” means the content of the argument. So, if I understand you correctly, your argument is that predicate logic is superior because it ignores the primary tool of reasoning (the sentence), or content of the argument, and instead focuses only the symbols involved? If this is your argument, then I fully agree and attempted to make that not only the theme of my argument but an indication of the philosophic corruption of the field.

If I may ask a couple of questions: what can you learn about the argument itself from only the form? How do you, using logic only, explain the difference between Rand and any other philosopher? Do you not find that you must concede to anyone who can keep the form right that their argument is valid?

Aristotelian (categorical)/Rand’s logic (as I read it) using the axioms of philosophy are able to invalidate the content of a bad argument which is the only important thing to me.

5

How would you use a syllogism to analyse Ayn Rand (or Kant, or anyone?). You could probably formalise Ayn Rand’s detailed arguments in predicate calculus if you really wanted to, although I’m not sure what the point would be. You almost certainly couldnt put them in syllogism form though (What would be the syllogism for “Existence is identity therefore instantaneous travel is impossible”?).

I’m not really sure what your statement about Kant’s logic means. Kant wrote before the development of modern logic, and was highly influenced by Aristotle (he even had a table of categories!). The syllogism form played a role in Kant’s work, as did subject-predicate analysis.

Aristotelian (Categorical) logic extends far past the syllogism (to my knowledge predicate knowledge does not extend past the form). For myself, while I read Rand when I come to a difficult passage, I often pick out the terms and look for their syllogistic relationship to each other. (I will post an example in a bit.)

As to Kant, if logic is, as Ayn Rand described, the art of non-contradictory identification, then being able to disprove Kant, or any other philosopher must be part of it if you wish to point out their contradictions.

6

How is apply logic to my study.

This is my answer to Hal’s question about Rand and syllogisms. One thing that is important to point out is that most of Rand’s writing is inductive, but normally she has a deductive section in the summary. In a particular hard passage, the first thing that I take apart are the propositions, and then I study their relationship to understand the argument being made. For instance, from “Objectivist Ethics” in Virtue of Selfishness:

In answer to those philosophers who claim that no relation can be established between ultimate ends or values and the facts of reality, let me stress that the fact that living entities exist and function necessitates the existence of values and of an ultimate value which for any given living entity is its own life. Thus the validation of value judgments is to be achieved by reference to the facts of reality.

Here is the propositions that I wrote down:

  1. What is the relation between ultimate ends or values and the facts of reality?

  2. Living entities exist and function necessitates the existence of values and of an ultimate value.

  3. The ultimate value for any given living entity is its own life.

  4. [Missing] The conditional nature of your life is a fact of reality.

  5. The validation of value judgments is to be achieved by reference to the facts of reality.

There is a missing premise here (offered an example of what it might be), but when I take the premises apart on paper, I can chew and agree or disagree with them much better. The syllogistic aspects are automated for me, so I look for the term being distributed (ultimate value, and life in my example) and look to see if it has been or what would be required for it to?

7

This is known as a enthymeme, a missing premise. There is no syllogistic problem with this statement. You can easily find the premises missing, such as “humans have heads”.

I edited the example in my previous post to highlight its incompatibility with syllogistic logic even further. But sticking with the “Objectivist heads” example, why should we say there is a premises misssing - we dont need to add any premises in predicate calculus. And adding that premises could cause problems:

“All Objectivists are humans, therefore all Objectivist noses are human noses”

Would you want to add the premises “All humans have noses”? But this would be false - there are humans that dont have noses (eg ones that have been in accidents). Yet the above argument is still valid.

This is not an argument, but if you used it as a proposition to an argument, then I do not see what the problem is.

Two plus two is four. Four is two times two. So Two plus two is two times two.

Which syllogism do you think this is? (theres a table here listing the various forms)

I am curious how you handle it in predicate logic?

This is another example of a enthymeme – some horses are pink. There is no problem in syllogistic logic with moving from all to some.

No. “Some horses are pink” is a false statement - there are no pink horses. The problem is that this syllogism (the Darapti form on the page I linked above):

All pink horses are pink

All pink horses are horses

Some horses are pink

has true premises yet a false conclusion. This is because Aristotlean logic doesnt handle existential statements properly - it assumes that “some” implies “at least one”. The predicate calculus formalization of this argument would be:

(all x)(Horse(x) & Pink(x) => Pink(x))

(all x)(Horse(x) & Pink(x) => Horse(x))

(exists x)(Horse(x) & Pink(x))

This is clearly invalid - the conclusion does not follow from the premises because we have introduced an existential quantifier out of nowhere. But this is not shown in the syllogistic version because existence is taken as being a logical predicate just like any other.

All sentences are subject-predicate.

Either it is raining or it is not raining

Two plus two equals four

In both cases you are talking about an attribute of an existent or set of existents.

Well, in the statement “all objectists are philosophers”, the subject is not the set of Objectivists. The set of Objectivists is not a philosopher. “All Objectivists” is not a proper name in the same way that “Bob the Objectivist” is.

8

In order to handle these new operators, and to later fully integrate with set theory, predicate (now known as “formal”) logicians created a new field called “metalogic”. This field was highly axiomatized in place of induction and allowed for the resolution of the problems caused by the new operators with the creation of a series of logical systems. Each system was designed to handle a specific category of problems with a specific set of operators and rules. Examples of such systems are free logics, multivalue logics, supervaluations, infinite values logics, fuzzy logics, and intuitionistic logics (Nolt, chapters 15 and 16).

Hal, if you would, I wonder if you could comment on your thoughts about the notion of predicate logic of multiple specialized logics. Much of my why I think the field is completely bankrupt rests on the premise that they use “logic” for whatever they want it to be so that their arguments are “valid”. Such as intuitionistic logic which denies truth and seeks “confirmation”, or relevance logics which have a special symbol (added to true and false) to indicate contradiction?

9

The “particular interpretation of the symbols” means the content of the argument. So, if I understand you correctly, your argument is that predicate logic is superior because it ignores the primary tool of reasoning (the sentence), or content of the argument, and instead focuses only the symbols involved?

Yes, but syllogistic logic does the same thing, only in an inferior way. The purpose of the syllogisms is to describe the FORMS of valid arguments, not their contents. The argument:

All X are Y

M is X

M is Y

is valid under all interpretations of M, X and Y. Syllogistic logic is just as ‘form focused’ as predicate calculus.

If I may ask a couple of questions: what can you learn about the argument itself from only the form?

You can learn whether the truth of the premises guarantees the truth of the conclusion (just like in syllogistic logic).

Do you not find that you must concede to anyone who can keep the form right that their argument is valid?

Yes, but their conclusion dont need to be true unless their premises are true. The following argument is valid:

All Objectivists have 4 heads

Ayn Rand was an Objectivist

Ayn Rand had 4 heads

If the premises are true, then the conclusion is guaranteed to be true. But the premises are not true. Hence this argument is valid, but not sound.

10

Either it is raining or it is not raining

Two plus two equals four

Are you proposing that a grammatically correct sentence does not contain a subject and predicate? Do you believe that these are not sentences? The first is a compound sentence, and a contradiction. The subject is two plus two, and the predicate is equals four.

11

Hal, if you would, I wonder if you could comment on your thoughts about the notion of predicate logic of multiple specialized logics. Much of my why I think the field is completely bankrupt rests on the premise that they use “logic” for whatever they want it to be so that their arguments are “valid”. Such as intuitionistic logic which denies truth and seeks “confirmation”, or relevance logics which have a special symbol (added to true and false) to indicate contradiction?

“Predicate calculus” refers to the specific logic invented by Frege and refined by others in the early 20th century. It is independent of other specialist logics like intutionistic and non-classical logics, and hence even if you thought the latter were founded on bad premises, this would not effect predicate calculus. As for my thoughts, some specialist logics are useful, and some arent.

12

Are you proposing that a grammatically correct sentence does not contain a subject and predicate? Do you believe that these are not sentences? The first is a compound sentence, and a contradiction. The subject is two plus two, and the predicate is equals four.

Yes, the majority of sentences are not of subject-predicate form. Here are some random examples;

Stop theif!

I now pronounce you man and wife

I promise to pay you back that $10 I borrowed

Are you going to the shops?

but assuming you want to restrict the discussion to declarative sentences (propositions), then I would continue to hold that these do not necessarily have subject predicate form. The two I mentioned above are grammatically correct sentences, yet neither is subject-predicate. I suppose this is a good place to give our definitions of ‘subject-predicate’ form, so here is mine: A sentence is subject-predicate if it has the logical form F(x), where x is the subject and F is a logical predicate.

Another good example of a non-subject-predicate sentence is “horses exist”, since existence is not treated as a logical predicate within predicate calculus (well, to be technical its a second order predicate).

13

In reference to the Ayn Rand quote you provided earlier

In answer to those philosophers who claim that no relation can be established between ultimate ends or values and the facts of reality, let me stress that the fact that living entities exist and function necessitates the existence of values and of an ultimate value which for any given living entity is its own life. Thus the validation of value judgments is to be achieved by reference to the facts of reality.

Here is the first part of the argument decomposed (the second part has no logical relation to the first without a LOT more premises, so you couldnt use a syllogism either):

Living entities exist and function implies that an ultimate value exists.

This ultimate value for a given entity is its own life

Here is the predicate calculus formalization

(all x)(Alive(x) & Functions(x) .=>. Ultimate_Value(x,Life(x)))

Why is this harder to use than the equivalent syllogistic form (whatever that is)?

14

Yes, the majority of sentences are not of subject-predicate form. Here are some random examples;

Stop theif!

I now pronounce you man and wife

I promise to pay you back that $10 I borrowed

Are you going to the shops?

but assuming you want to restrict the discussion to declarative sentences (propositions), then I would continue to hold that these do not necessarily have subject predicate form. The two I mentioned above are grammatically correct sentences, yet neither is subject-predicate. I suppose this is a good place to give our definitions of ‘subject-predicate’ form, so here is mine: A sentence is  subject-predicate if it has the logical form F(x), where x is the subject and F is a logical predicate.

Another good example of a non-subject-predicate sentence is “horses exist”, since existence is not treated as a logical predicate within predicate calculus (for good reason).

I seem to not be able to connect. Subject-predicate is the definition of a sentence. Aristotelians study sentences. Grammar is the precursor to logic and logic reinforces good grammar. This is what I mean by the depth of the logic. Having clauses in a sentence is exciting it is a sub-premise (if the writer is good) giving details to the main clause.

I do find your example interesting. It reminds me of the most quoted Objectivist quote that I am aware of:

Existence exists – and the act of grasping that statement implies two corollary axioms: that something exists which one perceives and that one exists possessing consciousness, consciousness being the faculty of perceiving that which exists.

(Italics mine)

Existence exists is of course a proposition. One that is used in the next sentence of the quote.

If nothing exists, there can be no consciousness: a consciousness with nothing to be conscious of is a contradiction in terms.

I still see predicate logic as a weak subset with no room to grow and nothing to do excite math-like exercises with small arguments. (which is what I did for a year in predicate logic classes)

My goal, my only goal, is to study actual arguments and understand them, and with the exception of some funny sophists, I doubt any famous philosopher has gapping holes in form.

Now, I do not deny that valid form is important. But it is a minor issue is analyzing an argument, and the individual only applies it argument by argument and not, as my predicate logic book seems to indicate, by converting the whole paper into a series of symbols.

15

Here is the predicate calculus formalization

(all x)(Alive(x) & Functions(x) .=>. Ultimate_Value(x,Life(x)))

Why is this harder to use than the equivalent syllogistic form (whatever that is)?

I have no idea what that means.

For all x that is alive and functioning there is an ultimate value x and life x?

Can you give me a couple ‘x’ examples?

16

I seem to not be able to connect. Subject-predicate is the definition of a sentence.

No, it isnt. A subject predicate sentence is a sentence in which something is predicated of a grammatical subject. The sentence “John is a farmer” assigns the predicate of ‘farmer’ to John, and the the sentence “Ayn Rand was a philosopher” assigns the predicate “philosopher” to “Ayn Rand”. But a sentence like “Stop theif!” has no grammatical subject or predicate. Neither does the sentence “Are you going to the shops?”.

Existence exists is of course a proposition. One that is used in the next sentence of the quote.

This is a good example. This sentence does not assign the predicate ‘exists’ to the subject ‘existence’. ‘Existence’ is not a subject.

I still see predicate logic as a weak subset with no room to grow and nothing to do excite math-like exercises with small arguments. (which is what I did for a year in predicate logic classes)

If predicate logic is a subset of syllogistic logic, this implies there are syllogistic arguments that cannot be expressed in predicate calculus. But this is false. There are however predicate calculus arguments that cannot be expressed as syllogisms (I gave some in my first post).

Now, I do not deny that valid form is important. But it is a minor issue is analyzing an argument, and the individual only applies it argument by argument and not, as my predicate logic book seems to indicate, by converting the whole paper into a series of symbols.

Yes, I agree. But your issue is not ‘syllogistic logic vs predicate calculus’ - its ‘informal logic vs formal/symbollic logic’. The syllogisms are a type of formal/symbolic logic. I would never use predicate calculus to formulate and analyse an argument in a philosophy text. But I wouldnt try and cram it into a syllogism form either.

17

I have no idea what that means.

For all x that is alive and functioning there is an ultimate value x and life x?

Can you give me a couple ‘x’ examples?

Life(x) is a function which would return the life of the object x (so Life(ayn_rand) would return “ayn rand’s life”, and Life(john) would return “john’s life”).

Ultimate_Value(x,y) is a two-place function which says that the ultimate value of x is y. So Ultimate_Value(john, mary) would say that john’s ultimate value is mary.

So, Ultimate_Value(x, Life(x)) says that the ultimate value of x is x’s life.

Adding a universal quantifier, “(all x)(Ultimate_Value(x, Life(x))” says that everything has its own life as its ultimate value. But we need to restrict ourselves to living things which function, so we get:

“(all x)(Alive(x) & Functions(x) .=>. Ultimate_Value(x, Life(x)))”

18

double post

19

But I wouldnt try and cram it into a syllogism form either.

This is probably be my last post tonight (bedtime).

I do not use the term “cram” to refer to syllogisms. My understanding, partly through introspection, is that syllogisms are the mechanism by which our minds put arguments together. I work on mastering syllogisms because I see them in a number of places such as when I do an outline for an article. Or when I feel like I have something missing in an argument in my mind. Or just when I get to a difficult passage in some work.

I also wish to add, before I am off, that I do respect your arguments Hal and will give them much thought in the coming weeks and months as I further my study in logic. Though I still disagree with you, I respect many of the reasons that you defend your position and I understand better why someone would hold your position.

20

I’m off to bed too, but before I go, I’ll summarize my general position.

The purpose of a formal logic (in this context) is to allow the forms of arguments to be analysed by studying the general patterns of correct reasoning using variables which could refer to anything (the 'x’s). This is what both syllogistic logic and predicate calculus aim at doing - I dont think there is any basis for saying that syllogistic logic is more focused on content than predicate calculus. You have a table of syllogisms specifying the permitted rules of inference (“All P are Q and All Q are R implies all Q are R”) without any notion of what P,Q and R are meant to be - they can be anything. Therefore, the standard for comparing formal logics is how well they perform this task of formalising arguments - namely their expressiveness (can they formalise ALL arguments?), and their constency (do they have internal contradictions?). Predicate calculus beats syllogistic logic on both counts - it is more expressive, and unlike syllogistic logic, it handles existential statements in a way that doesnt lead to false conclusions being deduced from true premises.

Beyond this, there can be deeper metaphysical issues involved, such as what ‘universals’ actually refer to (syllogistic logic forces them to function as proper names which leads Aristotle to conclude that universals are a special type of substance, whereas predicate logic allows you to use a Bertrand Russell style ‘description’ theory), and whether existence should be treated as a predicate (Aristotle says yes, Frege says no).

This last point is actually a very important technical issue, and I’ve hinted at it several times in this thread without stating it explicitly. Before Kant/Frege, it was customary to treat ‘existence’ as being a predicate. Saying “horses exist” was thought to be similar to saying “horses are brown” - we are attributing the predicates ‘existence’ and ‘brown’ to the subject ‘horses’. However this leads to problems with statements like “Unicorns dont exist” - what is the subject of this sentence? The subject cannot be ‘unicorns’, because unicorns dont exist - how can a non-existent thing be a subject? (we cant point to a thing and say “that thing doesnt exist!”). And the subject cant be our idea/concept of unicorns, because the statement is not that “our idea of unicorns doesnt exist”. This type of thing baffled philosophers for centuries, and led to bizarre theories involving semi-real objects which have “being but not existence”.

It was Kant who first noted that existence shouldnt be treated as a predicate in this way. When we say that horses dont exist, we are not adding anything to our concept of ‘horses’ like we do when we say that they are brown. If all horses died tomorrow, our concept of ‘horse’ would remain unchanged. When we say that horses exist, we are making the statement that there is something in the real world which our concept of ‘horse’ refers to - ie we are saying that our concept has referents. So the statement “horses exists” doesnt assign the predicate ‘exists’ to ‘horses’ - it assigns the predicate “refers to something” to our concept of “horse”. Similarly “unicorns dont exist” says that our concept of unicorns doesnt have any referents. This discovery is one of the things that led to predicate logic - existence stopped being treated as a logical predicate, and started being treated as a quantifier (a second order predicate - ie a property of concepts). Because syllogistic logic necessarily treats existence as just being a standard predicate, it is unable to formalise arguments which refer to fictional/non-existent objects (such as the pink horses example above) and often leads to strange metaphysical theories. If you want to treat existence as being a predicate like Aristotle did, you need to give a coherent account of the subject of a statement like “unicorns dont exist”. To the best of my knowledge, this has never been done.

But all these issues refer to our choice of a formalization scheme - they will only arise when we wish to formalise our arguments and analyse their form (or get confused about the logical form of something). For standard reading, we dont do this - we rely on what is generally called ‘informal logic’, or ‘critical thinking’. This is neither syllogistic nor predicate calculus - its something else entirely. Ayn Rand used the term ‘logic’ to describe both formal and informal logic (I think), whereas most modern philosophers reserve that term for symbollic arguments. But this is just a terminological issue - noone would deny that Ayn Rand’s “art of non-contradictory identification” is important. They would just refer to it by a different name than ‘logic’ to avoid confusing it with symbollic logic.