Owen Griffiths and A.C. Paseau. — Implication vs Inference

Griffiths, Owen and A.C. Paseau. One True Logic: A Monist Manifesto. Oxford: Oxford University Press, 2022.

 

"As just hinted, we take capturing implication and capturing reasoning as distinct applications. The implications of some premises are their logical consequences; they follow from them, whether or not one can deduce them from the premises. In contrast, an inference is what an agent does when she deduces a conclusion from some premises. Reasoning or inference tries to respect implication, though is distinct from it. Thus we write ‘implicational’ rather than ‘inferential’ whenever we are interested in what follows from what—as we typically will be—rather than what can be deduced from what. All this applies even to idealized notions of reasoning (which for example prescind from human subjects’ errors in reasoning). [Fn6: So long as the sense of reasoning/inference is not so idealized that it means nothing other than the ability to accurately reflect implicational facts.]

     It follows in particular that there is no reason to suppose at the outset that the correct foundational logic is completable by a (sound and effective) deductive system . . . Perhaps no deductive system can capture all logical entailments. (Incidentally, we use the words ‘implication’, ‘consequence’, and ‘entailment’ interchangeably, with the epithet ‘logical’ understood when omitted.) [Fn7: With the usual act/outcome ambiguity, resolvable by context; e.g. ‘entailment’ can mean the relation that holds between some premises and a conclusion, or the conclusion itself.] Implication is modelled by model-theoretic consequence (⊨) and derivability by deductive consequence (⊢)." (Prologue xx-xxi.)

Reductio ad Absurdum is preserved on glut theory

https://www.youtube.com/watch?v=xkMTC5TfpiE

One objection to dialetheism, or any kind of glut theory, is that by giving up the Law of Non-Contradiction* we give up the truth seeking tool of reductio ad absurdum. But this isn't true. In a reductio, we start with a conclusion, like God exists, and from there, combined with other necessary truths, conclude that God does not exist. If negativity and positivity cannot overlap, and thus all contradictions are strictly false, then showing that A leads to ~A shows that A is strictly false. Given glut theory, you lose the strictly part of the reductio. Showing that A leads to ~A, on glut theory, shows either that A is strictly false or that A is true and false.

 

But there never has been a case where someone has had orange juice in their fridge and also has not had orange juice in their fridge. There never has been a case where someone is both in Sidney, Australia and not in Sidney, Australia. There has never been a case where someone has both casted a ballot and not casted a ballot. Etc. It seems that for virtually any imaginable contradiction, there's no good reason to think that it's anything other than strictly false. If the only plausible candidates for true contradictions are found in self-reference, law, motion, mathematics, maybe quantum mechanics, and maybe ineffability, then that leaves nearly all imaginable contradictions strictly false. So showing something to be a contradiction is to show that it's a priori almost certainly false. The theist who wants to escape the above reductio's conclusion that God does not exist is forced to refute the reductio or argue that God's existence is a true contradiction – that it's true and false that God exists. But 1) the vast majority of theists would not feel comfortable with this conclusion (probably because their intuitions say that it's impossible for something to be true and false), and 2) this is quite the burden to bear for the theist, considering that God's existence is not found in the list of plausible dialetheias. 
 
So reductio ad absurdum is still a powerful tool on glut theories, and indeed it retains nearly all of its strength. In fact, reductio could be a key tool in discovering further plausible dialetheias. First, use reductio to establish a contradiction. Second, argue that this contradiction is a dialetheia. So while I don't subscribe to any glut theory, the "loss of reductio ad absurdum as a truth seeking tool" objection to glut theory is not a viable objection at all.
 
*Classical logic is typically associated with three laws: The Law of Non-contradiction, the Law of Excluded Middle, and the Law of Identity. You might also include the Law of Bivalence. But why can't we simply reduce all of that to the Law of Excluded Middle? 
 
LEM: All propositions are either true or false.  
 
This is equivalent to: For any proposition, either it is true or its negation is true.
 
∀x(Px⟶Tx∨Fx) "For all x, if x is a proposition then x is true or false."
 
But if something is true or false, then it's a proposition. So:
 
∀x(Tx∨Fx⟶Px)

 

So: ∀x(Px⟷Tx∨Fx)
 
But this means that something is a proposition if and only if it is true or false. But that's not right. A proposition is more than just being something that is true or false. A proposition will include other elements like bundling references to properties and relationships between properties (or whatever your favorite theory of propositions says).

 

Easy fix: ∀x(Px⟷Hx∧(Tx∨Fx))

Something is a proposition if and only if it has [insert preferred theory of propositions here] and is true or false. This assumes acceptance of LEM and separates out the LEM from the other parts of the theory of propositions.

 

On the theory I'm playing around with, a proposition is true when it is comprised of references to properties and relationships between properties that we experience or that explains our experiences or can explain someone's experiences.

 

So: ∀x(Rx⟷Tx) = If and only if when something is comprised of references to properties and relationships between properties that we experience or that explains our experiences or can explain someone's experiences, then it is true.

 

∀x(~Rx⟷~Tx) = If and only if when something is not comprised of references to properties and relationships between properties that we experience or that explains our experiences or can explain someone's experiences, then it is not true. 

 

∀x(Qx⟷Fx) = If and only if when something is comprised of references to properties and relationships between properties that we do not experience and that do not explain our experiences and cannot explain anyone's experiences, then it is false.

 

∀x(Fx⟶~Tx) = When something is false, it is not true.

 

∀x(Fx⟶~Rx) = When something is false, it is not comprised of references to properties or relationships between properties that we experience or that explains our experiences or can explain someone's experiences.
 
A rock is not true, because rocks are not propositions and only propositions can be true, but a rock is not false, because only propositions can be false. So 'not true' and 'false' are not equivalent, but are mutually entailing when it comes to propositions.  
 
This by itself entails a) there are no gaps, as no propositions can be neither true nor false; b) there are no gluts, as no propositions can be both true and false; c) that A=A (because truth and falsity are assumed to be different things) and d) that there are only two truth values: true and false (otherwise, the law would read: Propositions are true or false or a secret third thing, with that third [or fourth, fifth, etc.] thing spelled out by a many-valued logic).

Non-overlap theory has greater explanatory power

Non-overlap: Truth and falsity are inverse operations that cancel each other out and do not and cannot overlap. You can no more have a proposition that is both true and false than you can have a number that is both positive and negative. This theory is in direct opposition to any glut theory, including dialetheism, which will say that truth and falsity can overlap and are not inverse operations that cancel each other out.

 

Support for Non-overlap: This theory explains, where gut theories fail to explain, the data of my intuitions (and I bet the intuitions of most people) surrounding the impossibility of everyday contradictions. If there is orange juice in my fridge, this is not merely strong evidence that it is not also the case that there is no orange juice in my fridge. Rather, this is proof that it cannot also be the case that there is no orange juice in my fridge. If a barber has shaved himself, this is proof that he has not also not shaved himself. If a person has casted a ballot, this is proof that they cannot also have not casted a ballot. And so on.

 

But why would the positive be a proof of an absence of a negative in these cases? Why do I feel so forced to believe that positivity entails an absence of negativity in these cases? Suggestion: Because Non-overlap is true and positivity and negativity are inverse operations and where truth picks out positivity, falsity picks out negativity.

 

Graham Priest says accounts of negation are contentious. (See: https://plato.stanford.edu/entries/dialetheism/#ArguNega.) Are they? Isn't Non-overlap the common account of negativity?

First Pass: Liar Paradox

I saw someone react to the Liar Paradox on YouTube so I'd thought I'd share some old thoughts of my own that form something of a first impression or first pass at the problem. I haven't read much on this topic, so I still have much to learn.

 

Part 1: Start with the Barber Paradox

 

A barber shaves all and only those people in the town that don't shave themselves. Question: Does the barber shave himself? If the barber shaves himself, then the 'only' part of the rule is violated, because the barber only shaves those people who don't shave themselves. If the barber does not shave himself, then the 'all' part of the rule is violated, because the barber shaves all people who don't shave themselves. So either way, the rule will be violated. The rule is impossible.

 

If we force the rule to go through, then the barber does and does not shave himself. Contradiction. Though there is another option: Keep the rule, but it applies to everyone except the barber. So we end up with the following inconsistent set:

 

1) Self-reference;

 

2) The Rule;

 

3) The Law of Non-Contradiction.

 

You can only keep two out of these three. One of them has to go.

 

Option A: If you keep 2 and 3, then self-reference is impossible.

 

Option B: If you keep 1 and 3, then the rule is impossible.

 

Option C: If you keep 1 and 2, then there are true contradictions.

 

Part 2: Option C doesn't work 

 

I have the following issue with Option C: You can verify that it doesn't work! At the end of the day, the barber will have either shaved himself or not. That's undeniable. So no matter what, Option C doesn't work, at least not in this case. But whatever necessary truths there are that would ground (or constitute) the necessary truth that Option C doesn't work, I don't see why those same necessary truths wouldn't apply in parallel cases.

 

So let's consider two parallel cases: Set Theory and Propositions. These are known as Russell Paradoxes after Bertrand Russell.

 

Part 3: Russell Paradoxes 

 

Set Paradox: There is a set that contains all and only those sets that don't contain themselves.

 

This is why it's nice to start with a concrete example like the barber paradox. With the set paradox, it's harder to see that Option C must be rejected. But if we can and must reject Option C in the barber case, I don't see why we shouldn't do the same in this case. So again, either the set does not self-refer OR the set's rule is not true because it's impossible. Just as there can be no self-referenced barber that satisfies the barber rule, for the exact same reason there can be no self-referenced set that satisfies the set rule. The barber is impossible, and so is the set. What's good for the goose is good for the gander.

 

Proposition Paradox: There is a proposition about all and only those propositions that aren't about themselves.

 

Same deal. Thanks to the barber paradox we know that Option C doesn't work. And there are no relevant differences between the barber paradox and the set or proposition paradoxes. At least, this is true if we discover the principle behind the impossibility of Option C for the barber case and see that it applies just as well to these parallel cases. 

 

Part 4: The principle behind the impossibility of Option C 

 

The question now is, why is it the case that Option C is impossible? I can see that it is the case that Option C is impossible, because I can see that at the end of the day the barber will have either shaved himself or not, which means that either the rule has been violated or that the rule doesn't apply to the barber. But why is it that I can see such a thing? What makes Option C impossible?

 

Here's one explanation: Option C is impossible because true contradictions are impossible; the Law of Non-Contradiction is necessarily true.

 

Part 5: Option A (eliminate self-reference) or B (eliminate the rule), or leave it open?

 

Now how do we decide whether to go for Option A and say that self-reference is impossible, or B and say that the rule is impossible? One option would be to leave it open and say you can pick either one.

 

But we can force the issue by forcing self-reference:

 

Barber*: There is a barber that, including himself, shaves all and only those people in the town that don't shave themselves.

 

Set*: There is a set that, including itself, contains all and only those sets that don't contain themselves.

 

Proposition*: There is a proposition that, including itself, is about all and only those propositions that aren't about themselves.

 

Now my hand is forced and there is only one option: Option B. The rule itself is impossible.

 

It doesn't seem like there is a similar way to force the rule. Consider:

 

Barber**: There is a barber that, including himself, shaves all and only those people in the town that don't shave themselves, and this rule is necessarily true.

 

This doesn't force my hand in a similar way, because the rule is still just false. By tacking on the "necessarily true" part at the end, you're just adding another false bit to the rule.

 

But if self-reference is impossible, then the "including himself" bit is impossible. If that bit is included in the original rule, then you'd have a case where both self-reference and the rule are impossible, which amounts to choosing both Option A and Option B, which is not possible.

 

You could say that the "including himself" bit adds a second rule, and so Option A involves denying the self-reference rule while Option B involves denying the original rule.

 

So as far as I can tell, Option A and Option B are both available. Insofar as self-reference is necessary, then Option B is selected, and insofar as the original rule is necessary, then Option A is selected. But I don't see a way to force the truth of either the self-reference rule or the original rule. A couple thoughts:

 

Thought a) Intuitively, it feels like self-reference is easier to accept over the rule, because the rule itself is ad hoc while self-reference is a more stable or familiar concept. But I don't have a strong enough grasp of this to make an argument for treating self-reference as necessary.

 

Thought b) You could combine the self-reference rule with the original rule to get one rule with three components: the self-reference component, the 'all' component, and the 'only' component. This changes the Options from above to what's below:

 

Option A*: Deny the rule by eliminating the self-reference component.

 

Option B*: Deny the rule by eliminating the 'all' component.

 

Option C*: Deny the rule by eliminating the 'only' component.

 

Option D: Accept all components of the rule and reject the Law of Non-Contradiction.

 

Just as with Option C, Option D is necessarily rejected in the barber case. The only explanation I can think of that explains why Option D is impossible is because contradictions are impossible. But if the impossibility of contradictions is what explains the necessary rejection of Option D, then that same explanation carries over to parallel cases.

 

There is a theoretical Option E, which is to accept A*, B*, and C* and deny all components of the rule. But that would fail to accurately describe what happened to the barber. The options can be framed as descriptions:

 

Description A: The barber successfully shaves all and only those people who don't shave themselves other than himself. The barber is an exception to the rule because self-reference is impossible in this case, because self-reference in this case entails a contradiction and contradictions are impossible, and anything that entails an impossibility is itself impossible.

 

Description B: The barber shaves all and only those who don't shave themselves, including himself. The barber does not shave himself. So really, he shaves almost all and only those people who don't shave themselves.

 

Description C: The barber shaves all and only those who don't shave themselves, including himself. The barber shaves himself. So really, he shaves all but not only those who do not shave themselves.

 

Description D: The barber shaves all and only those who don't shave themselves, including himself. The barber is both shaved and unshaved in the same sense at the same time.

 

(I suppose there is a bonus option: The barber is excluded from the category 'people' so that he does indeed shave all and only those people who do not shave themselves. But I take that to be clearly false: the barber is necessarily a person.)

 

Description D necessarily gives a false description. But do any of the other descriptions necessarily give a false description?

 

Following 'Thought a)', you could argue that as a matter of description, self-reference is taking place. You could imagine a kind of phenomenal self-reference in the form of an intentional self-reference. The barber says to himself, "I am going to shave all and only those people who don't shave themselves, including myself." There is now a mental, intentional moment of self-reference. Because of that, if you describe the situation as lacking self-reference, your description is incorrect. But does an intention of self-reference necessitate self-reference? You could describe the situation in terms of the barber intending and failing to self-reference. 

 

Purely intuitively, I suspect there's a way to get necessary self-reference out of the description, which would mean that Option A gives a necessarily false description. For example, if intentional self-reference entails necessary self-reference, then you can have your necessary self-reference at least that way. But I don't have a way to show this at the moment. So as far as I can tell, Description A, B, or C can apply to the barber paradox, the set paradox, and the proposition paradox.

 

Part 6: The Liar Paradox

 

Liar: This sentence is false.

 

The Law of Non-Contradiction explains why Description D is necessarily false. So if we take the idea seriously that there are no true contradictions, what becomes of the liar paradox? I see a number of options:

 

L1: Failed Reference Problem: 'This sentence' fails to refer to anything, because, like with Option A above, self-reference is impossible here because it entails a contradiction. So the liar is claiming that some indeterminate thing is false. But indeterminate things cannot be false. So the Liar is false.

 

Or, because 'This sentence' fails to refer to anything, the Liar sentence fails to be about anything, and thus fails to express a proposition, because propositions are necessarily about something.

 

L2: Fragment Problem: 'This sentence' refers to the literal sentence fragment 'this sentence', in which case the liar is saying that the sentence fragment 'this sentence' is false. But sentence fragments do not express propositions, and only propositions and their expressions can take on a truth value. So the Liar is false.

 

L3: Nesting Problem: 'This sentence' refers to the sentence 'This sentence is false.' But that means we can substitute any instance of 'this sentence' with 'This sentence is false.' But that means we have an infinite regress. When we unpack it once, we get: This sentence is false is false. When we unpack it again, we get: This sentence is false is false is false. When we fully unpack it, we get an infinite number of "is false" phrases at the end, and so the sentence never completes. Assuming propositions must complete, the Liar fails to express a proposition and so is neither true nor false.

 

L4: Aboutness Problem: This sentence is false. False about what? False about its being false. False about its being false about what? False about its being false about its being false. False about its being false about its being false about what? Again, this goes on forever, and the sentence has this infinite regress of intentionality. But propositions don't have regressing intentionality. So the Liar fails to express a proposition.

 

This applies to the 'Truth Teller': This sentence is true. True about what?

 

L5: The Meta Problem: For every proposition there is a meta proposition about that proposition. For example:

 

Proposition: The earth revolves around the sun.

 

Meta proposition: <The earth revolves around the sun> is true.

 

Meta meta proposition: [<The earth revolves around the sun> is true] is false.

 

Meta meta meta proposition: ([<The earth revolves around the sun> is true] is false) is true. Or: It's true that it's false that it's true that the earth revolves around the sun.

 

We can call these meta(1), meta(2), and meta(3). The truth value of a meta(1) proposition depends on the truth value of the corresponding proposition. That dependence relationship only goes one way; propositions can never have a truth value that depends on the truth of the meta proposition.

 

If 'This sentence is false' is translated as 'The meta proposition of this sentence is false', then the Liar is either false because it claims there is a false meta proposition when there isn't one, or fails to express a proposition because propositions cannot have their truth value depend on their meta proposition.

 

So it seems to me there is at least one way, if not several ways, that the Liar fails to express a proposition, or expresses a false proposition. I don't see any need to give up the Law of Non-Contradiction.

 

Part 7: Paradoxes of Law

 

While I'm at it, why not address paradoxes of law.

 

The following is my paraphrase of Kane B's formulation, wherever he got it from, from one of Kane B's videos:

 

Sidney is an Australian aboriginal. Like many aboriginals, she is a housekeeper. The owner of the house becomes fond of her and writes her into his will. After he dies, Sidney becomes the owner of his land.

 

There are two Australian laws:

 

Law(1): If you own land, you have a right to vote.

Law(2): If you are an aboriginal, you do not have a right to vote.

 

Normally, aboriginals cannot own land, as they are not allowed to purchase it. And normally, no one gifts land to aboriginals. But in this unique case, Sidney is both a landowner and aboriginal, and so Sidney both has and does not have a right to vote.

 

Because a right is a legal status granted by law, it's undeniable that Sidney has the legal status 'Can Vote' and it's undeniable that Sidney does not have the legal status 'Can Vote'. So we have a true contradiction: It's both true and false that Sidney has a right to vote.

 

But we see a similar problem as with the barber paradox: At the end of the day, it's undeniable that Sidney will either be allowed to cast a ballot or not. If Sidney is allowed to cast a ballot that counts, then she has a right to vote. If Sidney is forbidden from voting, or if her vote is tossed on account of her being aboriginal, then she does not have a right to vote.

 

What would be impressive would be showing Sidney casting a vote and not casting a vote in the same sense at the same time. But that can't be shown, because it's impossible, because contradictions are impossible. Instead, all we have is this suspicious notion of "legal status". 

 

I don't know how jurisprudence systems interpret the notion of 'legal status' or how they interpret what laws are really saying or doing. At the very least, we can say that if a law makes a false claim or implies acceptance of a false claim, then that doesn't make that claim true.

 

If we interpret the laws as saying:

 

Law(1*): If you own land, you have a right to vote. That is, you can cast a vote without being physically stopped or you can vote without having your vote tossed.

Law(2*): If you are an aboriginal, you do not have a right to vote. That is, you will be physically stopped from casting a vote, or if you do vote your vote will be tossed.

  

Now it's undeniable that one of the laws will be proven false when Sidney goes to vote. 

 

Instead of interpreting rights as these magical legal statuses that people have, if rights are interpreted as descriptions of what will happen given varying circumstances or how the state will treat a citizen, then the paradox resolves. 

 

Another interpretation:

 

Law(1**): If you own land, you have a right to vote. That is, you can vote without violating any laws. 

Law(2**): If you are an aboriginal, you do not have a right to vote. That is, there are no laws granting you the right to vote.

 

In this case, both laws are false. Sidney cannot vote without violating any laws and there is a law granting Sidney the right to vote. So both laws contain falsehoods with Sidney acting as the exception to both laws.

 

Another interpretation of law is that laws are something like performative commands under fictional canons. So if I say "Harry Potter defeated Voldemort", I say something true relative to a fictional canon, but not true in any absolute sense. If I write my own fanfic where "Voldemort defeated Harry Potter" is true, then that claim is true relative to a fictional canon as well. But that doesn't mean there's a true contradiction that Harry Potter defeated and failed to defeat Voldemort.

 

So Sidney could have a right to vote according to a canon established by one law, and not have a right to vote according to a canon established by another law. But just like fictions, there is no absolute truth to either canon. I'm aware of Kathleen Stock's idea of collective fiction. So law might work in that way, as a collective fiction. State and country borders, and legal entities being treated as persons, are examples of legal fictions. Legal fictions correspond to social facts, facts about how people behave as if something is true. So while there is no objective border that delineates one state from another, we will still behave as if such a border exists. Likewise, while there is no objective fact that someone has or does not have a right, we will behave as if that person has or does not have that right.

 

Another way to impress me would be this: When lawmakers discover that Sidney's situation creates a contradiction within the law, they leave the contradiction. After all, if contradictions are not fundamentally problematic, lawmakers can just leave the contradiction there. Sure, it's true that Sidney has a right to vote. And sure, it's true that she doesn't. What's the problem? Haven't you heard, there are true contradictions, and this is one of them. Of course, that's not what happens. Lawmakers or a judge would issue a ruling and resolve the contradiction, either by having one law trump the other or by changing the laws. If laws are meant to dictate behavior, then it is exactly because we cannot behave as if a contradiction is true that is why the contradiction cannot be left in the law. The idea that contradictions are fundamentally problematic makes sense of all of this.

 

(Imagine what that behavior would look like and how absurd it would be. Sidney casts her ballot. When folks are asked whether Sidney voted, they will say "Yes she did, but also no she didn't." And imagine that her vote happens to be the tie-breaking vote. "Yes, the election was a tie. It's also the case that the election was not a tie." The elected official would be both elected and unelected, and they would have to both implement and not implement a tie-breaking procedure because there both was and was not a tie. Imagine if through that procedure the rival candidate wins the tie-breaker. Then you'd have two officials, both elected and non-elected, both serving and not serving the same governing role. Any laws they veto would be both vetoed and not-vetoed. And so on.)

 

The point is that what laws are really doing metaphysically needs to be explained. Given the surface-level explanations here, there's no true contradiction in the Sidney case.

 

(As an aside, I see how a representational theory of truth can work neatly with fictions: By treating a fiction as a backdrop for external properties, then you can measure internal properties against that backdrop. So the internal properties within the sentence "Harry Potter marries Hermione" fails to match with the "external" properties of the canon established by the Harry Potter books. But the properties of the canon itself fail to match the external properties of the real world. So that's how fictional claims can be true and false in different ways at the same time.)

 

I don't have a citation for this but I think it's a fun story. There's an interview on YouTube featuring JC Beall where he mentions that Graham Priest doesn't take Liar paradoxes to be the best candidate for true contradictions, but takes paradoxes of law to be better candidates. Beall implies that he disagrees with this and thinks paradoxes of law are no good, or at least not as good as Liar paradoxes. So while it's tempting to think that all logicians who deny classical logic in favor of some kind of subclassical or paraconsistent logic are on the same team, in typical philosophy fashion there can still be a great deal of disagreement within that space. So even philosophers who defend subclassical logics can agree with my conclusions about one category of paradoxes while disagreeing with my conclusions about the other category.

Distinction: Law of Bivalence vs Law of Excluded Middle

Alex O'Connor uploaded an interview with Joe (Unsolicited Advice). Joe studied logic in college, and he mentions that he got in trouble for confusing the Law of Bivalence with the Law of Excluded Middle. I totally thought these were the same thing, but apparently they are not. And yet, trying to spell out these laws is surprisingly annoying. Consider:

Law of Bivalence(1): For any proposition P, P is either true or false.

Expressed in logic: ∀P(P∨¬P)

Law of Excluded Middle(1): For any proposition P, P is either true or false.

Expressed in logic: ∀P(P∨¬P)

These are identical. Sometimes both bivalence and excluded middle are described as saying "P is either true or false", causing the confusion. If we define both laws as nothing more than the above, then they are the same law. But LEM is usually taken to be saying something slightly different. Consider:

Law of Excluded Middle(2): For any proposition P, either P is true or its negation is true.

So now we have explicitly added in the idea of negation. But how would we logically express this version of the law? This version is saying that disjunction is always true between a proposition and its negation. So for any situation involving P or ~P, that system is logically true. In other words: ∀P(P∨¬P).

...But that's just the same thing again.

Another source of confusion, for me anyway, is referring to these as laws. But how does a law differ from an axiom, a theorem, or a principle? Are the above laws in the sense that they are basic axioms, or are they conclusions derived from axioms?

Also, if bivalence is a 'semantic principle' and LEM is a 'logical truth', which is another thing I've heard, then how are these different? Maybe this means that when we are talking specifically about "P or ~P" we are talking about LEM, and when we are talking about bivalence we are not using logic sentences at all and are only, in that context, using English. But then that makes LEM nothing more than the logic version of bivalence, which doesn't sound right.

Putting these questions to the side, if we define LEM in conditional terms we get:

Law of Excluded Middle(3): If a proposition is true then its negation is false, and if a proposition is false then its negation is true.

If we take 'false' and 'not true' to be logically equal, then "If a proposition is true then its negation is false" means: P ⟶ ¬¬P (Double Negation Introduction). This is equivalent to: ¬¬P ⟶ P (Double Negation Elimination). Or: P ⟷ ¬¬P.

And "If a proposition is false then its negation is true" means: ¬P ⟶ ¬P. This shows LEM being a tautology when we equate 'P is false' with 'Not-P is true'.

Note: Consider the standard belief terms for God:

(1) God exists.

(2) God does not exist.

Theism = the view that affirms (1) and rejects (2).

Atheism = the view that affirms (2) and rejects (1).

Agnosticism = the view that neither affirms nor rejects (1) or (2).

According to LEM, "God exists or God does not exist" is true. So it might seem like agnosticism is the denial of LEM. But an agnostic can affirm that it will, in the end, either be the case that God exists or does not, but it's just that the agnostic is not able to epistemically affirm one proposition or the other. So one can accept "P or ~P" while at the same time withholding belief in either P or ~P. Compare: I might know that either my lottery ticket is a winner or is not a winner, but I won't actively believe one way or another until I check it.

Another problem: 'Not true' applies both to false propositions and non-propositions. So 'not true' and 'false' are not always equivalent. However, in many contexts not being true entails being false. If it's not true that God exists, then it's true that God does not exist. If it's not true that I want cereal for breakfast, then it's true I want not-cereal for breakfast (either I want nothing for breakfast or something other than cereal, assuming that wanting nothing and not wanting anything are equivalent).

This is related to the issue of "neg raising" and the scope of negativity. Basically, we need to be careful with where we place our 'nots' so that we know exactly what it is we are claiming.

A neat way to do things:

Law of Bivalence(2): There are exactly two truth values, 'true' and 'false.' 

Law of Excluded Middle(4): All propositions take on exactly one truth value.

This makes things much more clear. Bivalence(2) is about the number of truth values (there are exactly two) while LEM(4) is about propositions and the truth values they can take on (they can take on exactly one).

Notably, LEM here implies that if something does not have a truth value, then it doesn't count as a proposition.

What does denying bivalence look like?

There are three-valued logics, such as the one proposed by Łukasiewicz, which uses true (1), false (0), and possible (1/2).

Łukasiewicz also proposed what would come to be known as fuzzy logic, which says that truth is not a binary, but a spectrum, and so there are an infinite number of truth values ranging from 0 to 1.

See: https://plato.stanford.edu/entries/lukasiewicz.

A third truth value could be proposed as something like 'undefined' or 'indeterminate' or 'unknown.'

In theory you could deny bivalence by saying there is only one truth value, but I don't know of any reason to do this.

What does denying LEM(4) look like?

You can deny it by saying there are propositions with no truth values, or deny it by saying there are propositions with more than one truth value.

This is where 'gaps' and 'gluts' come into play. A gappy proposition is one that is neither true nor false, and a glutty proposition is one that is both true and false. Gluts are also known as dialetheias, also known as true contradictions. So LEM(4) amounts to saying "There are no gaps or gluts."

See: https://plato.stanford.edu/entries/dialetheism/. 

Note: In that SEP entry, Graham Priest says: "It is a subtle issue . . . whether a gap should count as lacking any truth value, or as having a non-classical value distinct from both truth and falsity, and similarly whether a glut has two truth values, or some third ‘glutty’ value."

So it's not necessarily clear which law is being violated when we posit gaps and gluts. It depends on how we define things.

Consider the Sorites paradox. "This is a heap of sand." You could say that this proposition is a genuine proposition, but is vague, and we cannot determine whether it's true or false. And so you deny bivalence and add a third truth value, 'undetermined,' and apply it to vague propositions. 

Or you could affirm bivalence and say that "This heap of sand" is neither true nor false, denying LEM(4).

So on bivalence(2) and LEM(4), it's clear that the two laws are not logically the same and it's clear that one doesn't imply the other. This way, gaps and gluts are clearly compatible with bivalence and incompatible with excluded middle, and multi-valued logics are clearly compatible with excluded middle and incompatible with bivalence.

This is not standard though. What's more standard is to take the three laws of classical logic...

(1) Law of Identity = ∀x(x=x) (Anything is numerically identical to itself.)

(2) Law of Excluded Middle = ∀P(P∨¬P) (For all propositions, either they are true or false.)

(3) Law of Non-Contradiction = ∀P(~(P∧~P)) (For all propositions, it's not the case that they are true and false.)

...and say that gaps deny (2) but do not deny (3), while gluts deny both (2) and (3). Confusingly, Law of Bivalence is called a law and yet is not counted among the "three laws of classical logic."

Recap: Alyssa Ney's Metaphysics: Preparatory Background

Propositional Logic

AND: P∧Q

P and Q.

OR: P∨Q

P or Q.

IMPLIES: P⟶Q

If P then Q.

NEGATION: ¬P

Not P.

IFF: P⟷Q

If P then Q and if Q then P.

THEREFORE: ∴Q

Therefore Q.

Modus Ponens (Affirming the Antecedent): 

1. P⟶Q

2. P

3. ∴Q

Modus Tollens (Denying the Consequent):

1. P⟶Q

2. ¬Q

3. ∴¬P

Disjunctive syllogism:

1. P∨Q

2. ¬P

3. ∴Q

First-order Predicate Logic

Existential Quantifier: ∃x

This can be read in the following ways (pg 20):

There exists an x such that...

There is at least one x such that...

Some x is...

Something is...

e.g. There is at least one philosopher = ∃xPx (There is an x such that x is a philosopher.)

e.g. There is at least one happy philosopher = ∃x(Px∧Hx) (There is an x such that it is a philosopher and happy.)

We can call a line of symbolic logic a sentence.

Note 1: Sentences with the existential quantifier do not always imply that something exists. (pg 26)

e.g. Fa ⟶ ∃x∃Px (If a is F, then there is at least one thing that is P.)

You're only committed to something being P if there is an a that is F.

Note 2: For proper syntax, make sure your variables are bound to quantifiers. (pg 21)

e.g. ∃xFx ∧ Gx

The variable x in Fx is bound to a quantifier, but the variable x in Gx is not. To fix this, use parentheses:

∃x(Fx∧Gx)

Universal Quantifier: ∀x (For all x's...)

e.g. All philosophers are happy = ∀x(Px⟶Hx) (For all x, if x is a philosopher, then x is happy.) (pg 23)

Note 3: ∀x does not entail that x's exist, only that if they do exist, then all of them have the features stated.

Existential Quantifier Introduction (EI)

Fa ⟶ ∃xFx (If a is F, then something is F.)

e.g. Humility is a virtue. Therefore there is something that is a virtue. (pg 25)

Existential Quantifier Elimination (EE)

∃xFx ⟶ Fa (There is something that is F and we will label a as one of those things.)

Universal Quantifier Introduction (UI)

Fa ⟶ ∀xFx (If an arbitrary a is F, then all a's are F. So all of something is F.)

e.g. Show that a is an arbitrary apple. If a is affected by gravity, then all apples are affected by gravity. So all of something is affected by gravity. (example validated by Chat GPT)

Universal Quantifier Elimination (UE)

∀xFx ⟶ Fa (If everything is F, then a is F.)

 

e.g. ∀xIx (Idealism: Everything is an idea in a mind). Therefore, Disneyland is an idea in a mind. (pg 27)

Mostly, EI and UE are used.

(The quantifier characters show up as slightly bigger or smaller versions. On the back end it's all one symbol; I don't know what causes the change.)

There's more to this chapter but these are the highlights that interest me.