allzermalmer

Truth suffers from too much analysis

Posts Tagged ‘Logic’

False Hypotheses & True Predictions

Posted by allzermalmer on June 23, 2016

In logic, a conjunction is a logical connective that connects two separate propositions. For example, say we have the propositions ‘The Golden State Warriors won the Western Conference Championship of the NBA in 2016’ and ‘The Cleveland Cavaliers won the Eastern  Conference Championship of the NBA in 2016’. We can represent each of those propositions, respectively, as P and Q.

The logical connective of conjunction would combine each of these two separate propositions together. Each of these propositions would be known as a conjunct that makes up a conjunction. Conjunct of P and conjunct of Q make up the conjunction of ‘Both The Golden State Warriors won the Western Conference Championship of the NBA in 2016 & The Cleveland Cavaliers won the Eastern Conference Championship of the NBA in 2016’. This can be represented as ‘P&Q’.

A conjunction is only true when each conjunct is true. A conjunction is false when either one of the conjuncts is false or both conjuncts are false. In the example presented, it is true that both teams won the Conference championships in 2016. So the conjunction is a true proposition.

Logic tells us that from false hypotheses, or hypothesis, that true predictions follow from it.  Suppose that P means ‘The Golden State Warriors won the NBA Championship in 2015’ and that Q means ‘The Golden State Warriors won the NBA Championship in 2016’. From these two propositions, we can form the conjunction of ‘Both The Golden State Warriors won the NBA Championship in 2015 & The Golden State Warriors won the NBA Championship in 2016’. This can be represented as ‘P&Q’.

Taking ‘P&Q’ as a hypothesis, we can prove that some propositions follow from that hypothesis. One of these propositions that follow is P.  So from the hypothesis of ‘Both The Golden State Warriors won the NBA Championship in 2015 & The Golden State Warriors won the NBA Championship in 2016’ that it necessarily follows by rules of logic that ‘The Golden State Warriors won the NBA Championship in 2015’.

Suppose ‘P&Q’ then necessarily follows ‘P’.

The hypothesis presented is false, P&Q is false. One of the conjuncts is false, Q is false. One of the conjuncts is true, P is true. So the conjunction is false. But from this false hypothesis, we find that a true conclusion follows from it.

Advertisements

Posted in Philosophy | Tagged: , , , , | Leave a Comment »

Defense of Skepticism

Posted by allzermalmer on December 28, 2013

This post is based on a paper that was done for a class on epistemology. The paper was about a paper called “A Defense of Skepticism“, authored by Peter Unger, and appeared in the philosophical journal A Philosophical Review, Vol. 80, No. 2 (Apr., 1971), pp. 198-219.

“Both propositions of “Forty-five  plus fifty-six is the same as one hundred and one” and “45+56=101” are propositions that hardly anyone knows, under Peter Unger’s skeptical thesis. Under the skeptical thesis presented by Unger, most people don’t know those propositions are true, though some might, & those people that don’t know those propositions are true typically talk as if those propositions are true. In fact, it will be given that there are a great many propositions that we may reasonably believe or a great many propositions that we may suppose to be true, while never knowing them to be true.

Unger’s skepticism wants to deals within common idea of language and knowledge. Skepticism presented by Unger deals with language, since that is the background in which some philosophers believe that they have solved some skeptical problems about knowledge. For example, the common idea of language and knowledge that Unger is working under is that the language we speak is adequate at expressing truths. From this common idea, Unger presents idea of how our language habits possibly serve us well in practical ways, while these language habits have us saying what’s false rather than true.

Our language habits that involve positive assertions contain special features that Unger want’s to point out, and which also motivates his skeptical thesis. There are absolute terms within our languages or language habits, and some of these absolute terms are basic. From these basic absolute terms, we may build up other absolute terms. Examples of absolute terms are both “flat” and “certain”. The proposition “That is a cube” involves the basic absolute term of “flat”, since a cube’s surface is flat. If we don’t know a basic absolute term of “flat”, then we can’t know what is built up from those basic absolute terms, which is “That is a cube”.

“That is a cube” is a proposition that many people might reasonably believe, or “The road is flat”. But the basic absolute term of “flat” generally fails to apply to the world. A cube is a geometrical object which has no width or depth. However, those objects that we generally come across do have width or depth. So those objects that we call “cubes” aren’t something that applies to the world. We may talk as if “That is a cube” or “The road is flat”. It is reasonable to suppose or believe that they are true. From whatever is built, or has any part of, of that basic absolute term would also generally be false.

The skepticism presented by Unger implies some things that might be deemed impossible to accept, since it would imply something about the functioning of our language. It would attack a common idea about language being adequate to express truth. Unger’s thesis would have it that common terms of language involve the use of error systematically in expressing truth. Two common terms that would be found to use error systematically would be “know” and “knowledge”. While we believe what we say is true, what we believe is actually false.

Suppose that an individual believes that “this region of space is a vacuum” is true. Now further suppose that contrary to the individual’s belief, “this region of space isn’t a vacuum” since “this region of space has the slightest trace elements of gaseous stuff”. Now further suppose that for practical purposes principle, there isn’t an important difference whether you falsely believe the first supposition or truly believes the second supposition, i.e. “this region of space has the slightest trace elements of gaseous stuff”.

One might think that the first supposition implies the practical purpose principle, since there is no important difference. Things would go on as we practically expected with the region of space being a vacuum, since if the region is a vacuum then whatever gaseous content it has is none. Even giving such a thought, there is still something that is unique. The second supposition doesn’t imply the practical purpose principle. The principle may be derived from the first supposition and can’t be derived from the second supposition. In other words, from “this region of space is a vacuum” we can derive that the region of space doesn’t contain any gaseous elements or doesn’t practically contain any gaseous elements. The practical result of the first supposition would show that there aren’t any gaseous elements, so the practical result would have been no gaseous elements. However, by the second supposition, wouldn’t practically show up as a result. So our belief can still be reasonably supposed true, even though false.

The example given above is to help point out that we can have false beliefs, even though the beliefs are reasonable, and they don’t clash with experiences of life, or way we experience the world. We can have many false beliefs, based on some of the account that was given above. The practical purpose would come from positive assertions, since the first supposition was a positive assertion and the second assertion was a negative assertion. From the positive assertion we were able to derive the practical purpose principle, and we couldn’t derive the practical purpose principle from the negative assertion. The individual isn’t in position to determine if the region of space contains gaseous elements, given the practical limitation that was given from the derived principle from first supposition.

Those terms of knowledge belong to a class of terms in our language, and these are those absolute terms. “Flat” is an absolute term, so saying that “a surface is flat” is also saying that matters of degree are not instanced in the surface to any degree. Being flat means no degrees of being bumpy or having a curve, i.e. perfectly flat. “Bumpy” and “curve” are taken as examples of relative terms, and so we notice that there is some connection between absolute terms and relative terms.

Relative terms and some absolute terms, specifically those of basic absolute terms, have the special ability of being modified by many different terms. We can take a term such as “very”, can be applied to either relative terms or basic absolute terms, and we can obtain something like “the table is very bumpy” or “the table is very flat”. These modifications of either the relative term or basic absolute term are one based on matters or degree, or indicate matters of degree.

“Cube” is an absolute term, even though it isn’t a basic absolute term. This is because “Cube” is built up off of the absolute term of “flat” & “straight”. “Cube” doesn’t admit of matter of degree, even though the basic absolute terms used to build up the term do admit to matter of degree. In other words, it takes two basic absolute terms of “flat” and “straight” to build up the absolute term of “cube” & “cube” doesn’t allow for matter of degree while “straight” and “flat” do allow for matter of degree. Not all absolute terms can be modified by matters of degree, but some like “flat” or “straight” can be, and all relative terms can be modified by matters of degrees.

Some might think to take basic absolute terms as the same relative terms, but this won’t do. There is a special distinction between both types of terms when it comes to matter of degrees. When we say that “a surface is very flat” or “a surface is very bumpy”, we are saying how flat the surface is or how bumpy the surface is. Intuitively, there is a difference between these two. Flat being an absolute term, it has an ultimate location in which we may judge things. We are saying how close the surface is to being flat. We have perfectly flat, which is what the absolute term of flat means, and saying it is very flat allows us to say how close it is being perfectly flat since we have the index of perfectly flat to compare it with.

When it comes to relative terms, like “a surface is very bumpy”, since “a surface is bumpy” implies “a surface is very bumpy”. However, from “a surface is very bumpy” we can’t derive that “a surface is bumpy” because might be that the surface isn’t bumpy at all. We notice an asymmetry, and this comes from an intuitive level. Something would seem strange in our common language to express “a surface is bumpy” and deny that “a surface is very bumpy” is implied, or “a surface is very bumpy” and affirm that “a surface is bumpy”.

Either both the first surface is flat and the second surface is not flat or both the first surface is closer to being flat than the second surface. This is a more complex way in which Unger deals with the matter of degrees when they are applied to basic absolute terms. This helps to change things so that when we deal with relative terms and matter of degree, the same interpretation of common language doesn’t hold.

Basic absolute terms can at least be partially defined by relative terms, or the matter of degree terms. And since absolute terms are defined by basic absolute terms, this in turn means that absolute terms that aren’t basic absolute terms are also partially defined by relative terms. These relative terms help to point out the negative accept of the skeptical thesis presented by Unger. For supposing that “flat” is a basic absolute term, which is defined in part by a relative term, means that something isn’t flat at all, or not in the least, bumpy. This is the negative relative requirement that basic absolute terms have to meet when partially defined by relative terms.

Absolute terms and Relative terms are both part of our language, but we also have things that are part of our language that are neither absolute terms nor relative terms. Some of these terms are unmarried or married, true and false, or right and wrong. But Unger points out that some of these terms can be taken as “absolute” in some language.

Some terms of our language are followed by propositional clauses, and we may call these terms propositional terms. So one might wonder, are these propositional terms, like “certain”, absolute or relative terms. A term like “certain” has two things that need to be made clear about them, or interpretations that may be given to it. (1) Certain in which certain is not certain of anything, or (2) certain in which certain is certain of something. An example of (1) is “It is certain that it is raining”, since the term “it” doesn’t appear to have any reference. This is the impersonal context, which is the impersonal idea of certainty. An example of (2) is “He is certain that it is raining”, since the term “he” does appear to have a reference. This is the personal context, which is the personal idea of certainty.

Unger believes that certainty has to contain both of those conditions; it is both impersonal certainty and personal certainty. What comes from each of these on their own is that certainty involves no doubt. So we get that “If certain that p then isn’t doubtful that p” with impersonal certainty. With personal certainty we get that, “In his mind, if he’s certain that p then isn’t doubtful that p.” All doubt is absent in his mind.

Certain are now connected negative definitions of certainty, which is an absolute term. Certainty or Certain are common concepts with language, and they are built on absolute term which has a negative definition.

When an individual says that he is certain that p, they are saying they aren’t confident of p & more than confident of p. If they say they are confident that p then they are saying they are confident that p. So there is a difference between an individual being certain and being confident. So you can be as confident as you want because of the highest reasonable belief.

Take these two propositions, either (1) “He’s (really) very certain of p” or (2) “he’s very certain of p”. The second proposition says more about certainty than the first proposition. The second proposition gets ride of one matter of degree. Each of these modifiers is saying that they aren’t certain, but one less degree of modifiers saying that they aren’t certain. This might seem implausible at first, but there has been a pattern in which to place our languages of language to express truth.

Someone saying that “I’m more certain that p than I am that q” is the same as “I am either certain that p while not certain that q or I’m more nearly certain that p than I am that q”. The first part of the disjunction tells us that either p or q is certain and it was proposition p. The second disjunction tells us that aren’t certain of either proposition and one is of a higher degree that the other. However, someone saying that I’m more confident that p than I am that q” isn’t the same as “I’m either both confident that p and not confident that q or I’m more nearly confident that p than q.”, since confident of both.

Given these expositions of propositional terms, absolute terms v. relative terms, basic absolute terms, impersonal certainty v. personal certainty, we can come to skepticism about many things that we reasonably believe. It can be taken that “lots of surfaces of physical things are flat” is a reasonable belief. But this comes to contradict the experience of life. We take a microscope and we start to examine those objects that we commonly come upon, and we start to notice that they aren’t flat. They take on the form of being bumpy. So now the absolute term is one that is false and yet for practical purposes it is true. The absolute term was even a reasonable belief that is or was held.

One of the basic problems of absolute terms is that there are counter-examples to them, and the absolute terms are part of language to express truth. These absolute terms, at least basic ones, are shown to be false by experiences of life. However, these terms are very useful and they do express some truth. Absolute terms have reason to doubt, since we do have a counter-example. Most absolute terms would have counter-examples, but that doesn’t mean that all absolute terms have counter-examples.

Going back to previous example, experience of life presents what seems to be a smooth stone & look through a microscope at the smooth stone. The smooth stone is found to actually be bumpy, which means that it isn’t smooth. To account for the stone being bumpy, an inference to the best explanation could be used. It could be that the smoothness is built up from the finer part of the stone which has small bumps. It can further be better explained that the bumps are made of something even smaller, like atoms, which combine in a certain way in which smooth stone is final outcome, while there are no smooth stones in the combination to begin with, i.e. neither atoms are smooth nor atoms are stones. This belief would have plenty of evidence, and be a reasonable belief. However, the degrees of “deeper” explanations to account for the counter-example can eventually end in an absolute matter where there is no counter-example to be found. So we eventually come to the point that we should suspend judgment on this issue. We don’t know either.

The real sting of Unger’s skepticism comes down to this form: If person is certain of p, then not anything of which the person is more certain. So the individual can be certain of p, which means that person isn’t as certain, i.e. isn’t certain, of q. This comes back to this point, if more certain of another thing, then either certain of other while and not being certain of first or more nearly certain of other thing than of the first. Suppose that it is logically possible that there’s something an individual might be more certain of than they are now of a given thing, then the person wasn’t really certain to begin with.

Is it reasonable to believe that there are automobiles? It would seem to be an experience of life. However, a dilemma can be presented to them. Either more certain that there are automobiles or aren’t. This can be because someone is more certain that they exist than certain automobiles are an experience of life. Since they are more certain that they exist, then certain that they exist and aren’t certain that automobiles exist. So when someone is presented with something that they think they are more certain of than another, they are saying that they aren’t certain of second. If they hold open that what they hold to be more certain can possibly be false, then they aren’t certain of it to begin with, either.

So from this, we can know one thing, but most everything else that we think we are certain of isn’t certain of. So we can hold that we are more certain of our own existence than of 45+56=101. For practical purposes though, we are certain that 45+56=101. This would be us having a reasonable belief without it being true, i.e. certain that 45+56=101. Even something like some basic arithmetic can fall for skepticism, even though it is not a universal skepticism in which no one knows anything.

The skepticism that Unger presents deals with knowledge being certain, which has also been a common idea of epistemology. However, some suppose that knowledge requires just belief, or at least reasonable belief such that, if both believe that p implies know that p & believe that p implies p then p implies know that p. This would appear to lead to omniscience, which seems to contradict experience of life. So such a supposition would be an absolute term which is false but practically useful. This belief is reasonable is because it meets certain conditions, which helps allow us to say that we have knowledge. However, these conditions don’t exclude having a false belief.

So Unger’s skepticism is an attack on language is capable of expressing truth. Unger holds that language is capable of expressing truth, but that it also systematically expresses error as well. Language relies on positive assertions, which require negative definitions. These definitions are based on absolute terms that are basic, while there are absolute terms that aren’t basic and there are relative terms. The basic absolute terms help to define absolute terms that aren’t basic. These basic absolute terms end up having counter-examples. Our common way of expressing language would have error.

One of the words that we use for knowledge is “certain”, which is an absolute term. However, based on the definition of being certain, there can only be one thing that we are certain about. So we may use the knowledge term of certain, or that we have knowledge, but we truly don’t have knowledge. The skepticism that Unger presents show there isn’t much of knowledge that we have, since there isn’t much that we are certain about, but that doesn’t mean we don’t have any knowledge. Unger’s skeptical thesis is consistent that we don’t know anything, but it doesn’t imply that we don’t know anything. We just know less than we say that we know or we are more certain of one thing than we are of another, or we aren’t certain at all.”

Posted in Philosophy | Tagged: , , , , , , , , , , , , | Leave a Comment »

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)

Posted in Philosophy | Tagged: , , , , , , , , , , , , | Leave a Comment »

Categorical Propositions aren’t Same as Conditional Propositions

Posted by allzermalmer on September 20, 2013

It is sometimes held that categorical propositions are equivalent to conditional proposition. However, at least with propositional logic, this isn’t necessarily true.

Categorical proposition: All X are Y.
Conditional proposition: If X then Y.

In other words, it is sometimes held that, All X are Y if and only if X implies Y.

This can be shown false by a very simple method that Categorical propositions aren’t equivalent to Conditional propositions. All we have to do is replace the variables X or Y with Truth Values, and see what the Truth of the proposition as a whole will come out to.

Suppose that X is True and Y is True. Now we replace those variables with their Truth Values in the Statements.

Categorical Proposition: All True are True= True.
Conditional Proposition: If True then True= True.

Suppose that X is False and Y is False.

Categorical Proposition: All False are False=True.
Conditional Proposition: If False then False=True.

Suppose that X is True and Y is False.

Categorical Proposition: All True are False= False.
Conditional Proposition: If True then False= False.

Here is where the Fallacy comes in of thinking Categorical Propositions are equivalent to Conditional Propositions.Suppose that X is False and Y is True.

Categorical Proposition: All False are True= False.
Conditional Proposition: If False then True=True.

We immediately notice that their truth value’s are not equivalent when each variable has the same truth value. This shows that categorical propositions necessarily say something different from conditional propositions.

The only way that Categorical Propositions will say the same thing as Conditional Propositions is if the Subject of the Categorical Proposition isn’t False & the Predicate isn’t True. In other words, the Subject of the Categorical Proposition must Exist.

All Mermaids are creatures that swim in the Ocean if and only if Mermaids implies creatures swim in the Ocean. Mermaids can’t not exist for this equivalency to hold with the Conditional, while the Conditional doesn’t need that Mermaids exist.

Posted in Philosophy | Tagged: , , , , , , , , | Leave a Comment »

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]

Posted in Philosophy | Tagged: , , , , , , , , | Leave a Comment »

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 in Philosophy | Tagged: , , , , , , , , , , , | Leave a Comment »

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.

 

 

 

 

 

Posted in Philosophy | Tagged: , , , , , , , , , | Leave a Comment »

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))

Posted in Philosophy | Tagged: , , , , , , , , , , , , | 4 Comments »

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.

Posted in Philosophy | Tagged: , , , , , , , , , , , , , , , , , , | Leave a Comment »

Logical Analysis of Consciously Held Beliefs

Posted by allzermalmer on June 10, 2013

Axioms of Consciously Believing

Axiom: “I believe that p” if and only if “I believe that I believe that p”
EBpBBp = Bp <-> BBp

Axiom: “I don’t believe that p” if and only if “I believe that I don’t believe that p”
ENBpBNBp = ~Bp <-> B~Bp

Axiom: If “I believe that not p” then “I don’t believe that p”
CBNpNBp = B~p -> ~Bp

Axiom: If “I believe that p implies y” then “I believe that p implies I believe that y”
CBCpyCBpBy = B(p->y) -> (Bp->By)

Theorems of Consciously Believing

T1: “I don’t believe that both p and not p”
NBKpNp = ~B(p&~P)

T2: “I believe that p is equivalent to y” implies “I believe that p is equivalent to I believe that y”.
CBEpyEBpBy = B(p<->y) -> (Bp<->By)

T3: “I believe that p or I believe that y” implies “I believe that both p or y”
CABpByBApy = (BpvBy) -> B(pvy)

T4: “I believe that both p & y” if and only if “I believe that p & I believe that y”
EBKpyKBpBy = B(p&y) <-> (Bp&By)

T5: “I believe that p” implies “I don’t believe that not p”
CBpNBNp = Bp->~B~p

T6: “I believe that either p or y & I don’t believe that p” implies “I don’t believe not p”
CKBApyNBpNBNp = B(pvy)&~Bp -> ~B~p

T7: “I believe that either p or y & I believe that not p” implies “I believe that y”
CKBApyBNpBy = B(pvy)&B~p -> By

T8: If “I believe that p implies y” then “I believe that not y implies I believe that not p”
CBCpyCB~yB~p = B(p->y) -> (B~y->B~p)

T9: If “I don’t believe that p implies y” then “I don’t believe that not p & I don’t believe that y”.
CNBCpyKNBNpNBy = ~B(p->y) -> (~B~p&~By)

T10: If “I believe that I believe that p or I believe that y” then “I believe that either p or y”
CBABpByBApy = B(BpvBy) -> B(pvy)

T11: If “I believe that p implies y” then “I believe that I believe that p implies y”
CBCpyBCBpy = B(p->y) -> B(Bp->y)

T12: “I believe that I believe that p implies p”
BCBpp = B(Bp->p)

http://video.msnbc.msn.com/mitchell-reports/34510812#52158575

http://bcove.me/bqpor8gd

Posted in Philosophy | Tagged: , , , , , , , , , , , , , , , , , , , , , , , | Leave a Comment »