2. Says that we can do what we should do. That seems unwarranted. Say I am an alcoholic. I should quit drinking. But perhaps I cannot.

I think what's happening here is that two different meanings of "can be done" are conflated. 2. Would be true if expressed as "whatever should be done is theoretically possible to do". But 3. uses can in the sense of "what is practically possible". Even for a determinist, the set of theoretically possible events does not equal the set of actual events. — Echarmion

This is the author's justification for the second and third premises (which is in the full paper linked above):

My second premise is the "'ought' implies 'can'" principle: that is, to say that something should be done implies that it can be done. For example, suppose a student explains to me that he could not make it to class because his car broke down. One way I might respond would be by telling him that he could have made it to class some other way. But it would be nonsensical of me to say, "Yes, I understand that you could not have come to class, but you should have come anyway." In general, it is not the case that you should do the impossible. This is equivalent to saying that if you should do something, then you can do that thing.

The third premise states that, if determinism is true, then whatever can be done is actually done. This follows directly from the definition of determinism given above: determinists hold that any person, at any given time, has one and only one course of action open to him. Thus, according to determinists, if a person fails to perform an action, that means he literally was unable to perform it. Which implies that if a person is able to perform an action, then he performs it.

Maybe this can clarify a little.

If the domain of quantification is empty (there are no choices), that entails determinism and denies MFT, shortcircuiting the argument. — SophistiCat

Yes, this is right. Of course the domain of quantification isn't empty (by the premise 1), and I didn't deny that. What I said was that it doesn't follows from the second premise.

Thanks for mentioning that.
How does it implies the existence of anything? Premise 2 simply says that for any x, if x should be done, then x can be done. It doesn't even imply that there is something that should be done, nor that there is something that can be done. It is simply a universally quantified conditional sentence, without existential implications.

Well, the inferences made by the author aren't explicit, so I made a proof (which is kinda big) of of it and I changed 4 to "I believe in anything a". The conclusion remains the same. Actually, I think that it is exactly the fact that anything seems to be true under the assumption that determinism is true that implies that it is false. The proof goes as follows.

First, the proof of the presented argument:

(Here, I used sub-numbers to maintain the original numbers of the premises)

Second, the same proof but with "m" (the minimal free will thesis) changed to "a" (anything). The red arrows shows the changes made.


I used the following symbolism:

Fx = x is false,
S[x] = I should do x,
C[x] = I can do x,
D[x] = I do x,
Bx = I believe that x,
m = the minimal free-will thesis.
a = anything

Are there any errors?

Oh, sorry, I forgot that. It means minimal free will. "The minimal free will thesis (MFT) holds that at least some of the time, someone has more than one course of action that he can perform."

Hmm, think I got it. So you are talking about a necessary condition, right? If P is a necessary condition to Q, then if you have Q, you also have P, because it is necessary for Q. And vice-versa. So P is a necessary condition for Q if and only if Q⊃P.
Now, if we say that X is unconditional, then there is no Y such that X⊃Y, because, if there was, then Y would be a condition to X. Hence any proposition which is unconditional must not imply any other proposition. But this cannot be true, because for any proposition P, if it is true, then it implies every tautology, and if it is false, it implies anything. Also, for any proposition P, P always implies P. So for any proposition P, there is a proposition Q such that P⊃Q. So there can be no such thing as a proposition that is unconditional (in the sense treated here).
It's worth remembering that I'm using the formal definition of the implication. Of course, in common sense language, it is weird to say that a false proposition implies any proposition, and that any true proposition is implied by any proposition. So I don't know how useful this reasoning will be in your discussion.

You don't see the statement non-condition is a condition as contradictory? — TheMadFool

But you didn't say that non-condition is a condition. Your statement was "The state of being unconditional is itself a condition". Here, "condition" applies to "the state of being unconditional", not to "non-condition". Also, there are no problems with this. The property of being a cat is not itself a cat, and there is no contradiction here. As I said, if your statement was "The state of being unconditional is itself conditional", then it would be contradictory.

I think set theory can help us. The set of non-conditions contains no conditions. The set of conditions contains non-condition. The set of non-conditions must be empty i.e. it's the null set. The null set is a subset of every set, am I right? — TheMadFool

Well, the set of non-conditions is the complement of the set of conditions. I don't know exactly what do you mean by "condition", but the set of non-conditions contains everything which is not a condition. So a knife or the number 42, for example, if it's not a condition, is contained in this set. So it cannot be empty. If it was empty, then since the set of conditions is it's complement, it must be the universal set, which means that everything would be a condition, which is kinda weird (but I really don't know what a condition is here).

"The state of being unconditional is itself a condition". How would you translate this statement? — TheMadFool

Hmm, it's a hard sentence to translate... I thought that using definite descriptions we could have something like K((ιx)(Sx∧¬Cx)), with K(x) standing for the predicate "x is a condition" and (ιx)(Sx∧¬Cx) for the definite description "the thing x that is a state and is not conditional".
But then I realized that this formalization is wrong, because in the phrase "the state of being unconditional", "being unconditional" is not a property of this state itself, but rather of some other thing. It seems that this statement assigns a second order property, namely the property of being a condition, to the property of being unconditional, which doesn't seems to be contradictory, since it operates in diferent hierarchy levels.
I thought then that it may be simpler than that. Using the function f(x) for "the state x of being unconditional" and C(x) for "x is a condition", we can get ∀x(C(f(x))), which states that anything that is the state of being unconditional is a condition. But this formalization shows again that it is not a contradiction, and it doesn't entails one, since being unconditional isn't the negation of being a condition.

If you had the statement "The state of being unconditional is itself conditional", then it would be K((ιx)(Sx∧¬Kx)), which is obviously contradictory, since it is equivalent to Ǝx(Sx∧¬Kx∧Kx). But in the way you stated it, I think it isn't contradictory at all.

In my proof the rules I used were modus ponens (for step 3) and reductio ad absurdum (for step 4). — Pfhorrest

Since 'a' is an individual, your use of negation is not the same as the negation of natural deduction, since the latter is a truth-functional operator and individuals are not things that can be truth-evaluated. Also, modus ponens is to be used with propositions of the form "P" and "P⊃Q". You can't use modus ponens in "a" and "a=~a" because neither 'a' is a proposition nor '=' is a a truth-functional operator (and therefore it isn't the implication operator). You cannot use the conjunction to assert a∧~a for the same reason, since 'a' is not a wff (it's a term, not a formula), and you are supposed to use wff as arguments to the conjunction operator. Also, you can't use reductio ad absurdum for step 4 since step 3 doesn't express a proposition, and reductio is used only in propositions.

1. How do I use natural deduction to show that a contradiction (p & ~p) is entailed by (~a) = a? — TheMadFool

Well, (~a) = a is not a well formed formula because the negation operator has to be used before formulas, and 'a' is an individual, so you cannot use '~a'. But if with this you mean "~(a = a)", then it is a wff, since "a=a" is a wff. And is easy to derive p∧~p from that. Actually, you can derive anything from that, because it violates the rule of introduction of identiy. The proof goes as follows:

1. ~(a = a) [Hypothesis]
2. | a = a [=Introduction] // Law of identity
3. | (a = a)∨(P∧¬P) [2, ∨Introduction] // Addition
4. | P∧~P [1, 3, ∨Elimination] // Disjunctive Syllogism
5. ∴ ~(a = a) ⊢ ⊥ [1 - 4, Deduction]

Which is similar to your proof. But from this proof you can derive any contradiction P∧¬P, as you asked.

2. Is (~a) = a already a contradiction? How? — TheMadFool

3. How does (~a) = a differ from rejecting the law of identity like so: ~(a = a)? — TheMadFool

The answer is exactly as before. If by "(~a) = a" you mean "it is not the case that a equals a", i.e., ~(a = a), then it is not exactly a contradiction since contradictions have the form φ∧¬φ. But a contradiction is easily derived from this as I showed. Also, in this case, there is no difference between "(~a) = a" and "~(a = a)". But if you have another use for the negation in '~a' (which as I said cannot be the same as in P∧~P), then let us know.

Sure, but how would you define □A without using ◊ or accessibility? In both cases the circularity would appear again.

Well, ok, but I couldn't find a precise formal definition/explication of 'possible' and 'accessible' without being required to already know one of this concepts. In 'Modal Logic for Philosophers', by Garson, and in 'Basic Concepts in Modal Logic', by Zalta, the same circularity appears.

So what it means for a world to be accessible to another?

He explains both concepts, as I said before. And he uses possibility to explain the accessibility relation.

No, I understand what it means for a proposition to be possible. The whole point is that Kripke explains it in terms of a concept (accessibility) which requires the notion of possibility to be understood, which seems to be circular. I mean, you know what "bachelor" means, but if I say that bachelor is a unmarried man and that an unmarried man is a bachelor, i'm not saying anything usefull, it's circular, even though you know the meaning of this terms.

Well, I don't think I've conflated accessible and possible, for me it's very clear the difference, and I agree with everything you said. But I still cannot understand exactly how this two terms are to be formally expressed without requiring one another.

Hmm, it might not be exactly a definition, but anyway he expresses the accessibility relation in terms of possibility (H2 is accessible to H1 if every proposition true in H2 is possible in H1, that is, (h1Rh2 ↔ ∀p(V(p, h2)=T ⊃ V(◊p, h1)=T)), and expresses possibility in terms of accessibility (A is possible in h1 iff there is a world h2 accessible to h1 in which A is true, that is, (V(◊A, h1)=T ↔ ∃w(h1Rw∧V(A, w)=T))) (Here, V(x, y) is de valuation function of the formula x in the world y). I cannot understand exactly what he formally means with one term without understand another.
He previously (p. 2) defines (and here he actually uses the term 'define') □B as follows: V(□B, H)=T iff for every H' such that HRH', V(B, H')=T. He then says that a formula A is true in a model associated to the world G if V(A, G)=T.

Thank you for answering!

Yes, in natural language it's ok to state that. But I want to know if ithere is any way of formalising that using logical or set-theoretical language.

Well, with ∀p(p∨¬p) I say that p∨¬p is true for every proposition, because p is a variable, not a specific proposition, while ☐(p∨¬p) says that "p∨¬p", which is a truth about the proposition p, is true in every possible world.
I want to formalize "to any sentence p, p is a member os S if and only if ¬p is not a member of the set S", that is, ∀p(p∈S↔¬p∉S), but I don't know how to do this.
I think I could something like ∀x(Wx⊃(S(x)↔¬S(N(x)))), with W(x) meaning "x is a well formed formula", S(x) meaning "x is a member of the set S" and N(x) meaning the definite description "the negation of x", x being a sentence. But this seems quite artifitial to me.

I think a more adequate formalization would be ∀x∀y(◊(Gyx∧∃z(¬Gyz∧z≠x))), that it, for any x and y, is possible that (y is good for x and there exists a z different from x such that y is not good for z). This means that for anything that is good for you can be person for which it's not good.

"the difference between A and B does not subsist"
— EmaFort

references the fact that A=B is true. In other words, it's essentially just the negation of the claim that ~(A=B), i.e. ~~(A=B). — Mentalusion

Yeah, you're right, this is true. But, for Frege, the statement "the difference between A and B does not subsist" wouldn't be neither true nor false, simply because "the difference between A and B" does not have any reference. It has meaning, but doesn't points to anything. That is what his theory tells us.

Oh, I thought it could be the case, but since english isn't my native language, I used a translator and thought that "begat" could be applied to both paternity and maternity. Thanks!

Oh, yeah, misstype. Thanks!

Well, the problem is at the inference from 3 to 4. The contrapositive of a statement has its antecedent and consequent inverted and flipped. Therefore, the contraposition of (P->Q) is (¬Q->¬P).
Step 3 states (¬A => A). (Note that i put the antecedent in italic and the consequent in bold, to facilitate). Thus, the contrapositive of (¬A => A) is (¬A => A), which is the same thing.

"Introduction to Logic", Irving M. Copi. - It was the very first logic book I read. It's kinda old, and treat some subjects like basic logic, language, definitions, fallacies, arguments, validity, syllogism, symbolic logic, natural deduction, inference rules, propositional functions, quantifiers and induction. It has a lot of exercicies, with the answer in the end of the book.
"Introduction to Logic", Harry Gensler - Newest, this book has a more didatic approach, and contains everything that Copi's has, and relations; identity, modal, deontic, imperative and belief logic; metalogic and a logical formalization and deduction of the golden rule of ethics.
"Introdução à Lógica", Cezar Mortari - The author of this book is brazilian, so it's written in portuguese, and I don't know if there is an english version of it, but maybe some lusophone is reading this and I hope it will be useful. It's a very technic book, with precise definitions. It contains almost everything that Gensler has, except for deontic, imperative and belief logic and the ethics formalization, and it has a basic arithmethic theory formalized, something about axioms system and other things.
"Introduction to mathematical logic", Elliott Mendelson. - I didn't read this book yet, but i heard about and people say it's a good book to start studying math logic.

That's what I'm talking about! I just woke up, so I'll do some things and se your reply calmy later. Thanks for answering!
Yeah, as I said before, I just put this title because the book says that sentence is the representation of 1+1=2, but it isn't really important to what I was proposing. The title was just something that i thought it would call people's attention; maybe it's inadequate.

Hmm, you are right, I expressed myself in the wrong way. Thanks!

P⊃Q doesn't permit Q⊃¬P in a consistent logic — sime

Of course, but you were saying about the consequent only, not about the entire implication. That is why I said that in "P⊃Q", "Q", alone, permits "¬Q".

That case is different to the set-theoretic case, where Fx ∨ Gx permits Fx ∧ Gx and is therefore a weaker statement than the latter. — sime

Hm, I don't know if I understood. For instance, in the sentence "(¬(∃x)(Fx∧Gx) ∧ Fx) ⊃ (Fx∨Gx)", you would say that the consequent "Fx∨Gx" permits "Fx∧Gx"? (It's just an example for me to understand, I'm not saying this is the case)

Of course, in a sense your antecedent might be said to contain your "conclusion" as a weaker premise, but i think it is a mistake to think of your right-hand side as a conclusion because it must forever remain tied to the antecedent if it isn't to be misinterpreted as allowing F and G to be overlapping sets containing multiple members... assuming of course, that you want to represent the number 2 as a union of pairwise disjoint singleton sets. — sime

Well, why couldn't we treat it like so? I mean, I could say that the antecedent is the premise and the consequent is the conclusion, and since the conclusion follows from the premise, I could represent they in a conditional statement. I didn't understand part of your latter paragraph... English isn't my native language and i'm not familliar with a lot of terms you used.

I think I already answered it, but feelings and this kind of thing are something that were developed through the evolution of the species. I don't know exactly why, but for some reason thoses species which members had this feeling were more capable of surviving than the others, and this characteristic came to us, and it's probably passed through DNA, even if the enviromental interaction produces changes in the person's personality.
I think now I understood your point. You are saying that we can explain why are we so simillar, but aparently we do not have sufficient explanation to our differences. Is this? This is interesting, but I think you are treating primitive instincts/feelings as if it were like personality, which is a more complex thing. And for some reason you leading to believe that because we all feel the pain of losing someone, and this is not something learnt by nurture, then our personalities must follow this line too. I don't think it's a valid reasoning...

But why would the conclusion need to explicit something that already has been said in the premise? I mean, in "P⊃Q", for instance, if you analyze only the consequent, you'll see that "Q" permits "¬P", which is denied by the antecedent. I'm not quite sure what kind of analysis are you doing but I think that analyzing only the conclusion without considering what was stated in the premise isn't the right way.

Wtf are you doing on this forum?
This is said in the antecedent, not in the conclusion. Maybe this image can help the visualization.

Well, I guess it can be taken as a paraphrase that is an interpretation of the mathematical assertion. I really don't know, and I really don't care if it's really 1+1=2, I just wanted to see if someone could prove the validity. Anyway, he's e-mail is hgensleratlucdotedu . He usually answer quickly, so please let us know if you get an answer.

If you had read what I said you would notice that the title isn't anything than a flashy title.
Why does the conclusion permits "(x or y) or (x & y) to be F & G" if it is said that nothing is simultaneously F and G?
The reference is in the text, just read. And it's not 1=1=2, it's 1+1=2.

Are you reading anything i'm writing? I said at least twice that the "1+1=2" is absolutelly irrelevant to what i'm proposing, it's just a detail. The proposal is to prove that the sentence "(((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))" is true, valid, call it what you want. I'll not repeat, and I'll not reply to this kind of comment again.

Are you really saying that "(((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))" is self-evident?

But you example was about bachelors. Anyway, it doesn't matter, the logical formula in question have variables, and I still not understanding the reason of the difficulty to understand what was proposed...

Well, I guess you are confunding logic tautologies with linguistic tautologies. In logic, a tautology is a formula, or a truth-function, that returns always "true" to whatever interpretation you give to the variables. That is, to any possible combination of the truth-value of the variables, the tautology is always true (opose to the contradiction, which is always false to any combination). Linguistic tautologies are just redundant statements that are true due the meaning of the terms and their internal relations. Your example isn't true if I define a bachelor as being an apple (I guess apples aren't unmarried man). On the other hand, the formula ((P⊃Q)∧¬Q)⊃¬Q), for instance, is always true, regardless what you attribute to "P" and "Q", and you easily can verify it's a tautology with a truth table.

Why not?. Monozygotic twins are genetically identical, at least during the early development (which doesn't necessarily have to do with the two twins be physically identical). As I said, phenotypic differences between twins can occur due epigenetical events, or even due environmental stimuli.

I don't even know what are you talking about. I'm leaving the post for those who want to test, and possibly improve, their logical capacity. Thanks for answering.

Do you ask your teacher the reason to prove something on a test? Man, it's a challenge, I'm assuming that people who frequent this section of logic and math philosophy know what it means to prove something, to show that a statement entails another, using inference rules, as I said. Sorry, but I really don't know whats the difficulty to understand that.

Lol, what about interpretation? I'm not talking about mathematics foundings here neither about some philosophical trip. Actually, it doesn't matter if that sentence can really be equivalent 1+1=2 or not. The only thing that I wanted to propose is to prove the validity of that claim. If you want, you can forget what is written about 1 + 1 being equal to 2, whatever.
Oh, and where did you get that tautology should not be proven? One thing is a self-evident tautology like 1=1 or p->p; another thing is something like "(((∃x)(Fx∧¬(∃y)(y≠x∧Fy))∧(∃x)(Gx∧¬(∃y)(y≠x∧Gy)))∧¬(∃x)(Fx∧Gx)) ⊃ (∃x)(∃y)(((Fx∨Gx)∧(Fy∨Gy))∧(x≠y)∧¬(∃z)((z≠x∧z≠y)∧(Fz∨Gz)))", which you can't trivially say whether it's a valid inference or not only by reading it's terms and connectives.

Of course it could exists, but as I said, infering it from the fact that science aparently can't explain this or that is a logical jump. And difference of the physical structure can explain the differenct in personality! For instance, if a person has more quantity of a specific neurotransmiter in a specific region in the brain, the behaviour and personality of this person would be different of that if he had less neurotransmitter, or had in another area. In the 50's, for example, it was very common the lobotomy of some psychiatric patients, which is a surgery in which some connections of the frontal lobe to the subcortical area are broken, changing the personality of the patient.

First, humans cannot be reduced to a DNA code or something like this. The fact that monozygotic twins are genetically identical doesn't have to do with they exact physical constitution. For two physical thing to be identical, they have to have exactly the same characteristics (except by the spatio-temporal localization, since two bodies cannot occup the same space at the same time), that is, the exactly same physical composition. If a twin have one strand of hair more the other (which isn't necessarily a thing genetically determined), then they cannot be called "identical". Even the DNA of the cells can change in one twin but not in the other, due mutations or something (note that not every mutation is malefic; some are benefic and some doesn't fenotipic expression).
Second, even if we assume that monozygotic twins are exactly identical, which we know that is false, there is one more thing you didn't considered in your point. Human beings - actually, every animal, in general - are in certain way biologically shaped by the environment where they live in. There is a concept in neurology called "critical period", which is a period in life during which the nervous system is very responsive to environmental influence. For instance, if a kid isn't taught to learn some language until the critical period ends (about 12 years old) cannot - or, if so, very dificultly - learn language anymore. And, following Wittgenstein, as our thought depends on language, of course a twin for whom the language wasn't presented will think and probably behave very diferently from the other that learned a language.
Even if two twins are presented to the same stimuli, it doesn't have to do with theirs respondes to it.
In a manner analogous to the kantian transcendental idealism, what determines how the individual will behave isn't the stimuli itself, but the individual response to the stimuli, which isn't the same in several cases (for Kant, we are not supposed to say something about the object itself, the "noumena", but yet about the way that such an object is presented to us, the "phenomena"). But, you can ask, how can the response to the same stimuli be different? Well, it has to do with the disposition of the synaptic network on the brain of the person. If, for some unknown reason, a stimuli makes the neurons to react in a manner that makes the person feels good, then the corresponding synapses are strenghtened. For else, they are weakened. The human brain has the incredible capacity of regulate itself to suit the environment, due the neuroplasticity, and reguling itself makes it's further response to environment to be different from other person's one, even if they are genetically identical. I have to go bed now, but what I wanted to say is that we do have neurobiological fundaments to explain that, even though I couldn't bring they with required scientific rigor here, and that saying that because of this difference between twins there is something like a "mind substance" (which you didn't defined, but I suppose it's an immaterial thing) is a logical jump. Thanks for posting!

Well... No. As you quoted, if A is false, nothing is said about B.
But you need to notice that it is a existential claim. You are treating it like as if it were a propositional logic claim, like "if it's a robber, then it's american" (if it were the case, then you would be right), but it's an existential predicative logic claim. It is said that there exists an x such that If it's a robber, then it's american. (not simply that "robber implies american"). It doesn't means that all robbers are americans, but that there are at least one. It can be the case that all robbers are americans, but it can be the case that there are robbers that aren't americans. If there is an american robber and a mexican one, the claim "(∃x)[Rx⊃Ax]" is also true. You are commiting a logical jump by infering an universal claim from an existential one.