Posts tagged ‘logic’

I was thinking and doing som reserch about one of my kurent projects: Making an English languaj that is syntakialy limited such that it makes posible automatik translation into lojikal formalism. I stumbled akros som prety interesting artikles listed below:

One of them has a website wher one kan find an introduktion to the system. It is aktualy very good and worth reeding. It is a 80 paj powerpoint presentation turned into PDF.

The 1944 paper kritikal of Basic English: How Basic Is Basic English?

The benefits of using plain languaj ar rather obivus and konkreet. Using non-plain languaj makes komunikation take longer and proseed les optimal. This is mostly just waste of time but somtimes it is a mater of life and deth.

My (it is shared) projekt has som on-going diskusion in my forum. However, the languaj i hav in mind is mor similar to formalism than ACE is (the one linked to erlyr). I think that it is too problematik to handle nested konditionals with quantifyrs like:

F1. (∀y)(∀xFxy→Gxy)→Fy

in sylogistik languaj, i.e., as in sentenses like:

S1. “All men are human.”

Rather, one needs sentenses that ar harder to understand and les like ordinary English but beter for formalization like:

S2. “For any X, if X is a man, then X is a human.”

In simple kases, such as the example sentenses with the form:

F2. ∀xMx→Hx

ther is no need for mor advansed sentense syntax, but in the kase of the formalization F1 ther is need for such sentenses.


Let E refer to the propositions expressed by the sentence:

“Everyone wants beer”

“everyone” refers to the three people on the right. Let’s call them “a”, “b”, “c” from left to right. Let “Wx” =df “x wants beer”.

We could try to show this but let’s just take it intuitively that the following holds:

Everyone wants beer ↔ a wants beer and b wants beer and c wants beer

Formalizing we get:


Now, assume that to begin with, a, b, c does not know whether the two others want beer or not. This is technically ‘left open’ in the comic, but it is not irrelevant.

Now, assume every person knows if he wants beer or not, or rather, he either knows that he wants beer, or he knows that he does not want beer. Without this, it doesn’t work either. Introducing “Kx(P)” to mean “K knows that P” and formalizing:


Now, interpreting “logicians” to be a group of people being perfect at making inferences in at least this case. Such people are sometimes called “ideally rational” or similar. They never make a wrong inference and never miss an inference.

Let’s think about a’s position as he is first to answer the question. If he knows that he wants beer, then he does not know the truth of E. But if he knows that he does not want beer, he can know that E is false. Because: he is part of everyone, so if he does not want beer, it is not the case that everyone wants beer. If a is being truthful etc., he has to answer either “I don’t know” or “No”.

Now b is in almost the same position as a. Obviously, if a has already answered “No”, there is no reason to respond or at least he should respond the same. If a’s answer is “I don’t know”, b still lacks information about whether or not c wants beer [Wc or ¬Wc1]. Likewise, b knows his own state, so he knows either that he wants beer or that he doesn’t want beer. If he knows that he doesn’t, he can infer that E is false, similarly to a. If he knows that he does, then he can’t infer anything about E, and has to answer “I don’t know”.

Now, c has all the information he needs to answer either “Yes” or “No”. Obviously, if any of the previous answers are “No”, then he should also answer “No” or simply not answer. If both earlier answers are “I don’t know”, he can infer that both a and b want beer. He also knows his own state. If he does not want beer, he will answer “No”. If he does, he can infer that E is true, and thus answer “Yes”.


Pretty similar comic which does need set theory to properly formalize:


1Which could also be interpret as: To go to the bathroom or not go to the bathroom? :D

Paul Tomassi – Logic

Priest, Graham – Logic, A Very Short Introduction

Copi introduction to logic

I am taking an advanced logic class this semester. Som of the reading material has been posted in our internal system. I’ll post it here so that others may get good use of it as well. The text in question is John Nolt’s Logics chp. 11-12. I remade the pdfs so that they ar smaller and most of the text is copyable making for easyer quoting. Enjoy

John Nolt – Logics, chp 11-12

Edit – A comment to the stuff (danish)

Dær står i Nolt kap. 12, at:

”We have said so far that the accessibility relation for all forms of alethic

possibility is reflexive. For physical possibility, I have argued that it is transitive as

well. And for logical possibility it seems also to be symmetric. Thus the accessibil-

ity relation for logical possibility is apparently reflexive, transitive, and symmetric.

It can be proved, though we shall not do so here, that these three characteristics

together define the logic S5, which is characterized by Leibnizian semantic… That

is, making the accessibility relation reflexive, transitive, and symmetric has the

same effect on the logic as making each world possible relative to each.” p. 343

mæn min entuisjon sagde maj strakes, at dette var forkert, altså, at de er forkert at

R1. For any world, that world relates to itself. (Reflexsive)

R2. For any world w1 and for any world w2, if w1 relates to w2, then w2 relates to w1. (Symmetry)

R3. For any world w1, for any world w2, and for any world w3, if w1 relates to w2, and w2 relates to w3, then w1 relates to w3. (Transitive)

er ækvivalænt mæd

R4. For any world w1 and for any world w2, w1 relates to w2. (Omni-relevans)

Mæn de er ganske rægtigt, at ves man prøver at køre diverse beviser igænnem, så kan man godt bevise fx (◊A→☐◊A) vha. en modæl som er reflexsive, symmetrical og transitive (jaj valgte 1r1, 2r2).

Mæn stadig er dær någet galt. Di forskællige verdener er helt isolerede, modsat vad di er i givet R4. Jaj googlede de, og andre har osse bemærket de:

”Requiring the accessibility relation to be reflexive, transitive and symmetric is to require that it be an equivalence relation. This isn’t the same as saying that every world is accessible from every other. But it is to say that the class of worlds is split up into classes within which every world is accessible from every other; and there is no access between these classes. S5, the system that results, is in many ways the most intuitive of the modal systems, and is the closest to the naive ideas with which we started.”


I’m reading Irving Copi’s Introduction to Logic, very boring and not that well written, any suggestions? Just a college level book that gives a comprehensive intro to the topic. Thanks.


“Boring” is not a particularly apt criticism of a text. There are jazzier logic books out there, with bunches of cartoon and jokes (and maybe even recipes for peanut butter and jelly sandwiches) but Copi is a good, solid elementary logic text which has been through more reprints than I can count, and has set the standard for logic texts in English. If you seriously want to learn, you really have to give up the entertainment addiction.

Notice how applicable this is to anything. Simply substitute logic for any serious discipline such as physics, chemistry and sociology.


This will not involve many science facts as the discussion is wholly philosophical in nature. This is an epistemological, not scientific essay, it just happens to use some facts of science.

I want to show the value of thinking of things as inconsistent sets of propositions (or whatever truth carrier you like, but I like propositions) or at least implausible sets of propositions (when at least one inference is inductive (or non-deductive, if you like that term better)).

Consider this set of propositions:

  1. Newton’s physics is correct.
  2. Things are in a such and such way at time t1.
  3. That Newton’s physics is correct and things are in a such and such way at time t1, implies that things will happen in such and such way at time t2.
  4. A study found that such and such did not happen at time t2.
  5. The study is correct.
  6. That the study is correct implies that such and such did not happen at time t2.

This set is plainly inconsistent; it cannot be true. At least one proposition in this set is false. Suppose we are around the time when Einstein introduced his relativity theories. At that time physicists had pretty good reason to believe (1) (among others: good explanatory power, lots of empiric confirmation), and I’m sure some people called physicists who did not stop believing in Newton’s physics even when some studies found results that are contrary to the predictions of Newton’s physics given some antecedent state of affairs dogmatic. I’m fairly sure such a claim of dogmatism is often thrown around in similar cases.

My point is that it is unwise to claim someone is being dogmatic quickly. For there are many things other than some widely accepted theory that could be wrong (in this case (1)). We could be wrong about the antecedent states of affairs (in this case (2)), or wrong about what the theory predicts (in this case (3)) given that state of affairs; perhaps the scientist that make the prediction from the theory made a calculation error. Something similar applies the the study that ‘challenges’ the accepted theory (in this case (5)). So there are many things that could be wrong without the accepted theory being false. It is wise to consider that before calling people that are being epistemically conservative for dogmatic.

The method of putting the relevant propositions in an inconsistent set forces us to be made aware of some perhaps not normally discussed propositions without which the set would be consistent (or not-implausible). Usually in a moderately complex case such as the one with Newton’s physics, a set of propositions that form as inconsistent set (or implausible) will contain 5-10 propositions. In more complex cases, the sets can be much longer (such as very complex cases involving the impossibility of an infinite past which involves temporal and modal logic). In general, the more propositions we can find that together forms an inconsistent set (or implausible), the better overview, and the easier to is to make a justified decision about which proposition(s) to stop believing in in the case that one actually believes all of them. If we are to avoid inconsistent beliefs (=inconsistent objects of beliefs), then we should think of the many potentially epistemically justified ways there are to deal with a such inconsistent set.

In the above case, rejecting (4) would probably not be a wise decision, neither would it be to reject (6). If there is only one study and it is not exceptionally well done, then rejecting (5) is probably not a bad decision to begin with. If more studies (by competent scientists) confirm the first study, then sooner or later we should begin wondering if not our beliefs would have better coherence were we to reject the theory (1). But before we do that we should consider other alternatives such as (2) and (3). It would not be good if we rejected some theory and later found it that we had no grounds to do that because we were wrong about what the antecedent state of affair was (2).

This way of solving problems (which usually involve an inconsistency of we add together the relevant propositions to a set), is applicable to every topic that I have thought of. It is especially useful to very complex situations where it is hard to get an overview and it seems hard to settle on a specific solution (that is, hard to find out which proposition is the epistemically most justified to deny).

Using the formalization system I wrote of earlier, let’s take a look at this famous question.

First we should note that this is a yes/no question which is different from the questions that I have earlier formalized. The earlier questions sought to identity a certain individual, but yes/no questions do not. Instead they ask whether something is the case or not. So this time I cannot use the (x=?) question phrase from earlier, since there is no individual to identify similarly to the earlier cases.

One idea is to simply add a question mark at the end. Like this:

F1. (∃x)(∃y)(Wxy∧Bxy∧Cxy∧x=a)?

Wxy ≡ x’s wife is identical with y

Bxy ≡ x beats y

Cxy ≡ x used to beat y1

a ≡ you

But this fails to capture that it is not all of the things that are being asked whether they are the case or not. It is only the Bxy part that is being asked. The rest is stipulated as true. We can change the formalization to capture this, like this:

F1*. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧(Bxy)?

The question mark is now understood as a predicate that works on whatever is before/to the left of it. (In parentheses for clarity.) Not to the right like with the other monadic predicates and propositional connectors (¬, ◊, □, etc.). In this case the question mark only functions on (Bxy) and not the rest of the formula.

Translated into LE:

There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you and is it the case that x beats y?

Answering yes/no questions

When answered in the positive, the answered version simply removes the question mark. Producing:


Answering in the negative removes the question mark and adds a negation sign to the part of the formula that the question predicate is working on. Producing this:

F3. (∃x)(∃y)(Wxy∧Cxy∧x=a)∧¬(Bxy)

If the produced formula is true, then the question has been answered correctly. However since this question is loaded. Both of the produced formulas are false, that is, it is both false that:


There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you and x beats y.

and that:


There exists an x and there exists an y such that x’s wife is y and x used to beat y and x is identical with you and it is not the case that x beats y.

Since they both imply the falsehood:



1Alternatively one could deepen to formalization to formalize the temporal aspect of this predicate. Though it doesn’t seem important here so I will leave it out.


I earlier wrote of the logical interpretation of subjects.1 There I suggested, following Russell, that the subject of a descriptive, active, meaningful (DAM) sentence should be interpreted as an existential quantifier (∃x) but I now believe that that this seems to depend on who made the utterance and in which situation. Suppose for instance that a positive atheist2 makes this utterance:

E1. God is omnipotent.

Do we really want to interpret this as:

E1′. (∃x)(Gx∧Ox)3

If we did, then the atheist would have contradicted himself since from (E1′) the existence of God follows. From this I conclude that this interpretation is implausible.

The utterer and the situation

One idea is to let (E1) represent a conditional when uttered by an atheist:

E1”. (∀x)(Gx→Ox)

Such a conditional is consistent with an atheistic position; It is not possible to deduce that God exists from (E1”). How should we think of sentences that are like (E1)? Should they always be interpreted as existential claims, should they always be interpreted as conditional claims or should the interpretation depend on the utterer and the circumstances in which it was uttered? The first option has already been dealt with and found implausible. Let’s consider the second option.

Consider this everyday sentence:

E2. The door is open.

If I said this to my roommate while we were both out in the garden, I think that he would think that I was silly or talking about some door far away. He would never interpret this sentence as a conditional which in that case is true. Is it true not because there is a door and it is open but it is true because there is no door at all in the garden. I imagine that it is like this in many other everyday situations. Suppose that is true, that is, everyday sentences like (E2) are most often best interpret as existential claims. We may allow that sentences involving non-everyday terms like “God” are often best interpret as conditionals.

These considerations indicate that the same sentence form Subject – sentence verb – subject predicate may yield different logical forms depending on which words are used. So, there is a disconnection between language form and logic form. This is undesirable.

Return to the first example. Suppose that a theist said the same sentence. Should it be interpreted as an existential claim or a conditional? I suppose that it is best to interpret it as an existential claim. But for the theist it would not make much of a difference since he also believes that God exists, and from that God exists, and the conditional, it follows that God is omnipotent.4

But even an atheist’s utterance of (E1) may best be interpreted as an existential claim. Suppose that the current american president is a closet atheist, that he is making a public speech and that the public believes that he is a theist. In that case it would be best for the public to interpret his words as an existential claim and not a conditional.


2One who believes that there is no God or no gods.

3Where “Gx” means x is God, and “Ox” means x is omnipotent.

4In symbols: From (∃x)(Gx) and (∀x)(Gx→Ox), (∃x)(Gx∧Ox) follows.


I invent and explore a terminology about degrees of sentences, I explore how to negate sentences and sentence parts in english, I distinguish between verbs that can be used in sup-sentence parts and verbs that cannot, I discuss some problems with the verb “ought”, and lastly I explore the relationship between this terminology and predicate logic.

Negating sentences in english (PDF, 15 pages)