Logic and God’s Character

0. Introduction

Vern Poytress is professor of New Testament interpretation at Westminster Theological Seminary. He has a handy website, which he runs with John Frame, on which he has put a lot of his published work available for free. In particular, he has a copy of his book Logic: A God Centred Approach to the Foundation of Western ThoughtIn this post, I want to focus on a particular small section of the book, which is Chapter 7 (p. 62 – 68). The chapter is entitled ‘Logic Revealing God’, and in it Poytress addresses the question of whether logic is dependent on God, or if God is dependent on logic. As he says, “We seem to be on the horns of a dilemma” (p. 63).

I will go through the chapter quite closely, and it might be worth reading as it is not long (although I will provide plenty of quotes from the original). It is an instructive chapter because it highlights many of the key themes and ideas that we see presuppositionalists making in their positive arguments. It is also done by a professor in a theological seminary, with a very impressive resume, including a PhD in mathematics from Harvard, and a ThD in New Testament Studies from Stellenbosch, South Africa. Therefore, the presentation of the argument should be pretty strong. And I do think that the book is quite readable, and is packed full of great learning material for anyone wanting to study logic.

However, I think that the sections of the book which deal with the theological and metaphysical underpinnings of his view of logic, such as the one I will explore here, leave a lot to be desired. Hopefully, what I will say will be clear, and my criticisms will be justified.

  1. The Dilemma 

The dilemma that Poytress refers to is not spelled out explicitly, but it seems easily recoverable from what he does say. The opening line in the chapter is: “Is logic independent of God?” To start us off, it is quite natural to see logic as independent from the existence of human beings, as Poytress explains:

“Logic is independent of any particular human being and of humanity as a whole. If all human beings were to die, and Felix the cat were to survive, it would still be the case that Felix is a carnivore. The logic leading to this conclusion would still be valid … This hypothetical situation shows that logic is independent of humanity.” (p. 63)

The example that Poytress gives is slightly confusing, as the truth of the statement “Felix is a carnivore” does not seem to be merely a matter of logic, at least not a paradigmatic one. However, it is clear that the idea of independence that is in play involves the following sort of relation:

Independence X is independent of Y   iff   X would still exist even if Y did not exist

The logical relation he highlights (involving the cat) would hold even if people did not exist, and is thus independent from the existence of people. It follows that X is dependent on Y if and only if the independence condition above fails.

The cat example seems to be mixing up a few different things at the same time. The classification of Felix as a carnivore does not depend on the existence of humans, in that whether people exist or not will not change whether a cat eats meat or not. Yet this fact does not seem to be a purely logical fact, and so the independence that it establishes is not really of logic from the existence of human beings.

It seems to me that an example which makes the point he expresses with “logic is independent of any particular human being and of humanity as a whole,” would be the following. Consider the following inference:

  1. All men are mortal
  2. Socrates is a man
  3. Therefore, Socrates is mortal.

The conclusion follows from the premises, and it does so regardless of whether Socrates exists or not. As it happens, Socrates does not exist (any longer), but this does not make the inference any less valid than when he did exist. Even if Socrates turns out to have been entirely a fictional character who never existed at all, the inference is still valid.

And indeed, the conclusion follows from the premises, regardless of whether anyone exists or not; even if everyone were to die in a nuclear war tomorrow, the above inference would remain valid. Even if there had never been any people at all, the inference would remain valid. At least, that is the thought.

Part of the reason for this thought is that we do not need to refer to the existence of any particular thing when coming to determine whether an inference is valid. We consult what it is that actually determines the validity of the inference, and in doing so we do not have to check to see if any particular thing exists. And what it is that the validity of the inference depends on is something like one of the following candidate considerations:

  • An inference is valid if and only if it is possessing the correct logical form.
  • An inference is valid if and only if it is truth-preserving.

Exactly how we cash this out is contentious of course, but I take it that something like these sorts of example is going to be correct. In Aristotelian logic, for example, the forms Barbara and Celerant are simply given as valid (they are the so-called ‘perfect forms’), and so is any form which is transformable into either of one of the perfect forms via the conversion rules. Different logical systems have different conceptions of what the ‘correct logical form’ is, but one thing that seems obvious is that the existence or not of any particular person, or of humanity in general, is irrelevant to the question of whether a given inference is valid or not. It is a different type of consideration that is relevant.

But if this (or something like this) is what the validity of the inference depends on, then whether it is valid or not isn’t just independent from the existence of human beings, but is independent from the existence of any existing thing – including God.

Here is how Poytress explains this idea:

“Through the ages, philosophers are the ones who have done most of the reflection on logic. And philosophers have mostly thought that logic is just “there.” According to their thinking, it is an impersonal something. Their thinking then says that, if a personal God exists, or if multiple gods exist, as the Greek and Roman polytheists believed, these personal beings are subject to the laws of logic, as is everything else in the world. Logic is a kind of cold, Spockian ideal.” (p. 62)

As I have explained, it is not just that philosophers have postulated logic as being just there, without any motivation. There are reasons, like the independence considerations I outlined, for thinking that any given inference is valid or invalid independently from the existence of any particular thing. It follows from these considerations that logic is not itself dependent on any particular thing, and ‘just is’ (as Poytress puts it).

2. Conflict

As a Christian, such a conclusion brings Poytress into conflict with his core theological doctrines. As he explains:

“This view has the effect of making logic an absolute above God, to which God himself is subjected. This view in fact is radically antagonistic to the biblical idea that God is absolute and that everything else is radically subject to him: ‘The Lord has established his throne in the heavens, and his kingdom rules over all’ (Ps. 103:19).” (p. 62)

Thus, logic seems like it is independent of God, because it seems independent of the existence of anything, yet the doctrine of God being absolute (in Poytress’ sense) requires that everything is dependent on God. I take it that this is the dilemma that he faces:

  • On the one hand, logic is independent from the existence of God (as it seems independent from the existence of any entity whatsoever) but that compromises God’s absoluteness (God seems to be subordinate in some sense to logic).
  • On the other hand, logic is dependent on God, which restores the absoluteness of God, but then we are owed some kind of story about how it is that the validity of an argument depends on the existence of God.

This dilemma can be put as follows:

Is God dependent on logic, or is logic dependent on God?

Poytress takes the second horn, and part of his endeavour in the chapter is to bring out how it is that we see God in logic, how logic ‘reveals God’, as a way of bolstering the claim that logic depends on God.

As a first pass, he says:

“The Bible provides resources for moving beyond this apparent dilemma.” (p. 63)

He provides three examples, which are:

  1. “God is dependable and faithful in his character”
  2. “the Bible teaches the distinction between Creator and creature”
  3. “we as human beings are made in the image of God”

Let’s go through each of these and see what he has to say about each of them.

3. “God is dependable and faithful in his character”

With regards to 1, Poytress points to Exodus 34:6, which mentions that God is faithful, and he then explains:

“The constancy of God’s character provides an absolute basis for us to trust in his faithfulness to us. And this faithfulness includes logical consistency rather than illogicality. God “cannot deny himself” (2 Tim. 2:13). He always acts in accordance with who he is.” (p. 63)

It is not clear to me how this engages with our question, which was whether logic depends on God or God depends on logic. Poytress is identifying the faithfulness, logical consistency and inability to deny himself as three special properties that God has, but to me the possession of these properties is irrelevant to the question at hand. I will try to explain my worry with a thought experiment:

Imagine I were to build a robot. And let’s say that I build the robot in such a way that it could not knowingly lie. This would mean that I program it in such a way that it cannot provide any output which is the contradicts any of the stored data it has in its memory banks (or something like that). If so, then my robot would be analogous in some sense to this description of God. It is, in effect, programmed to be honest. Given that a robot cannot do anything which it is not programmed to do, I would be able to trust in its ‘faithfulness’, in that I could know for sure that any output it generates is consistent with its data banks. Arguably, a robot like this is also logically consistent by definition (assuming the programming is consistent), and because it cannot lie, it cannot deny itself in the relevant sense either. Thus, my robot is perfectly faithful, logically consistent and cannot deny itself. Yet, this would not establish that the validity of any given inference was dependent on the existence of the robot, however. And if not, then it is not clear why these properties being possessed by God would be relevant to establishing anything like the horn of the dilemma that Poytress is going for either.

Perhaps you have some niggling objection here. The robot case isn’t really analogous to God, you might be saying. And that is quite true. For instance, no matter how advanced, my robot wouldn’t be all-knowing. And no matter how reliable its programming is, its programming might become corrupted. Either of these indicate the possibility of some kind of error. Because of the possibility of error like this I shouldn’t trust what it tells me with 100% certainty, and this makes the two cases unalike.

However, seeing as this is just a thought experiment, imagine that (somehow) I were to make a robot which did know everything, and couldn’t have its programming corrupted. Would this mean that logic now became dependent on the existence of the robot? Would the validity of an inference now depend on the existence of this robot? I see no reason for thinking that making these imaginary improvements to my robot could possibly have this effect.

As far as I can understand, an entity’s reliability, faithfulness, or inability to self-deny, etc, can never be relevant for making its existence something upon which the validity of an inference depends. If Poytress has some reason for thinking that the possession of these properties by God makes him the thing whose existence the validity of an argument depends, he spends no time explaining them here.

There are a few options at this point.

  1. By possessing these qualities, my robot becomes a thing that the validity of an inference is dependent on.
  2. The possession of these properties by my robot does not qualify it for being the thing that validity depends on, but they are what qualifies God for this role.
  3. The possession of these properties are not what qualifies anything for this role.

The first option seems prima facie implausible, and at the very least we have been given no reason to think that it is true. The second one leaves unanswered why it is that these qualities make God suitable for the role and not the robot, and implies that there are actually additional criteria for playing the role in question which make the difference (i.e. there must be something about God other than the possession of these qualities which distinguishes him from the robot). The third option is that these qualities are not relevant. Unless there is an additional option I cannot see, it seems like Poytress has to go with option 2, and owes us an explanation of the additional criteria.

4. The Bible teaches the distinction between Creator and creature”

So much for the first point. Let’s move on to the second one, which is about the creator/creature distinction. Poytress says the following:

“God alone is Creator and Sovereign and Absolute. We are not. Everything God created is distinct from him. It is all subject to him. Therefore, logic is not a second absolute, over God or beside him. There is only one Absolute, God himself. Logic is in fact an aspect of his character, because it expresses the consistency of God and the faithfulness of God. Consistency and faithfulness belong to the character of God. We can say that they are attributes of God. God is who he is (Ex. 3:14), and what he is includes his consistency and faithfulness. There is nothing more ultimate than God. So God is the source for logic. The character of God includes his logicality.” (p. 63)

This quote can be split into two sections. The first consists of the first five sentences (ending with “There is only one Absolute, God himself”). The first section really just affirms the doctrine of God being absolute. God alone is absolute; we are not absolute; being absolute, everything is dependent on God, including logic. This much is no help resolving the apparent dilemma we were facing earlier. It is just restating one of the two things we are trying to reconcile, i.e. the absoluteness of God. The question is how to fit this idea, of God being absolute, with the intuitive idea that the validity of an inference seems to have nothing to do with the existence of any particular thing. Simply repeating that God is absolute (in contrast to humans) does not shed any light on this issue.

The second part of the quote wanders back into the issue brought up in the previous point, by talking about the faithful character of God, and thus still seems irrelevant. Even if “[c]onsistency and faithfulness belong to the character of God”, how is the validity of an inference dependent on his existence? We are none the wiser.

Poytress does say that God’s ‘logicality’ is included in his character. And it might be thought that this is relevant somehow. After all, we are talking about logic, and ‘logicality’ is the property of being logical. Surely that is the link.

Well, I think it would be a mistake to think that. In some sense, my robot was already logical. It’s ‘brain’ is just a computer, which processes inputs and produces outputs according to some set of rules (its programming). This is a logical process; computer programming is just applied logic. It seems we are in precisely the same position we were in before. We are still left with no reason to think that if this thing did not exist, that an otherwise valid inference would be invalid. Why does being logical mean that logic depends on you? The answer, it seems, is that it doesn’t.

5. “We as human beings are made in the image of God”

On to point three. Here, Poytress is pointing to the fact that we are made in God’s image:

“God has plans and purposes (Isa. 46:10–11). So do we, on our human level (James 4:13; Prov. 16:1). God has thoughts infinitely above ours (Isa. 55:8–9), but we may also have access to his thoughts when he reveals them: “How precious to me are your thoughts, O God!” (Ps. 139:17). We are privileged to think God’s thoughts after him. Our experience of thinking, reasoning, and forming arguments imitates God and reflects the mind of God. Our logic reflects God’s logic. Logic, then, is an aspect of God’s mind. Logic is universal among all human beings in all cultures, because there is only one God, and we are all made in the image of God.” (p. 64)

The idea seems to be as follows. God makes plans, and so do we, although we only make plans on a ‘human level’. God has thoughts, and so do we, although his thoughts are ‘infinitely above ours’. So in this way, we are similar to God, without being the same as God. We are creatures, whereas he is the creator, and our likeness is only imperfect (or ‘analogical’).

The relevant section is when he explains that “our logic reflects God’s logic”, which is because it is us ‘thinking Gods thoughts after him’, in a process which “reflects the mind of God”. Just like with the planning and thinking examples, our grasp of logic is only analogical, which means that we have an imperfect, creaturely understanding in comparison with God’s perfect understanding. Nevertheless, we imitate of God’s thought processes.

The problem with this view is that it invites a Euthyphro-style dilemma immediately. God thinks in a particular way (a logical way) and we are to think in the same sort of way (to imitate and reflect his way of thinking). But, why does God think in this particular way? More precisely, does God think in this way because it is logical way of thinking, or is it a logical way of thinking merely in virtue of it being the way that God thinks? This is just another way of asking the same question we started with, namely: is God dependent on logic or is logic dependent on God? All we have done here is to rephrase it in terms of God’s thinking; is logic dependent on God’s thinking, or is God’s thinking dependent on logic? And there is no reason to think that rephrasing it in this manner will itself constitute any sort of solution to the initial problem.

What Poytress is actually giving us is a reason for why we (should) think in a logical way. We should think in a logical way because that’s the way that God thinks. And, whatever the merits of this point are, this plainly isn’t relevant to the initial question about the relation between logic and God. The best that can be said about this idea is that it is an answer to a different question altogether.

5. Sidebar – Logical Euthryphro 

But it is also rather hopeless as a solution, when we try to run the argument to its logical conclusion. Remember, the first horn was that God thinks in this particular way because it is (independently from him thinking it) a logical way of thinking. Presumably, Poytress would find just as “radically antagonistic to the biblical idea that God is absolute” as the initial claim that God depends on logic. It really just is the same claim. It just says that logic is independent of God. So, he has to opt for the second horn, which is that this way of thinking is logical merely in virtue of being the way that God thinks.

However, there is a problem with this; it makes God’s decision to think in this way (rather than some other way) inexplicable. To sharpen up the discussion, let’s use some examples. We know that there are lots of different logical systems, including classical logic, extensions of classical logic and non-classical logics, etc. Just to take two examples, there is classical logic and intuitionistic logic. They have different fundamental principles, e.g. intuitionistic logic doesn’t have excluded middle as a general law and classical logic does. God thinks in one of these ways and not the other (presumably). Let’s say he thinks classically, and not intuitionistically. If we were to ask why he thinks in this classical way, as opposed to the intuitionistic way, the one thing we cannot say as an answer is that thinking classically is (independently of God thinking like that) the logical way to think. If we tried to say this, then we would in fact be asserting the first horn of the dilemma, which is “radically antagonistic to the biblical idea that God is absolute”.

But what else could possibly be the answer to this question? God thinks classically rather than intuitionistically because … ? It might be that God has a preference for classical logic rather than intuitionistic logic, but this preference itself cannot be based on the idea that classical logic is (independently of God thinking like that) the logical way to think, or we are right back to the initial horn again. So even if he has a preference for classical logic, it can only be based on some other type of consideration, and not that it is itself the logical way to think. But there is nothing else which could be relevant. He may prefer it because he finds it simpler than intuitionistic logic, or because he likes sound of the word ‘classical’, or because he flipped a coin and it landed heads-up rather than tails-up. But whatever the reason, it can only be something which is irrelevant. His reason can only be arbitrary (which just means that it is a decision made without relevant reason). The one thing which could be relevant is ruled out as being the first horn of the dilemma. And that is what is so pressing about this sort of Euthryphro dilemma.

So let’s say we take this horn. It means that if God thinks classically (rather than intuitionistically), and if we were to imitate the way that God thinks (as Poytress urges), then this would produce some kind of explanation for why we think classically rather than intuitionistically. However, because there is no (non-arbitrary) reason why God thinks classically rather than intuionistically, there is correspondingly no real reason why we do either.

Imagine you find me performing a series of actions, walking to and fro in my house, picking things up and putting them down again seemingly at random. If you ask me why I’m doing this, I might say that I have a reason for doing so. Maybe I say to you that these actions performed together will culminate in an effect which I desire. So, maybe I am building something, but I am in the early stages of doing so, just setting out my tools and clearing a space. To you it looks like a random set of actions, but it has a purpose. I have reasons for doing each of the things that I am doing. Maybe once I have explained my purpose, then the series of actions stops looking so random to you.

Now imagine that you come across me performing a series of actions which again seem random to you. You ask me why I am doing these things, and this time I point to the TV, where you see a figure who is performing the very same sorts of actions. I say that I am acting out this person’s actions after him, and reflecting his actions. ‘Well, why is he doing these particular actions?’, I ask. ‘Oh, no reason’, you reply.

I think that in this second situation, we would have to conclude that you are doing something which is different in type to the first example. There your actions had a reason behind them and were not arbitrary, whereas now, you are just mirroring the random actions of the figure on the TV. Really, your actions are just as random as his; there is no reason why you are doing one thing rather than another, because there is no reason why the figure on the TV is doing one thing rather than another. This is what happens if we follow through on the idea that a) we think logically because we are thinking God’s thoughts after him, and b) if logic is not independent of God. Poytress is committed to b), as the other option would be “radically antagonistic” to his idea of God, and he is also urging that we accept a) in the passage we just looked at. Thus, if we go where Poytress urges, we become like the person imitating the random actions of the figure on the TV.

But, surely, this is where God’s characteristics come into play? God is consistent, and faithful, and cannot deny himself. Surely this is relevant. He couldn’t think in an irrational way, because this would mean being inconsistent. In this way, his consistency grounds the type of logic he opts for.

This may seem like a promising rebuttal. However (no surprise), I don’t think it is. Intuitionism is consistent, and many people have found it to be rational. Michael Dummett, for example, argued strongly for intuitionism. It is not the case that someone who prefers intuitionism to classical logic is committed to any contradictions as a result (intuitionistic logic is not inconsistent). They are not necessarily going to deny themselves, or be irrational, or be ‘illogical’ (partly because they would advocate for intuitionism being the correct logic!). None of the considerations that Poytress presents give us any reason to think that God would have any real reason to prefer classical logic over intuitionism based off the character traits that he has identified.

It might even be the case that God likes paraconsistent, or even dialethic logic. If the principle of explosion really were invalid, then God would be dishonest to say that it was valid. If there really were a true contradiction somewhere (and who knows, maybe God has a morally sufficient reason to create one), then God would deny his own act of creation to say that there was not one. Thus, his honesty, truthfullness and consistency could be made to fit with there being contradictions. His characteristics could be retrofitted to be compatible with pretty much any outlandish logical or metaphysical proposal. And this is because they really just float free from, and are orthogonal to, the issues involved in the debates about non-classical logic.

6. Wrapping up

This post is already quite a lot longer than I had anticipated when I started, so I will finish up by briefly going through the final parts of the chapter we are looking at. Those are called ‘Attributes of God’, ‘Divine Attributes of Law’ and ‘The Power of Logic’. In them, Poytress makes the point that logic and God seem to share various properties:

Atemporality:

“If an argument is indeed valid, its validity holds for all times and all places. That is, its validity is omnipresent (in all places) and eternal (for all times). Logical validity has these two attributes that are classically attributed to God.” (p. 65)

Immutability:

“If a law for the validity of a syllogism holds for all times, we presuppose that it is the same law through all times … If a syllogism really does display valid reasoning, does it continue to be valid over time? The law— the law governing reasoning—does not change with time. It is immutable. Validity is unchangeable. Immutability is an attribute of God.” (p. 66)

Immaterial yet effective:

“Logic is essentially immaterial and invisible but is known through its effects. Likewise, God is essentially immaterial and invisible but he is known through his acts in the world.” (ibid)

True/truthful:

“If we are talking about the real laws, rather than possibly awed human formulations, the laws of logic are also absolutely, infallibly true. Truthfulness is also an attribute of God.” (ibid)

These properties initially do seem to be drawing a close similarity between logic and God. They seem to share a lot of properties together. And initially, this might seem to be reason to think that their doing so is significant. However, consider that the same case could be made for the rules of chess:

There is nothing in the laws of chess which refer to any times and places. If it is true that, according to the rules of chess, a pawn can move two spaces on its first move, then this is true if you play chess in Bulgaria, or in China, or on the moon. It’s truth is independent on location, which means that that rule, if true anywhere, is equally true everywhere else. But also, if we went back in a time machine to prehistoric times, and if we had taken a chess set with us, we would not have to consult the local tribe to see if they had a different set of rules for chess. It would still be true that a pawn can move two spaces on its first move, regardless of what year we are playing in. And this means that the rule’s applicability is independent of time. If it is true at one time, it is true at all times. The rules of chess, it seems, are omnipresent and atemporal as well.

Chess is also immutable. You might be thinking that chess used to be played differently. In the past, people had different rules for chess, so it isn’t immutable – chess has a history. Quite true, chess does have a history. But so does logic (trust me, I am editing a book about it). People have changed how they have thought about logical laws. For instance, the idea of existential import is present in Aristotelian logic, but not in classical logic. If we can sidestep this issue with logic, by saying that the historical development of logic is not relevant for undermining the claim that logic is immutable, then we can also do the same for chess.

The rules of chess are immaterial. We cannot touch them or measure them, etc. Yet they govern how actual pieces of material get moved about on actual chess boards. So the rules of chess are immaterial yet effective.

The rules of chess are true. It is true that a pawn can move two spaces on its first move. That is a truth.

So the rules of chess are omnipresent, atemporal, immutable, immaterial yet effective and true. Therefore, God thinks ‘chessly’? God’s nature reflects the rules of chess?

We could run the same sorts of considerations for any different (consistent) logical system, like Łukasiewicz’s three valued logic. It also has all the same sorts of properties. But can God think classically and also with three truth-values at the same time? Only an inconsistent God could do that. So if God thinks classically, rather than non-classically, there must be something about non-classical logics which means that their possession of the properties that Poytress identifies is not indicative of anything significant. Again, if there is something which makes this difference, we are not given it.

7. Conclusion

I have no doubt that Poytress is a very smart guy. I don’t have it in me to get a PhD in mathematics from Harvard. And he clearly understands logic very well. It is puzzling then that his discussions on the area I have focussed on in this post are so weak. There is really nothing he has said which helps make the case that logic is dependent on God, rather than being independent from God. I can only conclude that this part of his book was not thought through very well. The only other possibility is that he is so determined to fit together certain doctrines that he is unable to see that his arguments are weak in this area. I may look further at other aspects of the same book in later posts, but from what I have read of it so far, I don’t imagine he will change in any particularly significant way.

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Does the scientific method rest on the fallacy of affirming the consequent?

0. Introduction

There have been some rather strange suggestions from certain apologists recently about the nature of the scientific method, such as here and here. Prime among the criticisms is the claim that the scientific method rests on a fallacy called ‘affirming the consequent’. However, this is a strange claim for various reasons. Firstly, the criticism doesn’t engage with how philosophers of science actually talk about the scientific method. From around 1960, with the work of Thomas Kuhn, attempts at summing up the scientific method in a simple inferential procedure have been largely abandoned. It is now widely taken in the philosophy of science that there is no one simple pattern of reasoning that completely captures the scientific method – a phenomenon known as the ‘demarcation problem‘. So there is no simple logical model of inference which completely covers the everything in the scientific method. But this means that there is no simple model of fallacious inference which completely characterises the scientific method either. In short, the scientific method is too complex to be reduced to a simple informal fallacy.

However, if we pretend that this Kuhnian sea-change had not taken place, then we would most naturally associate the scientific method with the notion of inductive inference, with evidence being given in support of hypotheses (or theories).  However, induction is not guilty of the fallacy of affirming the consequent, as I shall show here.

After this wander through induction, I will try to explain what the motivations are for the apologetical critique and how that misses the mark by failing to appreciate that scientific advances are often made through falsification rather than verification.

  1. Affirming the consequent

The fallacy of affirming the consequent is any argument of the following form:

  1. If p, then q
  2. q
  3. Therefore, p

The inference from the premises to the conclusion is invalid, because it could be that the premises are true and the conclusion is false. For example, if p is false and q is true, then the premises are true and the conclusion is false. If you want a proof of this, let me know and I will provide it in the comments.

The reason it is a fallacy to use affirming the consequent is just that the argument is deductively invalid. The lesson is this: if you have a true conditional, then you cannot derive the truth-value of the antecedent from the truth of the consequent.

2. Science affirming the consequent

The idea that the scientific method commits the fallacy above can be explained very easily. We might think that theories makes predictions. This could be thought of like a conditional, where the theory is the antecedent and the prediction is the consequent; if the theory is true, then something else should be true as well. So, take a scientific hypothesis (such as ‘evolution is true’, or whatever), and a prediction that the theory makes (‘there will be bones of ancient creatures buried in the ground’, etc). Here we have the conditional:

If evolution is true, then there will be bones of ancient creatures in the ground.

Now we make a measurement, let’s say by digging in the ground to see if there are any bones there, and let’s say we find some bones. So the consequent of our conditional is true. The claim by the apologists is that when a scientist uses this measurement as support for the hypothesis, they are committing the fallacy of affirming the consequent, as follows:

  1. If evolution is true, there will be bones of ancient creatures in the ground
  2. There are bones of ancient creatures in the ground.
  3. Therefore, evolution is true.

This is the sort of reasoning that is being alleged to be constitutive of the scientific method, and, as it is stated here, it is an example of affirming the consequent.

The problem with this line of thinking is not that it isn’t fallacious (it is clearly fallacious), but it’s that it is not what goes on in science.

3. Induction

In 1620, Francis Bacon published a work of philosophy called the ‘Novum Organon‘ (or ‘new tool’), in which he proposed a different type of methodology for science than the classical Aristotelian model that came before (Aristotle’s collected scientific and logical works had been collected together under the title ‘Organon‘). One way of characterising the Aristotelian method was that one does science by applying deductive syllogistic logic to ‘first principles’ (which are synthetic truths about the world). An example of this sort of first principle in Aristotelian physics might be that all things seek their natural place. It is of the nature of earth to seek to be down, and air to seek to be up, etc. This is, supposedly, why rocks fall to the ground, and why bubbles raise to the surface of water.

Part of Bacon’s dissatisfaction with this idea is that it provides no good way of discovering what the first principles themselves are; it just tells us what to do once you have them. Aristotle’s own ideas about how one discovers first principles are not entirely clear, but it seems that he thinks it is some kind of rational reflection on the nature of things which gets us this knowledge. Regardless, Bacon’s new method was intended to improve on just that, and is explicitly designed as a method for finding out what the features of the world actually are, of discovering these synthetic truths about the world. His precise version of it is a bit idiosyncratic, but essentially he advocated the method of induction.

Without going into the details of Bacon’s method, the idea is that he was making careful observations about the phenomenon he wanted to investigate, say the nature of heat, trying to find something that was common to all the examples of heat. After enough investigation the observation of a common element begins to be reasonably considered as not just a coincidence but as constitutive of the phenomenon under question. (He famously carried out just such an investigation into the nature of heat and concluded that it was ‘the expansive motion of parts’, which is actually pretty close to the modern understanding of it.)

In other words, starting with a limited number of observations of a trend, we move to the tentative conclusion that the trend is in fact indicative of a law. So the general pattern of reasoning would be that we move like this:

  1. All observed a‘s are G
  2. Therefore, all a‘s are G

The qualification of ‘all observed’ in premise 1 does most of the work in this argument. Obviously, just observing one a to be G would not count as much support for the conclusion. Technically, it would be ‘all observed’ a‘s, but it wouldn’t provide much reason to think that the conclusion is true. In order for the inductive inference to have any force, one must try to seek out a’s and carefully test them appropriately to see if they are always G’s. One must do an investigation.

So if we make a careful and concerted effort to investigate all a’s we can, and each a we come across happens to be G, then as the  cases increase, we will become increasingly confident that the next a will be G (because we are becoming increasingly confident that all a‘s are G). This is inductive inference.

With an inductive argument of this form, it has to be remembered that the conclusion does not follow from the premises with deductive certainty. Rather than establish the conclusion as a matter of logical consequence from the truth of the premises, an inductive argument makes a weaker claim; namely that the truth of the premises supports the truth of the conclusion; the truth of the premises provides a justification for thinking that the conclusion is true, but not a logically watertight one. Even the best inductive argument will always be one in which the truth of the premises is logically compatible with the falsity of the conclusion. The best one can hope for is that an inductive argument provides very strong support for its conclusion.

3. Induction affirming the consequent?

It is this inductive type of argument which the apologetical critique above is trying to address, it seems to me. They are saying that this type of scientific argument is really of the following form:

  1. If all a‘s are G, then all observed a’s will be G               (If p, then q)
  2. All observed a’s are G                                                         (q)
  3. Therefore, all a‘s are G.                                                      (Therefore, p)

Notice that the 2nd premise and the conclusion (2 and 3) is precisely the inductive argument from above; we have just added an additional premise (1), the conditional premise, onto the inductive argument. This fundamentally changes the form of the argument. Now the argument has the form of the deductively invalid argument ‘affirming the consequent’.

There are three problems with this as a critique of scientific inferences. Firstly, we have added a premise to an already deductively invalid argument, and shown that the result is deductively invalid, which is kind of obvious. Secondly, it characterises scientific inferences as a type of deductive inference, when there is good reason for thinking that they are not (at least if scientific inferences are supposed to discover synthetic truths about the world). Lastly, the addition of the first premise seems patently irrational, and obviously a perversion of normal inductive arguments. Let’s expand on each of these three problems:

Firstly

All the apologetical critique has demonstrated is that one can make a fallacious deductive argument by adding premises to an inductive argument. However, inductive arguments are already deductively invalid. There is a fallacy called the inductive fallacy. It consists of taking an inductive inference to be deductively valid. So if you thought that all observed swans being white logically entailed that all swans are white, then you have committed the inductive fallacy, because you would have mistaken the relation between the premise and the conclusion to be one of deductive validity, when it is merely that of inferential support. All observed swans being white does provide some reason to think that they are all white, but the fallacy is in thinking that it alone is sufficient to establish with certainty that they are all white.

The addition of the first premise does nothing to undermine an inductive inference. It doesn’t make it more fallacious than it was in the first place. In a sense, this analysis commits the essence of the inductive fallacy, in that it says that scientific inferences are deductive when they are not; the claim that scientific inferences are guilty of affirming the consequent is itself an instance of the inductive fallacy.

Secondly

We could, if we wanted to, add premises to an inductive argument to make it deductively valid, as follows:

  1. If all observed a’s are G,  then all a‘s are G        (If p, then q)
  2. All observed a’s are G                                             (p)
  3. Therefore, all a‘s are G.                                          (q)

Now the addition of the first premise has made the argument deductively valid, as it is just an instance of modus ponens.

The apologists were reconstructing scientific inferences as fallacious deductive arguments. Yet, even if we patched up the argument, as above with a deductively valid version of the inference, we still face a problem. This is that now we have a deductive argument, just like with Aristotle’s methodology. The very same reasons would remain for rejecting it, namely that as a methodology it provides no new synthetic truths; it only tells you what follows from purported first principles, not what the first principles are. We would be back to Aristotle’s dubious idea of introspecting to discover them. Thus, it isn’t desirable in principle to reconstruct an inductive argument as a deductive argument – even if the result is deductively valid. This means that the claim, that scientific reasoning is a failed attempt at being deductively valid, is implausible; even if scientific reasoning succeeded in being deductively valid that would be no help. The lesson is that they are a different type of inference, not to be judged by wether they are deductively valid or not.

Thirdly

Our original inductive argument went from the premise about what had been observed to what had not been observed. The whole point of inductive arguments is to expand our knowledge of the world, and so this movement from the observed to the unobserved is crucial. It is essentially of the form:

The observed a‘s are G ⇒ All a‘s are G

However, the first premise of the affirming the consequent reconstruction gets this direction of travel the wrong way round. They have it as:

All a‘s are G ⇒ the observed a‘s are G

If we keep clearly in mind that the objective of the scientific inference is to expand our knowledge, the idea of starting with the set (all a‘s) and moving to the subset (the observed a‘s) is weird. How could it expand our knowledge to do so? It is an inward move. This conditional though has been added to an inductive inference by our apologetical friends as a way of forming the ‘affirming the consequent’ fallacy out of an inductive inference.  But given that it gets the direction of travel exactly backwards, why on Earth would anyone ever accept this as a legitimate characterisation of a scientific pattern of reasoning?

This last concern highlights the cynicism inherent in the affirming the consequent critique. It isn’t a way of honestly critiquing a problem in science, but just an instance of gerrymandering an inductive inference, i.e. the change has been made just for the purposes of making the inference look bad, rather than as a way of highlighting a genuine issue. There is no independent reason for adding it on.

4. Or is it?

It might be claimed that I am pushing this objection too far. After all, there is reason to add the conditional premise on to the inductive inference. This is because theories make predictions. If a theory is true, then the world will have certain properties. And we do find examples of experiments being done in which the positive test result is used as a way of confirming the theory. And if this is right, then it looks like a conditional, and we are saying that the antecedent is true because the consequent is. So are we not back at the original motivation for the affirming the consequent critique?

Well, no. We are not. Here’s why. Let’s take an example. The textbook example. In Einstein’s general relativity, one of the many differences with classical Newtonian physics is that gravity curves spacetime. That means that there would be observable differences between the two theories. One such situation is when the light from a star which should be hidden behind the sun is bent round in such a way as to be visable from Earth:

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We already knew enough about the positions of the stars to be able to predict where a given start would be on the Newtonian picture, and the details of the Einsteinian theory provided ways to calculate where the star would be on that model. So, the Newtonian model said the star would be in position X, and the Ensteinian theory said it would be in position Y.

These experiments were actually done, and the result was that the stars were measured to be where Einstein’s theory predicted, and not where Newton’s theory predicted.

Is this an example of the affirming the consequent fallacy? It might look like it. After all, it may well look like we were making this sort of argument:

  1. If general relativity is correct, then the star will be at X  (if p, then q)
  2. The star is at X                                                                            (q)
  3. Therefore, general relativity is correct.                                (p)

However, the real development was not that general relativity was confirmed when these measurements were made, but that Newtonian physics was falsified. Corresponding to the above argument we have a different one:

  1. If Newtonian physics is correct, then the star will be at Y (If p, then q)
  2. The star is not at Y                                                                       (~q)
  3. Therefore, Newtonian physics is not correct.                        (~p)

The first argument is a logically invalid deductive argument; it is affirming the consequent. But the second argument is just modus tollens (If p, then q; ~q; therefore, ~p), and that is deductively valid.

What we learned with the measurement of light bending round the sun was not that general relativity was true as such, but that Newtonian physics, and any theory relevantly similar to it, was false. General relativity may still be false, for all the experiment showed us, but it showed us that whatever theory it is that does correctly describe the physics of our universe is going to be more along the lines of general relativity than Newtonian physics. We learned something about the world, even if we did not confirm with complete certainty that relativity was true. And this is what scientific progress is like.

5. Conclusion

It would be affirming the consequent if someone thought that the positive measurement deductively entailed that general relativity was true. If any scientist has gone that far, then they are mistaken. It doesn’t mean that the scientific method itself is mistaken however.

Revenge

0. Introduction

Recently, I had a conversation with my friends Matt Dillahunty and Ozy about philosophy. At about the 1:25:00 mark (the link above should be timestamped), we started talking about how there may be considerations which lead philosophers to rationally question the basic ‘laws of logic’, such as the law of non-contradiction (for all p: ~(& ~p)) and the law of excluded middle (for all p: (∨ ~p)). I brought up the liar paradox, as an example of this sort of thing. Matt objected that it is actually an instance of a ‘gappy’ sentence, which is neither true nor false. At the time, I knew there was a phenomena called ‘revenge’ which poses big problems for this strategy, but annoyingly I couldn’t bring the details to the bit of my brain that makes my mouth work. Here I want to right that wrong.

  1. The Liar

The liar sentence is of the following form:

a) This sentence is false.

The issue with a) is that it leads to a contradiction.

We only have to assume what seems like a very natural assumption about how the word ‘true’ works to get there. This is that if p is true, then p. We can think of this principle like this; if I say that it is daytime, and if what I say is true, then it is daytime. Alternatively; if I say a declarative sentence, and if it is true, then what it says correctly describes the thing that the sentence is about. This seems to be at the very core of idea of truth.

A corresponding idea is there for ‘false’ as well; if p is false, then it is not the case that p. If I say that it is daytime, and if what I say is false, then it is not daytime. If I make a declarative sentence, and it is false, then it incorrectly describes the thing that the sentence is about.

So let’s apply these principles to a):

If a) is true, then a) correctly describes what it is about. But a) is about itself, and it says about itself that it is false. So if it is true, then it correctly describes itself as false. So if it is true, it is false. And that is a contradiction.

So maybe a) is false. And if a) is false, then it incorrectly describes itself; yet what it says about itself is that it is false. If its self description is incorrect, then it isn’t false; and the only other option is that it is true. So if it is false, then it is true. Contradiction again.

So if it is true, it’s false; but if it is false, it’s true. Either way you go, you run into a contradiction. This is the paradox.

2. Gaps.

Yet, maybe there is a solution here. Matt certainly proposed a solution here. His idea was that a) is neither true nor false. So, let’s run through the options and see how it works.

a) says about itself that it is false. And we are now saying that it has no truth-value at all. Well, it certainly doesn’t correctly describe itself, because it says that it is false, and it is ex hypothesi neither true nor false. If something is neither true nor false, then it is not false. So it’s own self-description fails. This seems to leave no reason to consider it true. It says about itself that it is false, but we cannot derive that it is true. So far, no contradiction.

But, it says about itself that it is false, and this is incorrect (because being neither true nor false, it is not false). And it’s hard to see why this wouldn’t count as a case of a falsity. After all, it says that it is false, yet (ex hypothesi) it isn’t (because it’s gappy). It is certainly not true that it is false; it’s own self-description fails. But does this mean that it is false? Well, only if ‘not true’ means false. And, on this assumption, where we have some sentences which are ‘gappy’ (i.e. neither true nor false), there is a difference between being not-true and being false. If we listed all the not-true sentences, it would include all the ones which had no truth-value, and all the ones which were false. Thus, being not-true does not entail being false. Thus, we seem to have got out of the trap.

It is neither true nor false, and when it says about itself that it is false we can consider it’s incorrect self-description to be a case of being not-true, rather than false.

Strictly speaking, this does work as a consistent (i.e. contradiction-free) way to think about a).

So far, so good. However, things are not over. There is a second round.

3. Revenge

Consider the ‘strengthened liar’ sentence:

b) This sentence is not true.

We have, on our assumption of ‘gappyness’, three options. Either b) is true, or it is false, or it is neither true nor false. Let’s take them one at a time:

If b) is true, then it correctly describes itself. Yet it says about itself that it is not true. So if it is true, it is not true. This is a contradiction.

If b) is false, then what it says about itself is incorrect. Yet, if it is false, then it does come under the category of not-true, which is what it says about itself. So if it is false, then what it says about itself is correct, and so it is true. And we have another contradiction.

The only other option is the one we used for a), which is that it is neither true nor false. Yet, if b) is neither true nor false, then it is in the not-true category as well (because anything which is neither true nor false is not true). But, as it says about itself that it is not true, it would seem like it has correctly described itself. If it has correctly described itself, then it is not in the not-true category, but in the true category. So if b) is neither true nor false, then it is true! This is, again, a contradiction.

So, while the gappy proposal got rid of one liar sentence (i.e. a)), it fails to help us with another one (i.e. b)). As a strategy, gappiness won a battle, but it loses the war.

4. Conclusion

The problem that the liar paradox presents is subtle, and still an open question in philosophy and logic. It may be that a solution to the generalised problem exists which involves adopting a logic which has truth-value gaps. That may be the case for all I know. But it seems clear that simply adopting truth-value gaps does not solve the underlying phenomenon. It merely pushes the problem to somewhere else. Even if a) can be got around by postulating truth-value gaps, b) cannot be. The liar paradox has had its revenge.

As the philosopher Tyler Burge put it:

“Any approach that suppresses the liar-like reasoning in one guise or terminology only to have it emerge in another must be seen as not casting its net wide enough to capture the protean phenomenon of semantical paradox.” (Tyler Burge, Semantical Paradoxes, p. 173, (1979))