# Posts Tagged ‘Equivalence’

## Fundamental Tautologies

Posted by allzermalmer on September 29, 2013

First I shall list all the truth tables for basic logical operators. They shall each be given their own symbol as an operator. I will give both two different symbols for them, one for symbolic notation and one in polish notation.

Φ and Ψ will be used as meta-variables, which may be replaced by propositions at any time.

Meta-Variable for proposition Φ:
Given that Φ=True then Φ=True.
Given that Φ=False then Φ=False.

Symbolic (~) and Polish (N): Not..
Given that Φ=True then NΦ=False or (~Φ=False).
Given that Φ=False then NΦ=True or (~Φ=True).

Symbolic(&) and Polish (K): Both…and…
Given that Φ=True and Ψ=True, then KΦΨ=True or (Φ&Ψ)=True.
Given that Φ=True and Ψ=False, then KΦΨ=False or (Φ&Ψ)=False.
Given that Φ=False and Ψ=True, then KΦΨ=False or (Φ&Ψ)=False.
Given that Φ=False and Ψ=False, then KΦΨ=False or (Φ&Ψ)=False.

Symbolic (↓) and Polish (X): Neither…nor…
Given that Φ=True and Ψ=True, then XΦΨ=False or (Φ↓Ψ)=False.
Given that Φ=True and Ψ=False, then XΦΨ=False or (Φ↓Ψ)=False.
Given that Φ=False and Ψ=True, then XΦΨ=False or (Φ↓Ψ)=False.
Given that Φ=False and Ψ=False, then XΦΨ=True or (Φ↓Ψ)=True.

Symbolic (<->) and Polish (E): …if and only if…
Given that Φ=True and Ψ=True, then EΦΨ=True or (Φ<->Ψ)=True.
Given that Φ=True and Ψ=False, then EΦΨ=False or (Φ<->Ψ)=False.
Given that Φ=False and Ψ=False, then EΦΨ=False or (Φ<->Ψ)=False.
Given that Φ=False and Ψ=False, then EΦΨ=True or (Φ<->Ψ)=True.

Symbolic (v) and Polish (A): Either…or…both
Given that Φ=True and Ψ=True, then AΦΨ=True or (ΦvΨ)=True.
Given that Φ=True and Ψ=False, then AΦΨ=True or (ΦvΨ)=True.
Given that Φ=False and Ψ=True, then AΦΨ=True or (ΦvΨ)=True.
Given that Φ=False and Ψ=False, then AΦΨ=False or (ΦvΨ)=False.

Symbolic (↑) and Polish (D): Not both…and…
Given that Φ=True and Ψ=True, then DΦΨor (Φ↑Ψ)=False.
Given that Φ=True and Ψ=False, then DΦΨ or (Φ↑Ψ)=True.
Given that Φ=False and Ψ=True, then DΦΨ or (Φ↑Ψ)=True.
Given that Φ=False and Ψ=False, then DΦΨ or (Φ↑Ψ)=True.

Symbolic (->) and Polish (C): If…then…
Given that Φ=True and Ψ=True, then CΦΨ or (Φ->Ψ)=True.
Given that Φ=True and Ψ=False, then CΦΨ or (Φ->Ψ)=False.
Given that Φ=False and Ψ=True, then CΦΨ or (Φ->Ψ)=True.
Given that Φ=False and Ψ=False, then CΦΨor (Φ->Ψ)=True.

Tautologies:

Symbolic (&) and Polish (K): Both…and…
~(Φ&~Φ)=NKΦNΦ
~(~Φ&Φ)=NKNΦΦ

Symbolic (↓) and Polish (X):Neither…nor…
~(~Φ↓Φ)=NXNΦΦ
~(Φ↓~Φ)=NXΦNΦ

Symbolic (<->) and Polish (E):…if and only if…
(Φ<->Φ)=EΦΦ
(~Φ<->~Φ)=ENΦNΦ

Symbolic (v) and Polish (A):Either…or…both
(Φv~Φ)=AΦNΦ
(~ΦvΦ)=ANΦΦ

Symbolic (↑) and Polish (D):Not both…and…
(~Φ↑Φ)=DNΦΦ
(Φ↑~Φ)=DΦNΦ

Symbolic (->) and Polish (C): If…then…
(Φ->Φ)=CΦΦ
(~Φ->~Φ)=CNΦNΦ

Equivalence:

The order of these equivalence follow those above: (&), (↓), (<->), (v), (->), (↑)

(K) (Φ&Ψ): (Φ&Ψ), (~Φ&~Ψ), ~(Φ&~Ψ)&~(~Φ&Ψ), ~(~Φ&~Ψ), ~(Φ&~Ψ), ~(Φ&Ψ)

(X) (Φ↓Ψ): (~Φ↓~Ψ), (Φ↓Ψ), ~((~Φ↓~Ψ)↓(Φ↓Ψ)), ~(Φ↓Ψ), ~(~Φ↓Ψ), ~(~Φ↓~Ψ)

(A) (ΦvΨ): ~(~Φv~Ψ), ~(ΦvΨ), ~(Φv~Ψ)v~(ΦvΨ), (ΦvΨ), (~ΦvΨ), (~Φv~Ψ)

(D) (Φ↑Ψ): ~(Φ↑Ψ), ~(~Φ↑~Ψ), ~(Φ↑Ψ)↑(Ψ↑~Φ), (~Φ↑~Ψ), (Φ↑~Ψ), (Φ↑Ψ)

(C) (Φ->Ψ): ~(Φ->~Ψ), ~(~Φ->Ψ), ~((Φ->Ψ)->~(Ψ->Φ)), (~Φ->Ψ), (Φ->Ψ), (Φ->~Ψ)

## Knowability Paradox and Modal Realism

Posted by allzermalmer on August 17, 2013

Equivalency is defined this way: (p–>q)<–>(q–>p)

So if we assume (p–>q) is true and (q–>p) is true, then it necessarily follows that (p–>q) if and only if (q–>p).

(1) p–><>p
(2) <>p–>p [Modal Realism]
(3) (p–><>p)<–>(<>p–>p)

There is one thing that must be made clear.

(1) is necessarily true. It is not possible that it is not true. It is an axiom of modal logic. Now (2) isn’t necessarily true. It is possible that it is not true. It is not an axiom or theorem of modal logic.

(1) can be substituted with variable of x. So x if and only if p–><>p. (1) being necessarily true implies necessarily x. (1) and x, are analytically true.

(2) can be substituted with variable of y. So y if and only if <>p–>p. (2) being not necessarily true implies possibly not y. (2) or y, are contingently true.

(3), based on substitutions of (1) & (2), takes on the form of x<–>y. Or we can say analytically true if and only if contingently true.

x is analytic implies either necessarily x or necessarily not x. [](p–><>p) v []~(p–><>p).
y is contingent implies possibly y and possibly not y. <>(<>p–>p) & <>~(<>p–>p).

(3) shows that we have collapsed any modal distinction between possibility and actuality. There is no modal difference between possibly true and actually true. This specific proposition presents that possible if and only if actual.

(4) p–>Kp [Fitch’s Theorem]
(5) Kp–>p
(6) (p–>Kp)<–>(Kp–>p)

There is one thing that must be made clear.

(5) is necessarily true. It is not possible that it is not true. It is an axiom of epistemic logic. Now (4) is also necessarily. It is not possible that it is not true. It is a theorem of epistemic logic.

(4) can be substituted with variable of x. So x if and only if p–>Kp. (4) being necessarily true implies necessarily x. (4) and x are necessarily true.

(5) can be substituted with variable of y. So y if and only if Kp–>p. (5) being necessarily true implies necessarily y. (4) and y are necessarily true.

(6), based on substitutions of (4) & (5), takes on the form of x<–>y. Or we can say analytically true if and only if analytically true

x is analytic implies either necessarily x or necessarily not x. [](p–>Kp) v []~(p–>Kp).
y is analytic implies either necessarily y or necessarily not y. [](Kp–>p) v []~(Kp–>p).

(6) shows that we have collapsed any epistemic distinction between knowledge and truth. There is no difference between knowing something is true and something is true. What this specific proposition presents is that truth if and only if knowledge

Equivalency: ECpqCqp or ECNqNpCNpNq

(1) CpMp
(2) CMpp
(3) ECpMpCMpp

(4) CpKp
(5) CKpp
(6) ECpKpCKpp

Posted by allzermalmer on November 3, 2011

This blog is based on something brought up by philosopher of science Carl Gustav Hemple. It’s usually called the Ravens Paradox or Confirmation Paradox. The presentation of the problem originally arose in an article of his called Studies in the Logic of Confirmation (I.). It appeared in the philosophy journal Mind, New Series, Vol. 54, No. 213 (Jan., 1945), pp. 1-26.

“The defining characteristic of an empirical statement is its capability of being tested by a confrontation with experimental findings, i.e. with the results of suitable experiments or “focused” observations.”

This is to tell us that science is concerned with testing our theories against experience, and these experiences are typically of the sensory kind (sight, taste, sound, smell, and touch). This type of thing, empirical statements being able to be be tested based on experiment, is what separates it from statements of math and logic, or transempirical statements.

“Testability here referred to has to be understood in the comprehensive sense of “testability in principle”; there are many empirical statements which, for practical reasons, cannot be actually tested at present. To call a statement of this kind testable in principles means that it is possible to state just what experiential findings, if they were actually obtained, would constitute favorable evidence for it, and what findings or “data”, as we shall say for brevity, would constitute unfavorable evidence; in other words, a statement is called testable in principle, if it is possible to describe the kind of data which would confirm or disconfirm it.”

So the theory has to make predictions of what will happen, or is predicted to happen. This leads the theory to be an empirical statement. But now all empirical statements have to make predictions, whether we can bring them about now or sometime in the future. What they do is tell us that if we bring about certain conditions, then we should observe certain things to happen. But we can’t test them right now, and have to wait for some future time. Even though we can’t test it right now, it is still able to be tested in the future if the technology or situation arises that the statement makes clear. Knowing what the statement predicts, it allows to figure out if the event fits with the prediction (confirm) or doesn’t match with the prediction (disconfirm).

“no finite amount of experiential evidence can conclusively verify a hypothesis expressing a general law such as the law of gravitation, which covers an infinity of potential instances, many of which belong either to the as yet inaccessible future, or to the irretrievable past; but a finite set of relevant data may well be “in accord with” the hypothesis and thus constitute confirming evidence.”

There’s a reason why science can’t prove it’s theories. Science deals with some general statements. General statements, All A are B or If A then B, apply to all times. This means that it holds in the past, even if we weren’t there, or in the future or someplaces we haven’t yet observed or been to. The problem is that it holds for all time (past, present, and future), as well into all locations. But we’re finite beings with a finite amount of time, and finite searching capabilities, which means that we can’t find everything that meets that hypothesis to confirm all of it’s predictions.

“…relevance is a relative concept; experiential data can be said to be relevant or irrelevant only with respect to a given hypothesis; and it is the hypothesis which determines what kind of data or evidence are relevant for it. Indeed, an empirical finding is relevant for a hypothesis if and only if it constitutes either favorable or unfavorable evidence for it; in other words, if it either confirms or disconfirms the hypothesis. Thus, a precise definition of relevance presupposes an analysis of confirmation and disconfirmation.”

Now we go around accumulating all these data from our observations, or creating observation reports. “The squirrel had no tail” or “The counter clicked 15 times when we ran the experiment”, are observation reports or data. But we couldn’t make sense of these reports without some sort of hypothesis. We can’t say that the first sentence disconfirms a theory without comparing how that statement fits in with a theory. But we want to see how to confirm a theory, which involves the relevance of the observation statement in relation to the theory under question. And Hempel hopes to go on to show how we can do this.

“Take a scientific theory such as the atomic theory of matter. The evidence on which it rests may be described in terms of referring to directly observable phenomena, namely to a certain “macroscopic” aspects of the various experimental and observational data which are relevant to the theory. On the other hand, the theory itself contain a large number of highly abstract, non-observational terms such as “atom”, “electron”, “nucleus”, “dissociation”, “valence”, and others none of which figures in the description of observational data.”

He want’s to go on to point out that induction can only take us so far in forming scientific theories. Many times, in a theory, it contains terms that don’t come from observation itself. This means that we can’t invoke these terms just from observations, which is usually how induction work. Induction will go from what was observed, to what will be observed. It moves from past happenings to future happenings. This can mean that we predict that because the Green Bay Packers won the Superbowl last year that they’ll win it this year. Or, moving from particulars to a general conclusion (this raven is black to All ravens are black). So there’s another part of theory creation that isn’t based on observation/induction itself. This is based on imagination, and connect these imaginative terms up with observation. Connect the theoretical parts with the concrete parts of observation itself.

Now there is something called, or was called, Nicod’s Criterion of confirmation. It would follow as something like this:

“Consider the formula or the law: A entails B. How can a particular proposition, or more briefly, a fact, affect its probability? If this fact consists of the presence of B in a case of A, it is favorable to the law ‘A entails B’; contrary, if it consists of the absence of B in a case of A, it is unfavorable to this law. It is conceivable that we have here only two direct modes in which a fact can influence the probability of a law…Thus, the entire influence of particular truths or facts on the probability of universal propositions or laws would operate by means of these two elementary relations which we shall call confirmation and invalidation.”

This criterion is based on statements of the form “If X then Y” or “All X are Y”.

Take the simple example of a hypothesis of “All Ravens are Black” or “If Raven then Black”. The hypothesis is confirmed, under Nicod’s Criterion, if the antecedent (Raven) and the consequent (black) are satisfied. This means that we’ve confirmed the hypothesis if we find an object that is a Raven and Black.

The hypothesis “All Ravens are Black” is disconfirmed if the antecedent (Raven) is satisfied, but the consequent (Black) isn’t confirmed. This means that we find an object that is a Raven, but it isn’t Black, which means it could be white.

But this criterion has some shortcomings. One of them is that it only applies to universal hypothesis (All Ravens are Black: ), but it doesn’t work with existential hypothesis (Some Ravens are Black). This means that it doesn’t tell us the criterion to say that an existential hypothesis is confirmed or disconfirmed. It also doesn’t work with psychological hypothesis like “You can fool all of the people some of the time and some of the people all the time.” This means that the criterion doesn’t tell us when we’ve confirmed or disconfirmed an existential hypothesis in conjunction with a universal hypothesis.

But now let’s take a look at a second problem with Nicod’s criterion. Let us say that we have two Statements.

1. All Ravens are Black
2. All non-black things are non-Ravens

Now we have four objects before us to review, and if these statements confirm, disconfirm, or are neutral to the two statements, under Nicod’s criterion.

a. A Raven and is Black::      (confirms 1. but neutral to 2.)
b. A Raven but not Black::   (disconfirms 1. and 2.)
c. Not a Raven but Black::    (neutral to 1. and 2.)
d. Neither Raven or Black::  (confirm 2. but neutral to 1.)

But the problem with this is that is both 1. and 2. are equivalent hypothesis. They are logically identical in their truth-tables. All Ravens are Black is logically identical to All non-black things are non-Ravens. Or, If Raven then Black is logically identical to If not Black then not Raven. This is the logical equivalence of contraposition.

This seems to throw Nicod’s criterion into direct conflict with logic.The way that those statements should be in line with logic when it comes to confirmation or disconfirmation, it would look like this:

a. A Raven and is Black::     (confirm 1. and 2.)
b. A Raven but not Black::  (disconfirm 1. and 2.)
c. Not a Raven but Black::   (neutral to 1. and 2.)
d. Neither Raven or Black:: (confirm 1. and 2.)

This is what leads Hemple to present forward a logical principle to help out with.

Equivalence Condition: Whatever confirms (disconfirms) one of two equivalent sentences, also confirms (disconfirms) the other.”

This is just a logical principle being used to help us have a criterion for determine what counts as confirmation, neutral, or disconfirming for the theory being used.

One of the reasons is that the reasoning processes in creating empirical hypothesis is that they have to be logically consistent (no contradictions). From certain universal statements, you can logically deduce certain predictions for observation.

“an adequate definition of confirmation will have to do justice to the way in which empirical hypothesis function in theoretical scientific contexts such as explanations and predictions, they serve as premises in a deductive argument whose conclusion is a description of the event to be explained or predicted. The deduction is governed by the principles of formal logic, and according to the latter, a deduction which is valid will remain so if some or all of the premises are replaced by different, but equivalent statements; and indeed, a scientists will feel free, in any theoretical reasoning involving certain hypotheses, to use the latter in whichever of their equivalent formulation is most convenient for the development of his conclusions.”

Now here are certain things that follow from the Equivalence Criterion, which is a logical principle used.This principle brings about a necessary condition to say that we’ve achieved confirmation or disconfirmation of a theory.

If Raven then Black is logically the same as If not Black then not Raven.  X–>Y= ~Y–>~X. Now imagine that you hold the hypothesis “All Ravens are Black”. You want to go out and look for things that confirm your hypothesis that you hold. You are probably going to walk outside, and look for things that are Ravens. If it is not Black, then you’ve disconfirmed your hypothesis. Or you find a Black Raven and confirmed your hypothesis. But you don’t even need to go outside to confirm your hypothesis. Just look around your place and see if you can observe anything that is not black and not a raven. I can see a red cup, a blue pen, yellow notebook. This is all confirmation of the hypothesis “All Ravens are Black”. This is usually what becomes known as the Raven Paradox. It makes confirmation of a theory extremely easy, just walk through a grocery store and remember that’s all a confirmation of a All Ravens are Black.

Now there’s one way to try to get around this. One way is to conjoin a universal statement and an existential statement. This means that you put together a statement like “All X are Y” and “Some D are E”. But the problem that we come across is that this isn’t logically acceptable with equivalent statements. This would ruin the way of theoretical arguments, which science relies on itself with it’s hypothesis.

Another way is to say that scientific hypothesis implies an existential statements. This would mean that the Universal statement of “All Mermaids are Green” implies “Some Mermaids are Green”. A statement like “Some Mermaids are Green” means that something exists that is a mermaid and green. But this is no longer the case with the logical square of opposition, in modern logic. Universal statements don’t imply Existential statements. Like, “All Jedi have Midi-Chlorian in their blood system.” This would imply, if we take the suggestion seriously, that “Some Jedi have Midi-Chlorian in their blood system”, which implies that at least one Jedi exists, let alone that Midi-Chlorian exists. This is one reason that modern logic says that Universal statements don’t imply existential statements. They don’t say that what it talks about actually exists.

Another way of dealing with this problem is to say that we should focus only on the “Class of Ravens”. This means that we accept “All Ravens are Black” is our hypothesis, and accept that it’s equivalent to “All non-Black things are non-Ravens”. But we only look for things in the “Class of Ravens”, and see if they’re either black or not black. Now imagine that someone comes up to you and says, “I have a Raven behind my back. Would you like to see what color  it is?” You should say “Yes”, because that’s part of the “Class of Ravens” that your trying to confirm. But now imagine someone comes up to you and says, “I have a pen behind my back. Would you like to see what color it is?” You should say “No”, because that doesn’t fall under the “Class of Ravens”.

But the problem with this is that sometimes, a hypothesis goes beyond it’s class, and works in a purely negative way (i.e. outside of its “Class”). So let’s take a hypothesis All P’s are Q’s. The field of application would be P’s. Take the scientific hypothesis that “All sodium salts burn yellow”, and we focus on the class of “Sodium Salts”. We apply it to something that we don’t know to be a sodium salt or burn yellow. If this thing doesn’t burn yellow, we noticed the absence of Sodium Salt. If it burns yellow, then either we find out it’s Sodium Salt and confirm the hypothesis or find out it isn’t Sodium Salt and find a new thing to study and know it burns yellow.

“Our interest in the hypothesis may be focused upon its applicability to that particular class of objects, but the hypothesis nevertheless asserts something about, and indeed imposes restrictions upon, all objects (within the logical type of the variable occurring in the hypothesis, which in the case of our last illustration might be the class of all physical objects). Indeed, a hypothesis of the form “Every P is a Q” forbids the occurrence of any objects having the property P but lacking the property Q; i.e. it restricts all objects whatsoever to the class of those which either lack the property P or also have the property Q. Now, every object either belongs to this class or falls outside of it, and thus, every object-and not only the P’s-either conforms to the hypothesis or violates it; there is no object which is not implicitly “referred to” by a hypothesis of this type.”

This is what we’re logically faced with, but we come to think of Equivalence Criterion as leading to paradoxes. But these paradoxes have nothing to do with logic, or logical paradoxes, but psychological paradoxes.

Now imagine that we’re in a new situation. Imagine that you hold the hypothesis of “All Sodium Salts burn Yellow”. This time you decide to adduce an experiment where you take a cube of ice, and put it over a colorless flame, and find that the flame didn’t turn yellow. This would confirm “Whatever doesn’t burn Yellow isn’t Sodium Salts”, and this statement is equivalent to “All Sodium Salts burn Yellow”. Now this doesn’t obviously look paradoxical, but the other way seemed paradoxical. There’s no logical difference between them, but we take one to be paradoxical and ignore the other one as being a paradox if we accept the other to be a paradox.

“Instead, we tacitly introduce a comparison of H with a body of evidence which consists of E in conjunction with an additional amount of information which we happen to have at our disposal; in our illustration, this information includes the knowledge (1.) that the substance used in the experiment is ice, and (2) that ice contains no sodium salt. If we assume this additional information as given, then, of course, the outcome of the experiment can add no strength to the hypothesis under consideration.”

But we come to think that we have evidence that the hypothesis “Whatever doesn’t burn Yellow isn’t Sodium Salts” is supported by this instance, but it supports “All Sodium Salts burn Yellow”.

We, therefore, seem to be stuck in logical trouble and going beyond what scientist do with their theoretical arguments, if we want to change the confirmation/diconfirmation conditions of Equivalence Criterion.

This leads us to have confirmation for our hypothesis all the time, and have it all around us. With the example of “All Ravens are Black”, it means that everything we see that isn’t a Raven, is confirmation for “All Ravens are Black”. We literally have confirmation for our theories all around us, and see evidence everywhere. The logical point is we have confirmation for our hypotheses all the time.