Do Logic and Existence Go Together?

It's said by some (or most) logicians that “logic must handle every possible state of affairs” and hence that it “can't imply the existence of anything” (Dale Jacquette). That almost sounds like a non sequitur. Yes, logic must handle “every possible state of affairs”. Nonetheless, how does it follow from this statement that logic can't imply the existence of anything? Why can't logic be able to handle every state of affairs and imply the existence of something (or one thing)?

Is it because if logic is applicable to everything, then implying the existence of something would pollute its ability to handle all states of affairs (note the jungle of quantifiers here)? Or is is it that the case that something (or these things) would somehow make logic contingent (or empirical) in nature? Nonetheless, implying (or allowing) the existence of something that's contingent (or empirical) isn't the same as arguing that logic itself is contingent (or empirical). Logic can still be applied to the the contingent (or empirical) even if isn't itself contingent (or empirical).

Does it mean, instead, that if logic implies the existence of anything (or even something) that it would somehow depend on that something? And, if logic did imply the existence of anything, then its logical purity would somehow be sullied?

In that sense, quantificational logic (or first-order logic) is far from being pure. Quantifiers in logic have existential import (or have ontological commitment). That is, a quantificational proposition is about the existence (or non-existence) of something (or of many things). Even free logic accepts abstract objects of various kinds. It can also be said that logical statements about self-identity have existential import. That is, the proposition (x) (x = x) has existential implications. And, more obviously, so too does, (∃y) (y = y).

It seems to follow from the acceptance of quantificational logic that an empty universe is excluded – nay, it's logically impossible. However, do these facts about quantificational logic necessarily apply to the more generic “logic” we began discussing? Perhaps quantificational logic is actually a deviant logic!

Lewis Carroll's Premises Paradox

The British writer, mathematician and logician Charles Lutwidge Dodgson (which was Lewis Carroll’s real name) worked in the fields of geometry, matrix algebra, mathematical logic and linear algebra. Dodgson was also an influential logician. (He introduced the Method of Trees; which was the earliest use of a truth tree.) For some time Dodgson was Mathematical Lecturer at Christ Church, Oxford University.

And, of course, Dodgson (under the name Lewis Carroll) also wrote Alice in Wonderland, Alice Through the Looking Glass and many other books and poems.

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Lewis Carroll (I’ll use Dodgson’s better-known pen name from now on) formulated his premises paradox in 1895. (The argument was advanced in Carroll’s paper ‘What the Tortoise Said to Achilles’; which was published by the journal Mind.) This paradox refers to the possibility of infinite premises being required in order to reach a single conclusion.

Firstly, two or more premises are usually linked to a conclusion in a logical argument.

So how are they linked?

Now that can be a question of the philosophical nature of the link between premises and conclusion: whether it’s an example of entailment (or logical consequence), implication (or material implication) or whatnot. However, that wasn’t the point that Lewis Carroll was making. He was making a purely logical (not a philosophy-of-logic) point.

To put this simply: in order to justify (or explain) how any given premises are related to a conclusion (or how the premises entail or imply the conclusion), then a further premise will need to be brought into the argument in order to do so. Moreover, another premise will be required in order to tell us (or show us) how (or why) it’s the case that if the premises are true, then the conclusion must follow and also be true.

Now that added explanation (or justification) will itself be a premise within the argument.

Of course the same problem will repeat itself.

Just as we brought in a premise to link two, three or more premises to a conclusion: now we have to say how this new premise is itself linked to those previous premises. Or, alternatively, we’ll need to know how all the premises (when taken together) are linked to the conclusion.

A solution has been offered to this logical paradox.

That solution is simply to say that no added premises are needed in the first place. That is, the link between the premises and the conclusion doesn’t need to be explained (or justified) by a further premise.

So why is that?

The (or one) answer is to say that the premises and conclusion are simply linked by a rule of inference. That rule itself explains (or shows) the relation between the premises and the conclusion. Nonetheless, that sounds (at least at a prima facie level) like a cop-out. To simply say that premises are linked to a conclusion by a rule of inference doesn’t appear to be saying anything… much. Surely that rule -again! — will need to be explained by a further premise.

It can also be argued that saying that premises are linked to a conclusion by an inference rule is itself a further premise.

Yet that’s unless that rule of inference somehow works without any justification or explanation.

In other words, it’s not a justification: it’s just a rule. It’s something that “can be shown but not said” (to use Wittgenstein’s much-quoted phrase); just as the nature of the logical constants can’t be said, only shown.

Does the Mind-as-Computer-Programme Idea Support Mind-Body Dualism?

“Yes, I am a confirmed "dualist" at this point in my journey, i.e., I understand my mind to be separate from my body/brain, analogous to the way an application program and its data are separate from the computing infrastructure they depend on.” - Chasw

Does believing your mind to be separate from your body/brain automatically mean you're a dualist? That would depend on what the word 'separate' means. The mind simply being (fundamentally) different from the body wouldn't commit anyone to dualism.

Dualism is a commitment to the idea that mind and brain are fundamentally different ontological categories or 'substances'. That creates the problem of mind-body interaction. If mind and brain (body) were fundamentally different, then how could brain-to-mind and mind-to-brain interactions be explained?

Dualists are also committed to minds being separable from brains, not just being separate and different. I don't think the 

mind = programme/data 

and the 

brain = hardware

comparison works here because, after all, even programmes and data are physical. Programmes and data depend on physical syntactic devices which encode data/info and which themselves rely on physical processes to implement that data (as well as to send that data, electronically, around the hardware/infrastructure). The programme-hardware/mind-brain analogy works only to show us that the same programme can be instantiated in different kinds of hardware: from brains to computers to coke machines. In itself, it's not an argument for dualism; which is specifically about the mind being of a different substance/category than the brain. When it comes to computer programmes and their different hardware juxtapositions, there are no such deep ontological problems as there are for mind-brain/body dualism.

The Cartesian Circle

Rene Descartes' basic position can be summarized in the following manner:

 

i) I can only know (or be certain) as to whether or not some statement or proposition (p) is true if I already know (am certain of) that God exists and is not a deceiver.

 

ii) I can only know (or be certain) that (i) God exists and is not a deceiver if I already know (am certain of) that every proposition I clearly and distinctly perceive is true.

 

That encapsulates the "Cartesian circle".

In other words, i) depends on ii) and ii) depends on i). Neither i) nor ii) can get off the ground without the prior existence of i) or ii). Or, in other words, i) can't get off the ground without ii) and ii) can't get off the ground without i). It seems to follow, then, that neither can get off the ground.

 

In more detail, one would need to ask why Descartes has already assumed God's existence, let alone the fact that He's not a deceiver. It could be said that written into God's nature and existence, as it were, is the fact that He can't be a deceiver. But since Cartesian Doubt is supposed to begin at rock bottom, Descartes could hardly have assumed any of these things.

 

In terms of ii), we would need to know what Descartes mean by 'clear' and 'distinct'. How does clarity and distinctness guarantee truth? How did he know that he clearly and distinctly perceived certain propositions? And what did he mean by “perceive” here? (He wasn't talking here about external objects such as trees or chairs.)

 

Descartes, or at least philosophers defending Descartes, have said that metaphysical or logical certainly isn't required at the beginning of Cartesian Doubt. Or, more correctly, metaphysical or logical certainty isn't required to take one to the existence of God. Instead only psychological certainty is required: presumably to take you to God and be certain about the aforesaid propositions.

 

From psychological certainly you can reach metaphysical or logical certainty about both God's existence and the truth of clear and distinct propositions. This, of course, leaves out entirely what the nature of this psychological certainty is and whether or not it's meaty enough to take us to both God's existence and nature and the truth of these propositions.

 

It also leaves out the Cartesian circle: why talk about taking oneself to God's existence at all when God hasn't been proven in the first place? Surely even the assumption of God's existence can't be accepted in Cartesian Doubt. Yet here we are assuming God's existence in the very process of arguing about psychological certainly taking us to the existence of God. That means that God's existence is assumed not only before metaphysical certainty: it's also assumed before psychological certainty (whatever that is).

 

Nonetheless, philosophers have defined what metaphysical certainty is. This is one definition:

 

A proposition P is metaphysically certain if and only if there is no other proposition R that is a reason for doubting P.

Firstly, this definition is completely psychological in nature. The Cartesian wouldn't deny that. What I mean by that is that there could be a proposition R (or even many propositions) that would give S reasons to doubt P if he were aware of them. However, S isn't aware of them. In addition, R could be a logical reason to doubt P even though S doesn't take R as a logical reason to doubt P. Nonetheless, taken non-psychologically, R is logically a contradiction of P which S may not be aware of at all.

 

Indeed, in the post-Cartesian world, the very locution “metaphysically certain” has a strange psychological ring to it. Some philosophers or metaphysicians would ask: What has certainty (psychological or otherwise) got to do with metaphysical reality? In addition, R could contradict P without having any metaphysical reality as such; just as contradictions can be formulated in logical systems from symbols or formula which don't have any (metaphysical or empirical) content.

 

However, none of that is strictly relevant if all we're talking about is the Cartesian's move from psychological certainty to metaphysical certainty because that metaphysical certainty can be deemed to be just as psychological as the prior psychological certainty. That is, the metaphysical certainty that is derived from the psychological certainty is done so in an equally psychological manner – in what other manner could it be derived?

 

In an case, most of that is a sidetrack.

The Cartesian move is that, yes, we begin with only psychological certainty as to various clear and distinct propositions. The first thing we do with these psychologically certain propositions is prove that God exists. Then, with God's help, as it were, they become metaphysically certain as well. The question here, however, is that even if we accept psychological certainty, how is it, exactly, that such certainty “proves” God's existence? After all, psychological certainty alone can't prove God's existence or anything else for that matter. Only valid and true arguments can do that.

Then the Cartesian seems to move in another circle within the general Cartesian Circle. By proving God's existence from psychologically certain and therefore true propositions, we're giving a reason, apparently, for doubting all possible reasons for doubting those very propositions which prove God's existence. The propositions which took us to God are then blessed by God and therefore become certain. Thus:

 

God + psychological certainty = metaphysical certainty.

This could be seen as a virtuous, rather than a vicious, circle. Certain propositions lead us to God and God leads us the those very same propositions. And that, surely, is a classic case of the Cartesian Circle. I simply can't see how anyone has broken out of this very vicious Cartesian Circle. All that's has happened is we've expressed and codified it, not solved it.

Perhaps it all hinges on how psychologically certain propositions (if that's what they truly are) somehow “prove” God's existence. Clearly, simple psychological certainty of one or a hundred propositions can't prove anything, as I said. It all depends on what these propositions are. It also depends on whether or not these propositions assume the existence of God as well as the ability of God's existence to somehow give metaphysical legitimacy to the prior psychologically-certain propositions.

 

Another part of this Cartesian Circle is that reasons for doubting P also have to be certain. That is, not-P, or R, also has to be clearly and distinctly perceived. Clearly, if not-P is clearly and distinctly perceived to be true, then P is false, according to the Cartesian. This seems to imply that not-P or R is not clearly and distinctly perceived to be true. Therefore P must be true. Again, all this depends on what P is and what not-P (or R) is. In addition, the Cartesian requires more than one P and must defend itself against more than one not-P: there will be many not-Ps.

 

Of course one not-P is that God is a deceiver. Can I clearly and distinctly perceive that God is a deceiver? Not according to the Cartesian. (Really? Even if I'm an atheist?) That can't be true: there are many reasons for doubting clear and distinct propositions. One reason is their very existence as adumbrated by Descartes himself. Another thing we can doubt is that psychological certainty leads to metaphysical certainty, God's existence or anywhere else (as I have said).

 

Again, the Cartesian only accepts any not-P if it is certain. Alternatively, the Cartesian must be certain that not-not-P is certain (or -–P). Here again, and not surprisingly, it's all about psychological certainty, even if that can lead us to all sorts of other things. And, again, the only not-P that Descartes appears to consider is that God is a deceiver.