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.