Monday, July 12, 2021

The metaphysical presuppositions of formal logic

By “logic” we might mean (a) the rules that determine the difference between good and bad reasoning, or (b) some formal system that codifies these rules in a specific way, such as the systems of propositional and predicate logic that contemporary students of analytic philosophy learn as a routine part of their education.  These are not the same thing, and it is fallacious to confuse them. 

Most philosophers have at least a vague awareness of this.  For instance, they know from standard textbooks that traditional and modern logic differ in their interpretation of categorical propositions, the repercussions this has for their understanding of the square of opposition, and so forth.  They know that there has been much debate in contemporary philosophy over the status of modal logic, not to mention even more exotic systems like quantum logic.  They may be at least dimly aware that systems of logic were developed in the history of Indian philosophy that differ from those familiar to Western thinkers.  And so on.

All the same, contemporary philosophers tend unreflectively to utilize the formal methods they learned in graduate school, treating (b) as if it were for all practical purposes the same as (a).  In particular, they seldom consider that these methods might assume, or at least suggest, challengeable metaphysical presuppositions. 

When you think about it, it would be surprising if it were not so.  As I have argued many times (e.g. in this recent post and in greater depth in Aristotle’s Revenge), the mathematical abstractions of modern physics, for all their undeniable utility and power, can distort our conception of concrete physical reality if we are not careful.  For mathematical representations of their very nature both leave out aspects of the concrete reality they represent, and can also introduce features that are not part of that reality but rather merely reflect the mode of representation itself. 

But formal logic can do the same.  For one thing, qua formal, its aim is precisely to abstract from the specific nature of the subject matter being reasoned about.  (What is traditionally called material logic, by contrast, aims to reflect rather than abstract from that specific nature.)  At the same time, modern symbolic logic was developed precisely in a manner that would facilitate the expression of one particular subject matter, namely mathematics.  It would hardly be surprising, then, if the way that propositions concerning some other subject matter are expressed in modern formal logic might be potentially metaphysically misleading.

For example, John Bigelow has suggested that modern physics’ mathematical representations of local motion over time, together with the apparatus of modern predicate logic, tend to insinuate an eternalist rather than presentist conception of time.  For when we formulate the propositions of physical theory using predicate logic, we need to quantify over not only present events but also past and future events.  And if the existential quantifier asserts the existence of a thing, then physical theory is thereby made to seem to assert the existence of past and future events no less than present ones. 

Now, this fact does not by itself actually show that past and future events really do exist just as present events do.  For all we know just from what has been said so far, the result in question may reflect, not objective reality itself, but merely modern formal logic’s mode of representing objective reality.  To show that the eternalist conclusion really follows, and does not merely falsely appear to do so, would require independent metaphysical argumentation.  But in that case it is precisely this independent metaphysical argumentation itself, and not the system of formal logic, that is really doing the work.  (I would suggest that the “truthmaker” objection to presentism – which, as I have argued several times, is greatly overrated – may reflect this fallacy of reading off metaphysical conclusions from what is really nothing more than predicate logic’s mode of representation.)

Humean logic?

Rani Lill Anjum and Stephen Mumford, in chapter 5 of their book What Tends to Be: The Philosophy of Dispositional Modality (which earlier appeared as a separate paper), argue that modern formal logic reflects a metaphysical bias in favor of a Humean conception of the world and against an Aristotelian conception.  In particular, it is well suited to express causal propositions understood, as Hume would, as describing merely contingent relationships holding between “loose and separate” existents.  It is poorly suited to express causal propositions understood the way Aristotelians understand them, as describing necessary connections between intrinsically related existents. 

Now, among the crucial features of modern logic in this connection are that it is extensional and truth-functional.    In the context of predicate logic, extensionality has to do with the fact that co-referring terms can be substituted for one another without changing the truth value of a statement.  For example, since the statement that Spider-Man fights crime is true, and Spider-Man = Peter Parker, then it will also be true that Peter Parker fights crime.  (Statements involving propositional attitudes don’t fit this pattern, however.  They are famously intensional rather than extensional.  For example, if it is true that Aunt May believes that Spider-Man fights crime, then even though Spider-Man = Peter Parker, it does not follow that Aunt May believes that Peter Parker fights crime.  For if she does not know that Spider-Man = Peter Parker, the second statement might not be true even though the first is.)

In the context of propositional logic, extensionality has to do with the fact that a proposition that is a component of a compound proposition can be replaced by one having the same truth value without changing the truth value of the compound proposition.  For example, if it is true that water is wet and grass is green, and we replace the second of the component propositions with the true proposition that the sky is blue, then the resulting proposition that water is wet and the sky is blue will also be true.

Truth-functionality has to do with the fact that in propositional logic, the truth or falsity of a compound statement is a function solely of the truth or falsity of its component parts.  For example, if it is true that the sky is blue and it is also true that I am drinking coffee, then the conjunctive statement that the sky is blue and I am drinking coffee will also be true. 

Now, where these features have especially interesting implications – and implications relevant to Anjum and Mumford’s point – is with respect to material conditionals, statements of the form p É q or “If p, then q.”  In propositional logic, the only case where a statement of this form is false is when the antecedent p is true and the consequent q is false.  In every other case the conditional will be true.  This has some notoriously odd results (known as the “paradoxes of material implication”).  For example, the statement that if the sky is green, then robots rule the earth is true.  The antecedent and consequent are, of course, both false, but the statement as a whole still comes out true, as anyone knows who has worked through the relevant truth table.  Also true are the statement that if the sky is green, then 1 + 1 = 2 (since the consequent is true even though the antecedent is false) and the statement that if the sky is blue, then 1 + 1 = 2 (since both antecedent and consequent are true even though they have nothing to do with one another). 

Now, suppose you agree with Hume that there are no necessary connections between any things or events in the world.  Everything is, as Hume puts it, “loose and separate,” and in theory any effect or none might follow upon any cause – striking a match may cause it to turn into a cat, planting an acorn might cause a Volkswagen to grow out of the earth, and so on.  We don’t seriously believe such things will ever happen, but that has nothing to do with the natures of these things themselves.  It has instead to do only with psychological expectations on our part based on past experience, or at most with whatever laws of nature happen contingently to associate an event of one type with events of another type (where what a “law of nature” is on a Humean view is itself a problematic issue). 

In that case, say Anjum and Mumford, modern formal logic is well suited to convey any causal claim you want to make.  Weird conditionals like the examples given above are not prima facie suspect.  (True, there may be no law connecting, say, the sky’s being green with robots ruling the earth, and for that reason a contemporary Humean wouldn’t take the conditional in question to express a true causal claim.  But that would have nothing to do with anything intrinsic to the sky’s being green – nothing to do with there being no objective necessary connection between the sky’s color and robots ruling the earth.  Again, for the Humean there are no intrinsic or necessary connections between things in the first place.)

But suppose instead that you take the Aristotelian view that natural substances have inherent dispositions or powers by which they necessarily tend to generate effects of a certain specific kind.  Then, weird examples of conditionals like the ones in question are suspect.  They show that the connections between things that are captured by the material conditional are simply too weak to correspond to the strong connections posited by the Aristotelian metaphysics of causal powers.  You’re not going to be able to capture the truth of a causal statement like “Striking a match generates flame and heat” or “A planted acorn will grow into an oak tree” via the material conditional.  Indeed, attempts to capture such claims in terms of conditionals, or even in terms of counterfactuals, face notorious difficulties.  (See pp. 53-63 of Scholastic Metaphysics for an overview of the main arguments.)

Now, Anjum and Mumford note that adding predicate logic to propositional logic does not solve the problem, because predicate logic builds on propositional logic’s account of the material conditional.  But even adding modal operators, as modal logic does, does not solve the problem either, because the truth-functional character of propositional logic is preserved.  You’ll still get weird results (known as the “paradoxes of strict implication”), and in particular results that are going to be suspect from an Aristotelian point of view.  For example, you get the result that anything strictly implies a necessary statement:

 q ® (p ® q)

For instance, “If it is necessary that water is H2O, then this strictly implies that the fact that tomorrow is Taco Tuesday strictly implies that water is H2O.”  That weird sort of modal statement hardly captures the kind of necessary connections in nature that Aristotelians posit when they affirm the reality of causal powers.

Now, David Lewis famously held that every possible world is as real as the actual world.  And as Anjum and Mumford point out, this provides a way to read even statements in modal logic in a Humean manner that denies any intrinsic causal connections between things.  The truth of the statement that necessarily, if p then q requires only that in every possible world where p is true, q is also true.  It does not require that there be anything intrinsic to the states of affairs described by p and q (such as causal powers that follow upon the essence of a thing) that ties them together.  Of course, most people wouldn’t agree with Lewis’s view about possible worlds, but the point is that the mere possibility of interpreting modal logic in Lewis’s terms shows that it doesn’t capture the kinds of necessary connections that Aristotelians attribute to the natural order.

There is also the fact that on the Aristotelian account, causal powers tend toward generating certain outcomes, but still may not in fact generate them, because the manifestation of a power can be blocked.  Given the nature of the sulfur in the head of a match, it tends toward generating flame and heat when struck (as opposed to frost and coldness, or turning into a snake, or what have you).  But that doesn’t entail that flame and heat will always follow, because that tendency can be frustrated (for example, if the match gets wet). 

So, things are not “loose and separate” in the manner Hume supposes (e.g. it just isn’t true that striking a match might in principle bring about any old effect at all).  But at the same time there are not going to be exceptionless correlations between events (because the operation of a power can be frustrated, so that the event of striking a match might in some cases not be followed by the event of flame and heat being generated).  Anjum and Mumford propose positing a “dispositional modality” of tending toward that lies in between mere possibility on the one hand and necessitation on the other.  More traditional Aristotelians would speak of potencies that are distinct from actualities but are nevertheless really there in things themselves even if they are never actualized.  But however we describe the metaphysical details, Anjum and Mumford’s point is that they are not going to be captured in standard extensional and truth-functional formal systems. 

You might say “So much the worse for the Aristotelian,” but the point is that such a judgment would have nothing to do with formal logic itself.  Rather, it has to do with independent metaphysical assumptions that might lead one to favor a certain formal system.  A formal system may be useful for certain purposes and not so useful for others.  Our metaphysical predilections might lead us to judge that there is nothing more to the world than what the formal system captures, or they may lead us to judge that it leaves important things out.  Either way, the characteristics of the formal system itself don’t settle anything.  As Anjum and Mumford write:

Metaphysics is First Philosophy, prior even to logic.  And from that it would follow that one should first choose one’s metaphysics and then choose one’s logic, rather than the other way around. (p. 86)

I would qualify this by saying that metaphysics is prior to logic if “logic” is understood in sense (b) described above, though not if understood in sense (a).  Naturally, we have to presuppose certain canons of reasoning when reasoning about anything, including metaphysics.  But it doesn’t follow that we have to presuppose the codification enshrined in some particular formal system – such as, for example, modern propositional and predicate logic rather than traditional Aristotelian logic, or rather than some system that tries to capture the best of both worlds (such as that of Fred Sommers). 

Again, at some level most philosophers realize this, but it can be easy to forget if you and your colleagues all routinely learn and utilize a certain formal system, and questions about its underlying philosophical assumptions are considered only by the small minority of philosophers who specialize in such things.  Anjum and Mumford speculate that, despite his famous disagreement with Mill on matters of the philosophy of mathematics, Frege picked up a set of essentially Humean empiricist prejudices about logic from Mill’s A System of Logic.  These were then passed down from Frege to Carnap, then from Carnap to Quine, and then from Quine to Lewis and contemporary philosophers in general.  (Naturally, Russell and Whitehead played a major role too.)  Whatever one thinks of this hypothesis, it is certainly true that a dogmatic conventional wisdom can take root on matters of logic no less than it can with respect to any other area of intellectual interest.

What-logic versus relating-logic

Aristotelian complaints about the metaphysical prejudices enshrined in modern formal logic are not new.  Over fifty years ago, Henry Veatch addressed the issue at length in his book Two Logics: The Conflict between Classical and Neo-Analytic Philosophy (which was recently reprinted by Editiones Scholasticae). 

Veatch notes that we can distinguish what a thing is from the relations it bears to other things.  Now, Aristotelians are essentialists, who hold that there are facts of the matter about what things are and that we can at least to some extent discover those facts.  Logic, as understood in the Aristotelian tradition, is a tool for helping us to discover and express what things are.  A humble categorical proposition like “All whales are mammals” does precisely that, however little such a statement tells us all by itself.

However, Veatch argues, statements formulated in terms of the formal logic hammered out in works like Russell and Whitehead’s Principia Mathematica do not and indeed cannot, strictly speaking, tell us what a thing is.  They can express only relations.  Now, one of the advantages of modern predicate logic is precisely that it can represent relations in a way that Aristotelian categorical logic cannot.  It does so using multi-place predicates.  For example, the relation “___ loves ___” would be represented by the two-place predicate L ___, ___ where the spaces would be filled by lower-case letters naming individuals.  Hence “Harry loves Sally” would be represented as: Lhs.

But even one-place predicates, Veatch notes, are treated as representing relations, viz. relations between a thing and a property.  For instance, “Fred is bald” or Bf would represent the relationship between Fred and the property baldness.  One-place predicates are essentially treated as a limiting case of relational predicates. 

For this reason, Veatch argues, modern formal logic can really only ever express the relations between things, and not what a thing is.  Before you judge that that cannot be right, it would be a good idea to keep in mind that Russell himself held that even modern physics, when formulated in the language of modern logic, gives us knowledge only of relations and not of the intrinsic natures of anything.  (I discuss Russell’s views a length in Aristotle’s Revenge.)  One could, however, take this to show, not how little physics tells us, but rather how little formal logic tells us.

As Veatch also points out, the analysis of ordinary statements into statements of predicate logic tends to suggest an ontology of bare particulars and universals.  For instance, the statement “There’s a Ferrari parked outside” comes out as something like: ($x) (Fx • Px).  Any concrete attributes that might characterize the thing being described get analyzed as predicates, leaving just a bare something of which the universals named by the predicates (being a Ferrari, being parked outside) are predicated. 

Now, the notion of a bare particular is metaphysically dubious (cf. David Oderberg’s essay “Predicate Logic and Bare Particulars”), as is the notion of a world of which we can know only relations.  Of course, someone might nevertheless want to defend such philosophical exotica.  The point, however, is that even if the utility of predicate logic might suggest such views, it does not actually by itself give evidential support for them.  Again, if some apparent aspect of reality is difficult to describe using the apparatus of a system of formal logic, that may indicate merely the expressive limitations of the system, rather than the absence of those aspects from objective reality.  We cannot read a metaphysics out of formal logic without first reading one into it.  Metaphysics, as Anjum and Mumford insist, is in this sense prior to logic.

Faux rigor

Such considerations lend additional force to a point I have made before, which is that the use of formal methods in philosophical analysis and argumentation by no means guarantees that the results are more rigorously established, and indeed in some cases can even make them less so. 

For example, when analyzing an argument like Aquinas’s Third Way, some commentators like to reformulate it using the formal apparatus familiar from contemporary modal logic.  The reader easily impressed by such things thinks: “Wow, this is so much more rigorous than a less formal treatment!”  But in fact, such an analysis will simply change the subject, because the distinctively Aristotelian way in which Aquinas understands the relevant modal concepts cannot (as Anjum and Mumford point out) be captured in that formal language.  And an analysis that simply fails to capture what Aquinas is talking about is hardly rigorous.

In a post from a decade ago I discussed Robert Nozick’s treatment in Philosophical Explanations of the question why there is something rather than nothing, and noted that it affords another example of how semi-formal methods can obfuscate rather than illuminate.  Nozick speaks of various possible “states N [that] are natural or privileged” (one of which might be “nothingness” itself), of various “forces of type F” (one of which might be a “nothingness force”), of an “amount” there might be of such a force, and so on, and then proceeds to consider what relations may hold between N and various quantities of F, etc.  Because the discussion is couched in terms of symbols and variables, it gives the appearance of rigor.  But it is not prefaced with any treatment of the more fundamental and indeed crucial philosophical question of whether the proposed states and forces referred to are plausible (or indeed even coherent) in the first place.  Hence the apparent rigor is bogus.

None of this is intended to suggest that formal methods have no value, or to deny that sometimes they are even necessary.  The point is rather that their utility can be oversold and their neutrality overestimated.

49 comments:

  1. Ed, do you know if there is a parallel between these different types of logic (one of which prioritizes relations) and the notion in the "trinitarian" theologies of Zizioulas and others who reject a substance ontology in favor of a relation ontology?
    Tim Finlay

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    1. Interesting question, Tim. I don't think there is any direct connection, since theologians of the kind you're referring to seem primarily motivated by what they take to be the nature of relations of the interpersonal kind, specifically, whereas writers like Russell are concerned with relations of a much more abstract sort. But I suppose the former group could deploy the work of those of the latter group in the service of their own ends.

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  2. This is extremely important to know since it is sometimes very difficult to understand the implications related to formal logic and even more difficult to see some fallacies in some arguments. Thank you for that, Ed! May God bless you!

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  3. So does this mean that logic is an abstraction of the same particular system of metaphysics that it's meant to serve?

    Great Spider-Man analogy by the way.

    It's endlessly amusing how you make it seem like comic books and their content were invented by philosophers to express and teach philosophy.

    TEACHER: You see boys and girls, Stan Lee -like Nostradamus- was the greatest visionary of his time.
    But, the world was not ready; so he too had to cloak his wisdom in common mediums of the time where only those with wisdom could recognize them.

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  4. Can someone recommend a good book which gives a treatment of “ the rules that determine the difference between good and bad reasoning”? Want to have something for my kids before they hit late high-school or college.

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    1. I have Peter Kreeft's "Socratic Logic". It is touted as the best introduction to Aristotelian style logic in circulation.

      As with all of Kreeft's work it is very readable and not dry at all! It is also designed for high school use and includes lots of practice questions and the text itself includes answers to odd-numbered questions (I think you can email Kreeft for the even answers). It is hardcover, in-depth (but not too much). It will cost you a bit, being in the $40-50, but worth it.Good luck!

      https://www.amazon.com/Socratic-Logic-Questions-Aristotelian-Principles/dp/1587318083/ref=sr_1_1?dchild=1&keywords=Socratic+Logic&qid=1626211637&sr=8-1

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    2. Yeah it even gives logical examples from Winnie the Pooh!

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    3. Logic as a Human Instrument by Parker & Veatch and Logic: The Art of Defining and Reasoning by Oesterle

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    4. Memoria Press's Traditional Logic I and II are good for high school students. So is their Material Logic. Might as well get the Rhetoric while you're at it.

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    5. W. Stanley Jevons' Elementary Lessons in Logic is very readable and was the most popular logic text in the English speaking world for decades from late Victorian times well into the 20th century.
      If you need a hard-copy it's available on Amazon in several editions.

      Jevons was responsible for modifying Boole's original system into what we recognize today as boolean algebra (Boole's system was more complex and used ordinary high school algebra). His method is outlined in a chapter of the book. It's my favourite logical 'system' because it's so easy to use and is based very directly on Aristotle's three laws of thought, and the only inference rule is substitution of equals for equals.

      I also highly recommend his Magnum Opus - The Principles of Science.

      And if you're ever in Oxford, you should check out his 'Logic Piano' in the Museum of the History of Science.

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  5. There's a formal system that ameliorates at least some of the problems listed here, called "case-intensional logic", by allowing (and indeed requiring in some circumstances) intensional predication rather than merely extensional predication, and making it clear that the "existential" quantifier isn't ontologically committing. I've found it to be the most intuitive way of combining modal and predicate logic:

    http://philsci-archive.pitt.edu/9375/

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  6. Thank you for this post! I need to make reading time for Logic... after I finish the books on my list for Ethics.

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  7. Hi Ed,

    There is some justice in the Aristotelian-Thomistic complaint that modern predicate logic fails to capture the way we normally talk about dispositions, causal relations and necessary connections. But in that case, the obvious question is: why don't Aristotelians develop their own calculus, for the modern world? (It is universally acknowledged that Aristotle's 2,300-year-old logic is inadequate: for example, it is unable to demonstrate, from the fact that a horse is an animal, that the head of a horse is the head of an animal.) Do you know any Aristotelians who have attempted this task? Rob Koons, perhaps?

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    1. Check out Fred Sommers's work.

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    2. Vincent,

      This is a short helpful blog that answers that question.

      https://www.google.com/amp/s/vexingquestions.wordpress.com/2014/08/08/the-horse-head-argument/amp/

      Mr. Vecchio offers a particular syllogism:

      Every horse is an animal in virtue of being a horse.

      Every horse is that which has a head.

      Therefore: Some of those which have heads are animals, in virtue of being horses.

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    3. Vincent,

      I don't quite get how what you're saying makes sense. Take the following syllogism.

      All horses are animals.
      Each horsehead belongs to a horse.
      Each horsehead belongs to an animal.

      Is this not a valid syllogism?

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    4. I find Sommer's plus-minus calculus rather confusing, but his major insight was to recognize that the old Dictum de Omni et Nullo of traditional logic is applicable not only to simple 3 term syllogisms, but also to relations and multiply general sentences, such as 'Some boy loves every girl'. The dictum is a rule of substitution, and applying it to the horses head argument (using plain English) :

      H : horses
      A : animals
      h : heads

      1. All H are A
      2. All h of H are h of H (tautology)

      Now, by the dictum we can substitute A in premise 1 into the H in the predicate of premise 2, which gives :

      All h of H are h of A

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    5. @ Mister Geocon, strictly speaking, it isn't a valid categorical syllogism because there are too many terms. But it uses the dictum in a more general way, a la Sommers, and is actually easier to read than my solution because it hasn't pulled a tautology seemingly out of a hat.

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    6. Joe,

      There are only three terms: horses, animals, horseheads. How is this too many terms?

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    7. Well, I count 5 terms : horses, animals, horseheads, 'a thing which belongs to a horse' and 'a thing which belongs to an animal'.

      There are only 3 terms if the copula is 'belongs to', but you haven't used it in the first premise; you used 'is'. To be valid, you would have to use 'belongs to' as the copula consistently throughout. But 'All horses belong to animals' doesn't seem right.

      Actually, Aristotle's original formulation of categorical propositions did use 'belongs to' as the copula. The form was

      A does/doesn't belong to all/some B.

      But 'belongs to' in your argument doesn't convey the same meaning (that of subsumption). There, it's taken to mean 'a part of' or 'has', ie, a portion of a natural object.

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    8. Joe,

      'A thing which belongs to a horse' and 'a thing which belongs to an animal' aren't actual terms. To show why, let's reformulate the syllogism.

      Horses are a kind of animal.
      Horseheads are heads that belong to a horse.
      Horseheads are heads that belong to an animal.

      This says precisely the same thing as the other syllogism, except now, everything uses 'is.'

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    9. Mister Geocon,

      There are still 5 terms in your argument. So although not a valid categorical syllogism (which must have exactly 3 terms), it is valid.

      It contains an implicit substitution though. Let's make it explicit. The terms of your argument are:

      1. H = horses
      2. A = animals
      3. hh = horseheads
      4. hbH = heads that belong to a horse
      5. hbA = heads that belong to an animal

      Do you agree?

      The premises are :

      1. All H are A
      2. All hh are hbH

      Now, by deriving the conclusion

      3. All hh are hbA

      you have substituted 'A' in premise 1 into the 'H' in the predicate of premise 2. But this assumes that hbH can be interpreted as 'heads that belong to x', which isn't the kind of term allowed in standard categorical logic; it's more like a predicate in the predicate calculus, something like Bhx, where B = 'Belongs to', h = heads, and x is a variable.

      Your argument is an example of what traditional logicians called 'immediate inference by added determinants', except that the second premise is superfluous: you can go from :

      1. All H are A

      directly to

      2. All heads of H are heads of A

      But you have to be careful that the added determinant ('heads', in this case) doesn't qualify the predicate in a way which makes the conclusion fallacious. e.g. An invalid example would be :

      1. All elephants are animals
      2. therefore, 'All small elephants are small animals'.

      Actually there *is* a standard categorical syllogism which fits the bill :

      1. All [horses] are [animals]
      2. All [horses] are [that which have heads]
      3. => some [animals] (namely,horses) are [that which have heads]

      This is AAI, figure 3.

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    10. Sorry, I meant to start my 2nd sentence with 'BUT although not a categorical syllogism...'

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  8. Feser: "For when we formulate the propositions of physical theory using predicate logic, we need to quantify over not only present events but also past and future events. And if the existential quantifier asserts the existence of a thing, then physical theory is thereby made to seem to assert the existence of past and future events no less than present ones."

    An example here would be nice. I don't entirely follow.

    It seems rather like taking seriously the idea that to assert "unicorns don't exist" implies that unicorns exist, because nothing can be true about something (including unicorns) unless that something exists -- and furthermore claiming that this is a "metaphysical presupposition" of ordinary language. Quine takes this so seriously (or pretends to?) that he thinks we have to resort to the logic (in the narrow sense) of Russellian paraphrases to avoid the problem. Anyway, I do think "metaphysical predispositions of formal logic" would have been a better title here. Gyula Klima has a very nice paper on "Ontological Alternatives vs. Alternative Semantics in Mediaeval Philosophy" which I think treats the issue well. It think it puts more emphasis on the fact that metaphysics (ontology) doesn't really follow from logic (semantics).

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  9. I wonder how Aristotelianism would analyse indeterministic forms of causality in terms of the tendencies of things? Quantum mechanics is often cited as an example, but another one is a six-sided die which lands on any particular side in a non-deterministic stochastic fashion.

    In this case, the indeterministic die has a well-defined set of final causes (landing on 1, 2, 3, 4, 5, 6) and has a clear form. Its indeterminism may also be explained by its formal cause since it's like that by its very nature. The die landing on any particular number also has efficient causes - whatever threw the die, and maybe the die itself which lands on a number. Yet though the die has the tendency to generate a specific set of outcomes, and there are probabilities attached to each outcome, the way the outcomes are caused isn't deterministic.

    If the die were thrown and landed on 4, the fact it's indeterministic means that if you reversed time and played the event all over again, the die could also have landed on any other number instead. So no one particular outcome always follows - but not for reasons of interference of powers being frustrated.

    The fact that the outcomes are indeterministic and may not have a contrastive rather-than explanation is something intrinsic to the nature of the die because indeterminism is intrinsic to it.

    How would Aristotelianism view this type of causality, where any particular outcome doesn't always manifest, but not because of intereference - rather because of indeterminism?

    Does Aristotelianism even have better metaphysical tools for dealing with indeterministic causes than formal logic, since it avoids both there being no tendencies, but also those tendencies strictly always manifesting?

    Yet in thi

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    1. There aren't an infinite number of potentials for any given real object. Take the six-sided die. If you rewound time and threw the die again and again, you'd get a number between one and six, but however many times you rewound time and threw the die, you wouldn't get a number other than those six, nor would the die explode or turn into a live chicken. It's within the nature of the die to land on one of its six sides when thrown.

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    2. Those aren't the only potentials of a die. A die can land here, or it can land there, or it can be melted down a certain way, or... In the absence of gravity, it doesn't land.

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  10. When I took an intro to Logic class in college our teacher went to some lengths to point out that we were studying whether arguments had a valid *form*, but validity on its own said nothing about the truth of the arguments. It was also great that the formal logic section was followed by one on "informal" logic, introducing us to many of the more common fallacies, such as post hoc ergo hoc.

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  11. Isn't all this a reflection of the fact that merely asserting p -> q or □ p -> q doesn't tell us anything about WHY it is true (it may be explainable because of what p is, or, on the other hand, have nothing to do with the nature of p)? And can't these kind of relations also be incorporated into predicate logic? Something like □ (Np & ~Oq) -> (p -> q) where Np is the nature of p, and Oq is some other obstacle preventing q?

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  12. Any thoughts on "George Englebretsen - Robust Reality, An Essay in Formal Ontology (2012)"?

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  13. These are good points about the limitations of formal logic, Dr. Feser, but I have a different perspective on the limitations of formal logic. It isn't that they have an implicit metaphysics, but that, like any formal system, they have limited scope.

    In mathematics, 2+3=5 is only useful if 2 and 3 are the numbers of disjoints sets. If the sets overlap, then the answer is bogus. In probability, p(A&B)=p(A)p(B), but only if A and B are independent. Similar limitations apply to all forms of formal logic: they have constraints on how they can be used; for example, the law of the excluded middle is only true if you don't allow sentences like the Liar, and the predicate calculus only works for systems that have been idealized into something like bare particulars and universals (I don't care for that specific characterization, but it will do for purposes of discussion).

    However, this does not mean that predicate logic has a metaphysics; it only means that predicate logic has a restricted application.

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    1. "predicate calculus only works for systems that have been idealized into something like bare particulars and universals."

      Is that right? I'd have said that the things have to be expressed in terms that might be amenable to an interpretation along the line of bare particulars and universals; but that doesn't seem the same as the requirement of actually idealizing them into something like that. If it did, then the claim wouldn't be true, that "this does not mean that predicate logic has a metaphysics." Again, you say, "it only means that predicate logic has a restricted application." But why does it mean that? Why must it have just this restricted application, where it only 'works' on the supposition of a particular (seemingly metaphysical) idealization of the elements in its domain?

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  14. Spot on. Before logic and ratiocination can get to work, PERCEPTION must first take place. And, to do that accurately and adequately is truly an art that is difficult to attain. Athanasius, Augustine, and, more recently, Chesterton, Lewis, Balthasar, and DC Schindler come immediately to mind.

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  15. Great post. I can't help thinking that it was prompted by the recent exchanges with Joe Schmid?

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    1. It's something I've thought about writing up for quite a long time, actually, but the red herrings that came up in that recent exchange did indeed serve as an immediate inspiration.

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    2. I think Joe would be pleased knowing he had inspired you. ��

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  16. This was a great post on an interesting subject. What is Feser's favorite formal logical system? Aristotelian/syllogistic logic? I always found the case-by-case enumeration of validity strange in that system. That is, looking at each syllogism form and determining whether it is valid or invalid, rather than positing more general rules.

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  17. "David Lewis famously held that every possible world is as real as the actual world."
    Isn't that just an assertion of what the definition of "real" is?

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    1. Or it is just a bare assertion without a shred of basis, or truth.

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    2. Did not Lewis itself argued that his modal realism is the only way to make sense of modal truths or something?

      I mean, it is a silly view and we realists about universals can laugh at the argument, but it is not really a assertion.

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  18. Since the comment I had early on left praising this entry for its interest and value -probably shortly after it was put up - never took, let me second those who have commented on this intellectually satisfying yet very approachable and casual review.

    Someone had mentioned Sommers earlier, and it reminds me that I still have "The New Old Logic" edited by Oderberg, which you {Ed] may have recommended as worth study, still unread. The same, insofar as an uncompleted read goes for Heidegger's "The Metaphysical Foundations of Logic" which treats "judgments"; a term which always puts me off a bit when used in connection with logic, but which is common in European treatments of it.

    If anyone has made a careful study of that latter work, perhaps they can save me, and anyone else potentially interested, the full slog through it with a few insights.

    Leibniz ... "judgements" ... Gaia save me ...



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  19. Can I just remark how much better this combox is from just a few months ago. It's a shame it has taken Feser personally vetting comments to weed out the low value contributions from Gnus and trolls, but it does greatly improve the experience.

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  20. We developed a tape measure to measure this magic place we live in, only we forgot about the magic place and became enamored by the tape measure.

    The thing for which I am most indebted to Feser is the understanding that we cannot get more metaphysics out than we first put in. If people really understood that, it would, no doubt, end a lot of gibberish debates posing as intellectual.

    What kind of cool name is Veatch? Sup Veatches!

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  21. I recently managed to get a new copy of 'Two Logics' from Amazon for less than half price, and I'm hoping that Editiones Scholasticae will reprint Veatch & Parker's 'Logic as a Human Instrument'. Second-hand copies are very expensive, and although you can read it online at the internet archive library, it's not downloadable and you can only borrow it for an hour at a time.

    Another downside of modern predicate logic (Logic as Humean Instrument?), aside from those which Ed has pointed out, is that it's so damned hard to learn and apply,in comparison with term logic. The most difficult step is translating from natural language into 'Logicese'. True, it does have the characteristic of being 'ontologically explicit', in that it forces you to make clear just what does and doesn't exist, which is arguably an advantage for certain subjects such as abstract mathematics and computer science. The trade-off is that this generally makes proofs much longer than they would be in a term logic. For philosophy and everyday reasoning we need something more user-friendly. Logic should be for everybody, and in my opinion should be taught from grade 7 or 8 (it is, after all, simpler than arithmetic in many ways).

    Nor is the oft-touted superiority of predicate logic in terms of inferential power justified, as Sommers has shown. Indeed, there are sentences which can be formulated in term logic but not predicate logic, and the latter would find arguments involving numerical quantifiers, such as the following, incredibly long and tedious:

    At least 13 artists are dentists
    At most 3 beekeepers are not carpenters
    At most 1 carpenter is not a dentist
    Therefore, at least 9 artists are dentists

    But using the extended syllogistic it's no more difficult than solving an ordinary categorical syllogism.
    See here for an outline of Sommer's term logic, including an extension for numerical arguments.

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    1. Logic in sense (a) is for everybody. But in sense (b)?

      Also, I'm curious on what grounds you advocate the teaching of logic ((a) or (b)). I wonder how useful it is. I've spent enough time with students who have studied an intro to logic but who are completely useless at real reasoning to wonder if there is any real benefit in it. (Joe Schmid, for example, very bright young guy, probably got A+ in all his logic courses, but it seems to me his formal knowledge of logic helps him very little in real logical (sense (a)) tasks, that is, in attempting to actually engage in a constructive way with real arguments; and he's picked up some bad 'analytic' habits which positively hinder him from doing so.)

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  22. David, it's not quite clear to me that there is a significant difference between logic in sense (a) and sense (b). Just what are the rules that determine the difference between good and bad reasoning? There are many, and which to use depends on the kind of argument. But then, can you have such a set of rules which are NOT codified in a specific way?

    I guess you could take Aristotle's three laws of thought as the common ground of good reasoning, although there are logical systems which deny one or more of them (but of course, you can only deny them and not give reasons for denying them, otherwise you would be using them). Then there are the basic rules of validity for deductive logic, rules to ensure that inductive arguments are strong, etc, but are there not metaphysical presuppostions involved here, too? Don't the basic rules also constitute a formal system?

    I don't think you can separate metaphysics, epistemology and logic; they are in a sense different aspects of the same thing.

    Also, on reflection, it seems to me that Veatch exaggerates the difference between the two logics, at least in regard to his central argument. If, as he says, we "distinguish what a thing is from the relations it bears to other things" (and I agree), why is he so critical of modern logic's limitation (according to him) that it can only express relations? Any system of logic is about making connections. That is to say, establishing relations between terms (concepts) and/or propositions.

    As for teaching logic, I'm not sure whether it's ever been tried, but I believe if you start early enough -- grade 7, or maybe even earlier, when kids start learning arithmetic, it would pay huge dividends later in both their academic careers and life. What's so important is being able to grasp the principle that statements should be justified and explained through the process of argument, and that this is a virtue and the mark of a civilized society (so there is a moral element there, too).

    In my opinion, this what education should be about, not merely learning "stuff", which it too often is, even now in a time when we have never had such easy access to facts and information. In teaching mathematics, for example, there is the tendency to emphasize that the goal of getting the right answer is of supreme importance, rather than justifying the steps taken to reach it. There is a lot of logic involved in mathematics, but it's largely implicit. Making it explicit would enable students to understand what they are doing and give them more confidence. And actually, studies have shown that this is the case, with regard to mathematics, at least. I see no reason why it should not apply across the board. Logic should not be seen as a 'subject', primarily, but as a tool to be used and applied in every domain.

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  23. "What's so important is being able to grasp the principle that statements should be justified and explained through the process of argument, and that this is a virtue and the mark of a civilized society (so there is a moral element there, too)."

    Well I certainly agree with that. But then this general point should pertain to all of education, not just logic class; and if it does, then maybe the logic class per se is unnecessary; and if it doesn't, then maybe the logic class will be inevitably insufficient, and maybe not even helpful. The logic will end up being just more "stuff" that is learned. And I suspect that's indeed what happens in the real world when students take logic courses.

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  24. I was wondering what professor Feser thinks about Socratic Logic by Peter Kreeft and David Kelley's The Art of Reasoning. Also, just curious what you think of Aristotelian Logic( traditional square, existential import..etc.)

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  25. The question is what must already be assumed about logic in order to adjudicate different views of it.

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