allzermalmer

Truth suffers from too much analysis

Different Models of both Knowledge & Epistemology

Posted by allzermalmer on September 20, 2013

One of the differences in epistemology are different theories of knowledge. Renee Descartes helped to present one theory of knowledge, which followed a general form of Rationalism. There is another general form known as Empiricism.

We shall have two Categories and three Truth-values.

Category 1: Cognitive or Cognition
Truth Value: Either Cognition is always true or Cognition is sometimes true & sometimes false or Cognition is always false.

Category 2: Sensory or Senses
Truth Value: Either Senses are always true or Senses are sometimes true & sometimes false or Senses are always false.

Now we can combine both of these categories together to form Both Cognition & Senses, apply the Truth Values, and derive 9 different Models of Epistemology or Knowledge.

Hypothesis *1:
Both cognition is always true & senses are always true.
Both senses are always true & cognition is always true.

Hypothesis *2:
Both cognition is always true & senses are sometimes true and sometimes false.
Both senses are sometimes true and sometimes false & cognition is always true.

Hypothesis *3:
Both cognition is always true & senses are always false.
Both senses are always false & cognition is always true.

Hypothesis 1*:
Both cognition is sometimes true and sometimes false & senses are always true.
Both senses are always true & cognition is sometimes true and sometimes false.

Hypothesis 2*:
Both cognition is sometimes true and sometimes false & senses are sometimes true and sometimes false.
Both senses are sometimes true and sometimes false & cognition is sometimes true and sometimes false.

Hypothesis 3*:
Both cognition is sometimes true and sometimes false & senses are always false.
Both senses are always false & cognition is sometimes true and sometimes false.

Hypothesis *1*:
Both cognition is always false & senses are always true.
Both senses are always true & cognition is always false.

Hypothesis *2*:
Both cognition is always false & senses are sometimes true and sometimes false.
Both senses are sometimes true and sometimes false & cognition is always false.

Hypothesis *3*:
Both cognition is always false & senses are always false.
Both senses are always false & cognition is always false.

These models of knowledge, or epistemology, exhaust all logically possible positions given only these two categories and these three truth values. Some possible subdivisions could be made, especially when either categories, or both, take on the truth value of sometimes true and sometimes false.

One basic idea is that cognition, under Rationalism, would always be true & senses, under empiricism, would always be true.

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Proof of Modus Ponens

Posted by allzermalmer on August 29, 2013

There is a rule of Inference known as Modus Ponens. It is also sometimes known as rule of Detachment.

When it comes to rules of inference, some rules are taking as primitive and some are taken as derived. A primitive rule is one in which no proof is given for the rule, and a derived rule is one that is derived from the primitive rule.

The proof of Modus Ponens, which is to be given here, takes Modus Ponens as a derived rule and will use Disjunctive Syllogism as a primitive rule. So from Two premises and the primitive rule of Disjunctive Syllogism, Modus Ponens as a rule of inference can be derived.

Before we go through the proof, Modus Ponens form of argumentation shall be shown.

(1.) P–>Q [Premise]
(2.) P         [Premise]
(3.) Q         [Conclusion]

Here is an example of Disjunctive Syllogism.

(1.) ~PvQ [Premise]
(2.) P        [Premise]
(3.) Q       [Conclusion]

So Disjunctive Syllogism shall be used as the primitive rule, and from this primitive rule will be able to derive the inference rule of Modus Ponens.

Now there is one point that needs to be gone over first, which is that of Modus Ponens works with conditional statements, or if…then statements. Disjunctive Syllogism works with disjunctive statements, or ‘or’ statements. Conditional statements have equivalent forms of disjunctive statements. This is how conditional statements are known as material implications statements.

So, P–>Q is a material implication, and thus can be switched into it’s disjunctive form. So P–>Q, as a material implication, states that ~PvQ. This equivalence shall be used in the proof of modus ponens.

(1.) P–>Q [Premise]
(2.) P         [Premise]
(3.) ~PvQ [Definition of (1)]
(4.) Q       [(2)-(3) by Disjunctive Syllogism] [Conclusion]

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Fallacy of Evidentialism

Posted by allzermalmer on August 18, 2013

There are two philosophers, who are taken to be generally representative of Evidentialism. These two philosophers are David Hume and C.K. Clifford. These two philosophers have two quotes that are examplars of their Evidentialism thesis. They are, respectively, as follows.

“A wise man, therefore, proportions his belief to the evidence…when at last [a wise man] fixes his judgement, the evidence exceeds not what we properly call probability.” – David Hume in “Of Miracles” (Italics are Hume’s)

“We may believe what goes beyond our experience, only when it is inferred from that experience by the assumption that what we do not know is like what we know…It is wrong in all cases to believe on insufficient evidence” – W.K. Clifford in “The Ethics of Belief

Thomas Huxley,

Huxluy Evidence

Those quotes from these three writers are taken as representative of Evidentialism, and thus the Evidentialist Principle. The statements they make might appear to carry some validity & they might even seem to be sound.

However, Karl Popper holds that they are not valid. He also doesn’t hold that they are sound. They even contradict all empirical systems or all empirical propositions. They forbid us from ever believing or holding to any empirical system or empirical proposition, they forbid us from ever believing or holding to any scientific hypothesis or scientific proposition. But the problem of Induction applies to both the truth of this matter of fact assertion and the probability of the truth of this matter of fact assertion.

Both of the propositions contain signs of being based on Induction. Hume points out that a wise man will fix their judgements on a proposition when the evidence indicates that it is probable. Clifford points out that we may infer from experience what goes beyond our experience, but this is based on hypothesis that unknown is similar to the known.

Both of the propositions show that Evidentialism is founded on Induction, or inductive inferences.

Hume, supposedly, showed that it is logically impossible to infer the unknown from the known. It is logically impossible to derive the unknown from the known. Thus, Evidentialism is founded on a logical impossibility.

“The problem of the source of our knowledge has recently been restated as follows. If we make an assertion, we must justify it; but this means that we must be able to answer the following questions.

How do you know? What are the sources of your assertion?’ This, the empiricist holds, amounts in its turn to the question,

‘What observations (or memories of observations) underlie your assertion?’ I find this string of questions quite unsatisfactory.” – Karl Popper in “The Sources of Knowledge and Ignorance

Popper presents the Evidentialist Principle, in that quote, as saying that “If we make an assertion, we must justify it“. If you make an assertion, then you must justify it, or making an assertion implies must justify the assertion. You would have to answer one question, ‘How do you know? What are the sources of your assertion?’, and have to answer another question, ‘What observations (or memories of observations) underlie your assertion?’. 

As Popper points out, the Evidentialist Principle is an answer to The Problem of Source of Knowledge. So we may suppose that Evidentialism and Induction are to be based on the Source of a proposition or an empirical proposition. It seeks that the source of a proposition to be justified.

Criticizing or discrediting a proposition because of the source has some similarity to the Genetic Fallacy: “if the critic attempts to discredit or support a claim or an argument because of its origin (genesis) when such an appeal to origins is irrelevant.”

With the Genetic Fallacy, a proposition is being discredited, or supported, because it is “paying too much attention to the genesis of the idea rather than to the reasons offered for it”. The origin, or source, of the proposition is used to discredit, or support, the proposition.

Evidentialism would discredit a proposition because the source of the proposition is without justification.

We also find that David Hume presents an example of the questions that Popper finds to be unsatisfactory.

“All reasonings concerning matter of fact seem to be founded on the relation of cause and effect. By means of that relation alone we can go beyond the evidence of our memory and senses. If you were to ask a man, why he believes any matter of fact, which is absent; for instance, that his friend is in the country, or in France; he would give you a reason; and this reason would be some other fact; as a letter received from him, or the knowledge of his former resolutions and promises…All our reasonings concerning fact are of the same nature. And here it is constantly supposed that there is a connexion between the present fact and that which is inferred from it. Were there nothing to bind them together, the inference would be entirely precarious.

When it is asked, What is the nature of all our reasonings concerning matter of fact? the proper answer seems to be, that they are founded on the relation of cause and effect. When again it is asked, What is the foundation of all our reasonings and conclusions concerning that relation? it may be replied in one word, Experience. But if we still carry on our sifting humour, and ask, What is the foundation of all conclusions from experience? this implies a new question, which may be of more difficult solution and explication.” – David Hume in “Sceptical doubts concerning the operations of the understanding” (Italics are Hume’s)

David Hume himself goes down the line of questioning that Popper brings up. For example, suppose that some assertion is made like “all ravens are black”. This assertion is what Hume calls a Matter of Fact, i.e. Synthetic proposition or Contingent proposition. It is Possible that it is true that “all ravens are black” and it is possible that it isn’t true that “all ravens are black”. This starts a line of questioning once this assertion is presented.

Question: What is the nature of reasoning concerning that matter of fact?
Evidence: The assertion is founded on the relation of cause and effect.
Question: What is the foundation of reasoning and conclusion concerning that relation of cause and effect?
Evidence: The relation of cause and effect of that assertion is founded on Experience.

These two questions follow a basic form that Popper is bringing up, and the type of basic form that Popper finds unsuitable, or the type of basic form of Evidentialism that is unsuitable. The basic reason for this is because another question follows from the answer to the previous two questions.

Question: What is the foundation of that conclusion drawn from experience?

This new question is where the Problem of Induction arises, or what Popper calls The Logical Problem of Induction.

If all Ravens are Black then justified in the relation of cause and effect. If justified in the relation of cause and effect then justified by experience. If justified by experience then experience is justified by Induction. So if all ravens are black then justified by Induction. But, Induction isn’t justified. So assertion all ravens are black isn’t justified. Therefore, Evidentialism would make it so that the assertion all Ravens are Black isn’t justified. This applies to all matters of fact, and thus all empirical and scientific assertions.

“It is usual to call an inference ‘inductive’ if it passes from singular statements (sometimes called ‘particular’ statements), such as accounts of the results of observations or experiments, to universal statements, such as hypotheses or theories. Now it is far from obvious, from a logical point of view, that we are justified in inferring universal statements from singular ones, no matter how numerous; for any conclusions drawn in this way may always turn out to be false: no matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white. The question whether inductive inferences are justified, or under what conditions, is known as the problem of induction.” – Karl Popper in “The Logic of Scientific Discovery” (Italics are Popper’s)

The Problem of Induction comes about because Induction relies on statement that is a matter of fact assertion, but this matter of fact assertion cannot, in principle, be inductively justified. So either all reasonings concerning matter of fact seem to be founded on experience or not all reasonings concerning matter of fact seem to be founded on experience.

This is a logical problem because either Induction relies on a statement that is either a contingent proposition or necessary proposition. We can call this the “Principle of Induction”. But the Principle of Induction can’t be a necessary proposition because the negation of the Principle of Induction is possible to be false. A necessary proposition can’t be possible to be false. So it is possible that Principle of Induction is true and it is possible that isn’t true that Principle of Induction is true. Therefore, the Principle of Induction is a contingent proposition.

Hume points out that matter of facts about dispositions and universal propositions are matters of facts. Thus dispositional propositions and universal propositions are contingent propositions. Dispositional propositions describe law-like behavior and universal propositions describe lawful behavior or law-like behavior. These would both be contingent propositions, and so we wouldn’t be justified, based on Induction, in asserting those dispositional propositions or universal propositions.

We wouldn’t be justified, based on Evidentialism, when it came to assertions about dispositional propositions or universal propositions. Science wouldn’t be justified, based on Evidentialism, when it came to assertions about dispositional propositions or universal propositions. But science is full of assertions about dispositional propositions and universal propositions. Therefore, science wouldn’t be justified in asserting dispositional propositions and universal propositions.

“[Hume] tried to show that any inductive inference- any reasoning from singular and observable cases (and their repeated occurrence) to anything like regularities or laws- must be invalid. Any such inference, he tried to show, could not even be approximately or partially valid. It could not even be a probable inference: it must, rather, be completely baseless, and must always remain so, however great the number of the observed instances might be. Thus he tried to show that we cannot validly reason from the known to the unknown, or from what has been experienced to what has not been experienced (and thus, for example, from the past to the future): no matter how often the sun has been observed regularly to rise and set, even the greatest number of observed instances does not constitute what I have called a positive reason for the regularity, or the law, of the sun’s rising and setting. Thus it can neither establish this law nor make it probable.” Karl Popper in “Realism and the Aim of Science” (Italics are Popper’s)

The assertion “all ravens are black” isn’t justified as true under Evidentialism and “all ravens are black” isn’t jusified as probably true under Evidentialism. Hume himself points out that the wise man doesn’t fixate his judgement on an assertion in which the evidence exceeds what we properly call probability. In other words, the Evidentialist doesn’t hold to assertions in which the evidence exceeds what we properly call probability. So Evidentialist only hold to assertion in which evidence shows it is true or probably true. So “all ravens are black” is only held by an Evidentialist if evidence shows it is true or at least probably true.

Popper presents a solution to the Problem of Induction, and thus treats assertions differently from Evidentialism. Popper rejects Induction, and thus rejects Evidentialism. The source of an assertion has nothing to do with either discrediting the truth of a proposition or supporting the truth of a proposition.

Matter of fact propositions, or scientific propositions, don’t discredit or support the source of an assertion. Science doesn’t support the truth of a proposition or support the probability of a proposition. It, basically, seeks to discredit the truth of a proposition. Science seeks to show that the proposition is false, not that the proposition is true or probably true. Science always seeks to discredit it’s proposition and not to support it’s propositions. So scientific propositions are, in principle, possible to show they are false and never show they are true or probably true. This includes both dispositional propositions and universal propositions.

In other words, Evidentialism seeks both positive justifications for assertion and negative justifications for assertion. Evidentialism would be based on “full decidability”. Falsifiability, or Falsification, seeks only negative justifications for assertions. Falsifiability would be based on “partial decidability” . These negative justifications, for Falsifiability, basically state that scientific assertion hasn’t been demonstrated false as of yet. This never indicates a positive justification for the assertion being true or probably true.

“The problem of induction arises from an apparent contradiction between the basic empiricist requirement (only experience can decide the truth or falsity of a scientific statement) and Hume’s insight into the logical impermissibility of inductive decision (there is no empirical justification of universal statements). This contradiction exists only if we assume that empirical statements must be empirically “fully decidable”, that is, that experience must be able to decide not only their falsity, but also their truth. The contradiction is resolved once “partially decidable” empirical statements are admitted: Universal empirical statements are empirically falsifiable, they can be defeated by experience.” – Karl Popper in “The Two Problems of The Theory of Knowledge” (Italics are Popper’s)

For Falsifiability, the source of an assertion is irrelevant when judging whether the assertion is either true or false, and the source of an assertion is irrelevant when judging whether justified in believing that assertion is true or probably true. The source of an assertion is irrelevant for the justification of the assertion. Would have to rely on Induction, and Induction isn’t justified itself. The only justification of an assertion, specifically an empirical assertion, is that it is possible to show that assertion is false. An empirical assertion has the possibility to be shown false, but it doesn’t have the possibility to be shown true (or probably true).

Science, thus, doesn’t care of the source of an assertion. Science is justified in believing, or holding to, an empirical proposition because that empirical proposition allows for the possibility that can be shown that it is false, but hasn’t been shown that it is false yet. For example, science would be justified in believing the empirical proposition that “all ravens are orange” if wasn’t for “some ravens are black”. It would be a negative justification, since don’t have another empirical proposition that contradicts it, or shows that it is false.

One of the basic mechanisms of Falsifiability is that works by deductive inference. Modus Tollens forms an example of deductive inference that Falsifiability uses. Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.

If an empirical assertion is true implies another empirical assertion is true & the other empirical assertion is false, then original empirical assertion is false.

Principle of Modus Tollens:If all ravens are orange implies no ravens are not orange & some ravens are black, then not all ravens are orange. This is how the negative justification of empirical assertions works, which is deductive inference of modus tollens. It wouldn’t be possible for “not all ravens are orange” to be false. So it must be true.

The Principle of Modus Tollens is a necessary truth, which is different from the Principle of Induction. The Principle of Induction isn’t a necessary truth. It is possible that the Principle of Induction is false. So it might be true.

An assertion that is the conclusion of the Principle of Induction, or the assertion of a wise man that reviewed the Evidence, might be true. An assertion that is the conclusion of the Principle of Modus Tollens, or the assertion of a foolish man that never reviewed the Evidence, must be true.

The truth that the Principle of Modus Tollens always produces truth. It is similar to negative theology. It isn’t true that “all ravens are orange” & it isn’t true that “no ravens are not orange”. Each time saying what is true because true isn’t those false statements, since it is true that “not all ravens are black”.

The contradiction between “all ravens are orange” and “not all ravens are orange” are exclusive, they both can’t be true and no intermediary empirical propositions between them. If know that “all ravens are orange” is false then know that “not all ravens are orange” is true. All ravens are orange implied no ravens are not orange & some ravens are black. Therefore, it is necessarily true that not all ravens are orange. If Know that “not all ravens are orange” is true then “not all ravens are orange” is true. “Not all ravens are orange” is true.

Both the Principle of Modus Tollens are dealing with scientific propositions. The scientific propositions are possibly true or possibly false. If combine scientific propositions with the Principle of Induction, then scientific proposition infered might be true. If combine scientific propositions with Principle of Modus Tollens, then scientific proposition infered must be true. The negative justification allows for things that aren’t possibly not true & hold to statements that are only true, while positive justification allows for things that are only possibly true & hold to some statements that aren’t only true.

So Evidentialist like David Hume, or C.K. Clifford, would be justified in holding some scientific propositions that aren’t only true. Evidentialist would hold to both true statements and false statements. While the Non-Evidentialist, which follows Falsifiability or negative justification, would hold only to true statements. The Non-evidentialist wouldn’t be justified in asserting a scientific statement, even though conclusions drawn from it must be true.

Thus, Evidentialism is fallacious because the assertions that it concludes to be justified in holding, based on the evidence, aren’t truth-preserving. It’s conclusions of justified scientific propositions aren’t based on the evidence or derived by positive support it receives from the evidence. However, it is completely opposite with Non-Evidentialism of Falsification, or it isn’t fallacious.

The Evidentialist would be acting irrationally by seeking their justification, while the Falsifiabilist, which is necessarily a Non-Evidentialist, would be acting rationally by not seeking the Evidentialist justification.

Huxley’s assertion, in his examplar of Evidentialism, mentions that “merciless to fallacy in logic.” But we later find out that Evidentialism isn’t “merciless to fallacy in logic”, but is founded on a fallacy in logic itself. David Hume recognized this, even though exemplar of Evidentialism. Instead, he went about acting irrationally by seeking a (positive) justification of proposition by evidence & the rest of Evidentialism followed, like C.K. Clifford and Thomas Huxley. They would all go about by searching for evidence that proposition is true and end right back in the same place.

Finding Evidence

So we finally come full circle with the fallacy of Evidentialism, and find the source of the Evidentialist fallacy.

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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

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Gnostic and Agnostic Breakdown

Posted by allzermalmer on August 2, 2013

The main interest is of Agnosticism, and this by default can have some implication with Atheism and Theism.

It will be supposed that Agnosticism is about lack of knowledge or not knowing. Gnosticism will be about having knowledge or knowing. It will be supposed that to have knowledge of a claim, then that claim is Justified, True, and it is Believed.

(Gnostic) K=JTB
(Agnostic) ~K= (A1) NJTB v (A2) JNTB v (A3) JTNB v (A4) NJNTNB

There are four ways to agnosticism, but there is only one way to gnosticism.

(A1) Claim isn’t Justified & Claim is True & Claim is Believed.
(A2) Claim is Justified & Claim isn’t True & Claim is Believed.
(A3) Claim is Justified & Claim is True & Claim isn’t Believed.
(A4) Claim isn’t Justified & Claim isn’t True & Claim isn’t Believed.

(Gnostic Socrates) If Socratesl knows the claim p, then Socrates claim is Justified, True, and Believed by the Socrates.

(Agnostic Socrates) If Socrates doesn’t know the claim p, then…
(A1) Socrates claim isn’t Justified, but Socrates believes the claim and it’s True.
(A2) Socrates claim isn’t True, but Socrates claim is Justified and Believed.
(A3) Socrates claim isn’t Believed, but Socrates claim is Justified and it’s True.
(A4) Socrates claim isn’t Justified, isn’t Believed, and isn’t True.

Suppose that p is “there exists a deity”. So ~p stands for “there doesn’t exist a deity”.

(i)Kp= Socrates knows there exists a deity.
(ii) K~p= Socrates knows that there doesn’t exist a deity.

(iii) ~Kp= Socrates doesn’t know that there exists a deity.
(iv) ~K~p= Socrates doesn’t know that there doesn’t exist a deity.

Assume Socrates doesn’t know that the earth is flat. This is because Socrates knows that the earth isn’t flat. Socrates knowing that the earth isn’t flat implies that it is true that the earth isn’t flat. Socrates can’t know false things (but can believe false things), so Socrates doesn’t know that the earth is flat, especially because Socrates knows that the earth isn’t flat.

So it becomes obvious that:

(i) Kp doesn’t forbid ~K~p:: Socrates knows that there exists a deity doesn’t forbid Socrates doesn’t know there doesn’t exist a deity.

(ii) K~p doesn’t forbid ~Kp:: Socrates knows that there doesn’t exist a deity doesn’t forbid Socrates doesn’t know that there exists a deity.

(iii) ~Kp doesn’t forbid (ii) K~p :: Socrates doesn’t know there exists a deity doesn’t forbid Socrates knows there doesn’t exist a deity.

(iv) ~K~p doesn’t forbid (i) Kp :: Socrates doesn’t know there doesn’t exist a deity doesn’t forbid Socrates knows there does exist a deity.

(iii) or (iv) doesn’t imply that Gnostic, but can be Gnostic. (A1)-(A4) show some reasons on why (iii) and (iv) don’t necessarily imply, but don’t forbid, being Gnostic.

When it comes specifically to “there exists a deity”, it would mean that in order to be Agnostic on that claim, Socrates would have to take part of (iii) and (iv).

In order to be Agnostic, then Socrates doesn’t know there exists a deity and Socrates doesn’t know there doesn’t exist a deity.

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Logic without Existential Import or Free Logic

Posted by allzermalmer on July 31, 2013

Aristotle’s Logic
[(x)(F(x) → G(x)] → Ε(x)(F(x) & G(x))

Modern Logic
[(x)(F(x) → G(X)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))

Modern Logic can discriminate between inferences whose validity requires an existence assumption or doesn’t require an existence assumption.

Required Existence Assumpition
[(x)(F(x) →G(x)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))

Non-Required Existence Assumption
(x) F(x) → Ε(x) F(X)

When move from Quantification Theory to Identity Theory, Modern Logic’s new formula doesn’t hold with Identity Theory, because there is a counter example.

Assume that [(x)(F(x) → G(X)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))
Assume that F=y
then (x) [x=y → G(x)] → [Ε(x) (x=y & G(x))]

So Modern Logic Quantification Theory view of Existential Import would imply that all statements in Identity Theory of the form (x) [x=y → G(x)] carry Existential Import of Ε(x) (x=y & G(x)).

The source of this error in Modern Logic comes from Particularization.

(1) (x) [x=y → G(x)] → [Ε(x) (x=y & G(x))]
is deducible from valid formula
(2) [(x)(F(x) → G(X)) & Ε(x) (F(x))] → Ε(x)(F(x) & G(x))
by substituting “=y” for “F” and detaching with
(3) (E(x) (x=y)
(3) is valid in conventional Identity Theory and deducible from Identity Axiom
(4) y=y

Particularization is [Fy → (E(x)) (x=y)]

So Free Logic comes about to get rid of the Existential Importation that both Aristotle’s and Modern Logic allow for. This is a logic Free of existential import or assumption. It is built off of a modification of Quantification Theory by altering some axioms.

(x)(F(x) → F(y)) is an axiom of Quantification Theory which is replaced in Free Logic.

A1) (y)(x) (F(x) → F(y))
A2) [(x)(F(x) → G(x))] → [(x)(F(x) → (x)(G(x)]
A3) x=x
A4) (x=y → (F(x) → F(y))
A5) (E(x)(F(x) → F(x))

From these axioms, (x) F(x) → E(y) F(y) isn’t derivable. This means that existential import is not derivable from the axioms of the Free Logic in Quantifier Theory.

However, what can be derived is [F(x) & (E(x) (x=y)) (x)] → (E(y) (F(y)).

(x) [(F(x) → G(x)) → ((E(x) F(x)) → (E(x)G(x)))] can also be derived from the system.

(X) (x=y) can’t be derived and neither can [(x)(x=y → G(x))] → [(E(x)(x=y & G(x))].

This Free Logic allows the differentiation between singular inference patterns where the existence assumption is relevant or not.

Singular Inference Patterns Existence Assumption:
(F(x) & (E(x) (x=y & G(x))) infer (E(x) F(y))

Singular Inference Patterns Non-Existence Assumption:
x=y infer (F(x) → F(y))

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Impossibility Paradox of Computers by Curry Paradox

Posted by allzermalmer on July 29, 2013

There was a paper called “Computer Implication and the Curry Paradox”. It was authored by both Wayne Aitken and Jeffery A. Barrett, which appears in Journal of Philosophical Logic vol. 33 in 2004.

Suppose an Implication Program has two input statements that are about the behavior of the program, then it tries to deduce the second statement from the first statement by some specified rules in it’s library.

If program finds a deduction of the second statement from the first statement, then the program halts and has output of 1 to signal proof has been found.

The Implication Program can prove statements involving the Implication Program itself.

It is assumed throughout the paper that programs are written in a fixed language for a computer with unlimited memory.

The Impossibility Theorem basically states that “no sufficiently powerful implication program can incorporate an unrestricted form of modus ponens.”

One of the consequences of this Impossibility Theorem is that “modus ponens is an example of a valid rule of inference that can be defined algorithmically, but cannot be used by the implication program.”

Assume (1) that property C(X) is defined to hold if and only if X having property X implies Goldbach’s Conjecture. Furthermore, suppose (2) that C (C). Thus, by the definition of C (X) it implies that C (C) implies Goldbach’s Conjecture. Since C (C) is true by assumption, then it follows by Modus Ponens that Goldbach’s Conjecture is true.

This doesn’t prove Goldbach’s Conjecture yet. However, it does prove that C (C) implies Goldbach’s Conjecture. So by the definition of C (X), it follows that C (C) is true. And by the use of Modus Ponens, Goldbach’s Conjecture is true.

A statement is a list, i.e.  [prog, in, and out]. Prog is a program considered as data, and in is an input for prog, and out is an anticipated output.

A statement is called true if the program prog halts with input in and output out. A statement that isn’t true is called false.

“There is a program to check if but not to test whether [prog, in, out] is a true statement. Given [prog, in, out] as an input, it first runs prog as a sub-process with input in. If and when prog halts, it compares the actual output with out. If they match then the program outputs 1; if they do not match, the program does something else (say, outputs 0). This program will output 1 if and only if [prog, in, out] is true, but it might not halt if [prog, in, out] is false. Due to the halting problem, no program can check for falsity.”

So there is a program that check whether [prog, in, out] is a true statement, but the program can’t test whether [prog, in, out] is a true statement. From the Halting problem, no program can check for falsity. So the program can’t check for it’s own falsity, but can check for it’s truth.

It will use 1 as a signal of positive result, and 0 to signal a negative results. However, failure to halt also indicates, but is not a signal to a real use,  a negative result. So failure of the program to halt doesn’t signal a negative result, but it does indicate a negative result.

“A rule is a program that takes as input a list of statements and outputs a list of statements…A valid rule is a rule with the property that whenever the input list consists of only true statements, the output list also consists of only true statements.”

The output list will include the input list as a sublist, and that the rule halt for all input lists.

Take the program AND. This specific program expects as an input a list of two statements. These two statements are [A,B]. The AND Program first checks the truth of A in manner indicated above. If the program determines A is true, then it checks the truth of B. If B is true, then it outputs 1. Now if either A or B is false, the AND program fails to halt.

“A library is a list of rules used by an implication program in its proofs. We assume here that the library is finite at any given time. A valid library is one that contains only valid rules.”

“Consider the implication program ⇒ defined as follows. The program ⇒ expects as input a list of two statements [A, B]. Then it sets up and manipulates a list of sentences called the consequence list. The consequence list begins as a singleton list consisting only of A. The program ⇒ then goes to the library and chooses a rule. It applies the rule to the consequence list, and the result becomes the new consequence list. Since rules are required to include the input list as a sublist of the output list, once a statement appears on any consequence list it will appear on all subsequent consequence lists. After applying a rule, the program ⇒checks whether the consequent B is on the new consequence list. If so, it outputs 1; otherwise it chooses another rule, applies it to update the consequence list, and checks for B on the new consequence list. It continues to apply the rules in an exhaustive way until B is found, in which case ⇒outputs 1. If the consequent B is never found, the implication program ⇒does not halt.”

Take the Modus Ponens Program. This program expects an input list of statements, and from this it starts by forming empty result list. It searches the input list for any statement of the form [–>,[A,B],1] where A and B are statements. From all the statements, it searches to check if A is a statement on the input list. If A is found, then Modus Ponens Program adds B to the result list. The Program outputs a list that shows all the statements in the input list that are followed by all the statements of the result list (if any statements).

“The Modus Ponens program is a rule. A rule is valid if, for an input list of true statements, it only adds true statements. From the definition of –>, if [–>, [A,B],1] and A are on the input list and if they are both true and if the library is valid, then B will be true. So, MP is a valid rule if the library used by –> is valid. “

The EQ program expects an input list that contains [m,n], which are two natural numbers. Supposing m=n, then the EQ outputs 1, or outputs O. This is an example that some statements are clearly false. So let false be the false statement [EQ,[0,1],1]. If 0=1, then the EQ outputs 1, which is truth. This shows some statements are clearly false.

“Consider the program CURRY defined as follows. It expects a program X as input. Then it runs ⇒ as a subprocess with input [[X, X, 1], false]. The output of the subprocess (if any) is then used as the output of CURRY. If X checks for a particular property of programs, then the statement [X, X, 1] asserts that the program X has the very property for which it checks. The program CURRY when applied to program X can be thought of as trying to find a proof by contradiction that the statement [X, X, 1] does not hold.”

There is only way that CURRY can output 1 with input X. This is done by if –> outputs 1 with input [[X,X,1}, false].  This is what lies behind the Ad Hoc Rule (AH).

The AH expects a list of statements as input. From there it begins producing an empty result list. It than checks its input for statements that take the form of [CURRY, X, 1] where X is a program. For all such statements on input list, AH adds the statement [–>,[[X,X,1], false] to the result list. The AH will than construct a result list, which contains statements in the input list followed by the statements of the result list (if any).

AH is a valid rule because the statements on the input list are true and AH only adds true statements to form the output list. AH is ad hoc because it is specifically designed for the CURRY program.

“We now describe an algorithmic version of the Curry paradox. We assume that the library is valid and contains MP and AH. Consider what happens when we run ⇒ with input [[ CURRY, CURRY, 1], false]. First a consequence list containing the statement [ CURRY, CURRY, 1] is set-up. Next rules from the library are applied to the consequence list. At some point the Ad Hoc Rule AH is applied and, since [ CURRY, CURRY, 1] is on the consequence list, [⇒, [[CURRY, CURRY, 1], false], 1] is added to the consequence list. Because of this, when MP is next applied to the consequence list, false will be added to the list. Since the initial input had the statement false as the second item on the input list, ⇒will halt with output 1 when false appears on the consequence list.”

So the Implication Program outputs 1 with input of [[CURRY, CURRY, 1], false]. Based on the definition of CURRY Program, it implies that CURRY outputs 1 as CURRY is given as an input. Basically, the statement [CURRY, CURRY, 1] is true. A false statement is true.

Suppose that –> is applied to [[CURRY, CURRY, 1], false]. Because the antecedent [CURRY, CURRY, 1] is true, all statements added to the consequence list will also be true. But the statement false is added to the consequences list, which means that false is true, which is a contradiction.

The Curry Paradox has occurred in a concrete setting of a perfectly well-defined program and careful reasoning about the expected behavior.

The Curry Paradox proves that any library containing the Modus Ponens program and Ad Hoc Rule are not valid. AH is unconditionally valid, so we can conclude that MP is not valid in the case where all the other rules in the library are valid.

“We conclude from this that there are valid inference rules (including MP) that are valid only so long as they are not included in the library of rules to be used. Informally, we can say that there are valid rules that one is not allowed to use (in an unrestricted manner) in one’s proofs. It is the very usage of the rule in inference that invalidates it.”

In order to maintain a valid open library, one must check that the rule is valid itself and that it remains valid when added to the library. A rule is independently valid if it is valid regardless of which library is used by the implication library. The Ad Hoc Rule is an example of an independently valid rule. Any library consistent of only independently valid rules is valid.

The Mods Ponens rule isn’t independently valid. The Modus Ponens rule is contingent on the nature of the library. The Curry Paradox itself provides an example of libraries which MP is not valid.

It is thought that the source of the paradox can be considered to be the misuse of MP. It is suggested that modus ponens is the source of the classical Curry paradox

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Lack of Knowledge implies Knowledge

Posted by allzermalmer on July 28, 2013

Socrates was once opined to have said that all he knows is that he doesn’t know anything, or I know that I don’t know.

There is a formal system known as epistemic logic. It deals with an epistemic operator, K. One of the epistemic logic is known as negative knowledge, in some sense.

Negative Knowledge: ~Kp –> K~Kp or CNKpKNKp

If I don’t know p then I know that I don’t know P. Not knowing p implies knowing that don’t know p.

If I don’t know what it looks like down at the center of the Earth (or Sun), then I know that I don’t know what it looks like down at the center of the Earth (or Sun).

Furthermore, from this Axiom, we may easily show that not knowing something implies knowing something.

All we need is our axiom of negative knowledge, CNKpKNKp, and the law of contraposition. This law, basically, states that we switch the antecedent (i.e. NKp) with the consequent (i.e. KNKp), and we negate both of those propositions when we switch their places.

By the law of contraposition and negative knowledge, we obtain CNKNKpNNKp.
Now we use the law of double negation to the consequent (i.e. NNKp), and we obtain CNKNKpKp.

We obtain that if we don’t know that we don’t something then we know something.

 

 

 

 

 

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Proof of Disjunctive Syllogism

Posted by allzermalmer on July 28, 2013

anguage

(I) Symbols: Ø = contradiction, → = conditional, and [] = Modal Operator
(II) Variables: p, q, r, p’, q’, r’. (Variables lower case)

Well Formed Formula for Language

(i) Ø and any variable is a modal sentence.
(ii) If A is a modal sentence, then []A is a modal sentence.
(iii) If A is a modal sentence and B is a modal sentence, then A implies B (A→B) is a modal sentence.

* A, B, and C are modal sentences, i.e. upper case letters are modal sentences. These upper case letters are “variables as well”. They represent the lower case variables in conjunction with contradiction, conditional, or modal operator.

So A may possibly stand for p, or q, or r. It may also possibly stand for a compound of variables and symbols. So A may stand for q, or A may stand for p→Ø, and etc.

Negation (~) = A→Ø
Conjunction (&) = ~(A→B)
Disjunction (v) = ~A→B
Biconditional (↔) = (A→B) & (B→A)

Because Ø indicates contradiction, Ø is always false. But by the truth table of material implication, A → Ø is true if and only if either A is false or Ø is true. But Ø can’t be true. So A → Ø is true if and only if A is false.

This symbol ∞ will stand for something being proved.

(1) Hypothesis (HY) : A new hypothesis may be added to a proof anytime, but the hypothesis begins a new sub-proof.

(2) Modus Ponens (MP) : If A implies B and A, then B must lie in exactly the same sub-proof.

(3) Conditional Proof (CP): When proof of B is derived from the hypothesis A, it follows that A implies B, where A implies B lies outside hypothesis A.

(4) Double Negation (DN): Removal of double negation ~~A & A lie in the same same sub-proof.

(5) Reiteration (R): Sentence A may be copied into a new sub-proof.

Proof of Disjunctive Syllogism: Because at least one disjunct must be true, by knowing one is false we can infer tat the other is true.

If either p or q and not p, then necessarily true q.

Premise (1) p v q (Hypothesis)
Premise (2) ~p (Hypothesis)
(3) ~p implies q ((1) and Definition v)
Conclusion (4) q (Modus Ponens by (2) and (3))

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Proof of Modus Tollens

Posted by allzermalmer on July 28, 2013

Language

(I) Symbols: Ø = contradiction, → = conditional, and [] = Modal Operator
(II) Variables: p, q, r, p’, q’, r’. (Variables lower case)

Well Formed Formula for Language

(i) Ø and any variable is a modal sentence.
(ii) If A is a modal sentence, then []A is a modal sentence.
(iii) If A is a modal sentence and B is a modal sentence, then A implies B (A→B) is a modal sentence.

* A, B, and C are modal sentences, i.e. upper case letters are modal sentences. These upper case letters are “variables as well”. They represent the lower case variables in conjunction with contradiction, conditional, or modal operator.

So A may possibly stand for p, or q, or r. It may also possibly stand for a compound of variables and symbols. So A may stand for q, or A may stand for p→Ø, and etc.

Negation (~) = A→Ø
Conjunction (&) = ~(A→B)
Disjunction (v) = ~A→B
Biconditional (↔) = (A→B) & (B→A)

Because Ø indicates contradiction, Ø is always false. But by the truth table of material implication, A → Ø is true if and only if either A is false or Ø is true. But Ø can’t be true. So A → Ø is true if and only if A is false.

This symbol ∞ will stand for something being proved.

(1) Hypothesis (HY) : A new hypothesis may be added to a proof anytime, but the hypothesis begins a new sub-proof.

(2) Modus Ponens (MP) : If A implies B and A, then B must lie in exactly the same sub-proof.

(3) Conditional Proof (CP): When proof of B is derived from the hypothesis A, it follows that A implies B, where A implies B lies outside hypothesis A.

(4) Double Negation (DN): Removal of double negation ~~A & A lie in the same same sub-proof.

(5) Reiteration (R): Sentence A may be copied into a new sub-proof.

Proof of Modus Tollens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.

(If p implies q & ~q, then necessarily true that ~p)

Premise (1) p implies q (Hypothesis)
Premise (2) ~q (Hypothesis)
(3) q implies Ø ((2) and of Definition ~)
(4) p (Hypothesis)
(5) p implies q (Reiteration of (1))
(6) q (Modus Ponens by (4) and (5))
(7) q implies Ø (Reiteration of (3))
(8) Ø (Modus Ponens by (6) and (7))
(9) p implies Ø ( Conditional Proof by  (5) through (8))
Conclusion (10) ~p ((9) and Definition of ~)

Shortened version, with some steps omitted, would go as follows.

P (1) p implies q
P (2) ~q
(3) q implies Ø ((2) and Definition of ~)
(4) p (Hypothesis)
(5) q (Modus Ponens by (1) and (4))
(6) Ø (Modus Ponens by (3) and (5))
(7) p implies Ø (Conditional Proof by (3) through (6))
C (8)  ~p ((7) and Definition ~)

Here is an even shorter proof of Modus Tollens, and it only requires the rule of inference of Hypothetical Syllogism:

(1) p implies q (Hypothesis)
(2) q implies Ø (Hypothesis)
(3) p implies Ø (Hypothetical Syllogism by (1) and (2))
(4) ~p (Reiteration of (3) by Definition of ~)

So we have proved that If p implies q and ~q, then ~p is necessarily true.

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