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Posts Tagged ‘Possibility’

The Logic of Discussions

Posted by allzermalmer on June 7, 2013

This blog will be based on a logical system developed by Polish logician Stanislaw Jakowski. It was published in the journal Studia Logica: An International Journal for Symbolic Logic, T. 24 (1969), pp. 143-1960

Implication (–> or C), Conjunction (& or K), Disjunction (v or A), Equivalence (<–> or E), Negation (~ or N, Possibility (<> or M), Necessity ([] or L),  and Variables (P or p, Q or q, R or r).

P = <>P or p = Mp
<>P = ~[]~P or Mp = NLNp

Discussive Implication (D): P–>Q = <>P–>Q or Dpq = CMpq
Discussive Equivalence (T): P<–>Q = (<>P–>Q) & (<>Q–><>P) or Tpq = KCMpqCMqMp

D1: P–>P = Dpp
D2: (P<–>Q) <–> (Q<–>P) = TTpqTqp
D3: (P–>Q) –> ((Q–>P)–>(P<–>Q)) = DDpqDDqpTpq
D4: ~(P&~P) = NKpNp [Law of Contradiction]
D5: (P&~P) –>Q = DKpNpq [Conjunction Law Overfilling]
D6: (P&Q) –>P = DKpqp
D7: P –> (P&Q) = DpKpq
D8: (P&Q) <–> (Q&P) = TKpqKqp
D9: (P&(Q&R)) <–> ((P&Q)&R) = TKpKqrKKpqr
D10: (P–>(Q–>R)) –> ((P&Q)–>R) = DDpDqrDKpqr [law importation]
D11: ((P–>Q)&(P–>R)) <–> (P–>(Q&R)) = TKDpqDprDpKqr
D12: ((P–>R)&(Q–>R)) <–> ((PvQ)–>R) = TKDprDqrDApqr
D13: P <–> ~~P = TpNNp
D14: (~P–>P) –> P = DDNppp
D15: (P–>~P) –>~P = DDpNpNp
D16: (P<–>~P) –> P = DTpNpp
D17: (P<–>~P) –>~P = DTpNpNp
D18: ((P–>Q)&~Q) –>~P = DKDpqNqNp

D19: ((P–>)&(P–>~Q)) –>~P = DKDpqDpNqNp
D20:  ((~P–>Q)&(~P–>~Q)) –> P = DKDNpqDNpNqp
D21:  (P–>(Q&~Q)) –>~P = DDpKqNqNp
D22:  (~P–>(Q&~Q)) –> P = DDNpKqNqp

D23: ~(P<–>~P) = NTpNp
D24: ~(P–>Q) –> P = DNDpqp
D25: ~(P–>Q) –> ~Q = DNDpqNq
D26: P–> (~Q–>~(P–>Q)) = DpDNqNDpq

A formulation of Aristotle’s Principle of Contradiction would be: “Two contradictory sentences are not both true in the same language” or “Two contradictory sentences are not both true, if the words occurring in those sentences have the same meanings”.

In Two Valued Logic, there is a Theorem known as the Law of Overfilling, or Implicational Law of Overfilling, or Dun Scotus Law, or L2 Theorem 1.

L2 Theorem 1: P—> (~P—>Q)

If an assertion implies its contradiction, then that assertion implies any and all statements.

“A deductive system…is called inconsistent, if its theses include two such which contradict one another, that is such that one is the negation of the other, e.g., (P) and (~ P) . If any inconsistent system is based on a two valued logic, then by the implicational law of overfilling one can obtain in it as a thesis any formula P which is meaningful in that system. It suffices…to apply the rule of modus ponens twice[ to P—> (~P—>Q)]. A system in which any meaningful formula is a thesis shall be termed overfilled.”

  1. Assume: P—> (~P—>Q)
  2. Modus Ponens: P
  3. Conclusion: ~P—>Q
  4. Modus Ponens: ~P
  5. Conclusion: Q

“[T]he problem of the logic of inconsistent systems is formulated here in the following manner: the task is to find a system of the sentential calculus which: (1) when applied to the inconsistent systems would not always entail overfilling, (2) would be rich enough to enable practical inference, (3) would have an intuitive justification. “

This means that Discussive Logic does not have the theorem of implicational law of overfilling. The theorem is not always true in Discussive Logic. Discussive Logic does not entail that a contradiction does not always entail any and all assertions. So Discussive Logic rejects the truth of the theorem P—> (~P—> Q), which is a theorem is two value logic, i.e. been proven true under conditions of two value logic.

Kolmogorov’s System

Here are Four axioms from Hilbert’s positive logic, and one axiom introduced by Kolmogorov.

K 1: P—> (Q—>P)
K 2: (P—> (P—>Q))—> (P—>Q)
K 3: (P—> (Q—>R))—> ((Q—> (P—>R))
K 4: (Q—>R)—> ((P—>Q)—> (P—>R))
K 5: (P—>Q)—> ((P—>~Q)—>~P)

Under these axioms, Two valued logic cannot be proved. Implicational Law of Overfilling not being provable in Discussive Logic implies that Two Valued logic cannot be proved in Discussive Logic. This entails that there might be overlap between Two Valued logic and Discussive Logic, but there is not a total overlap between Two Valued logic and Discussive Logic. Not all theorems of Two Valued logic will be theorems in Discussive logic (like law of overfilling), but some theorems of two valued logic are theorems in Discursive logic.

From these Axioms and the rule of inference known as Modus Ponens, there is one theorem which has some similarities to implicational law of overfilling.

K 9: P—> (~P—>~Q)

It is not the only Theorem that can be derived from the Axioms and Modus Ponens. Here is a list of some Theorems that can be derived by using  Modus Ponens on the Axioms.

K 6 : (P—>Q)—> ((Q—R)—> (P—>R))
K 7:  ((Q—>P)—>R)—> (P—>R)
K 8:  P—> ((Q—>~P)—> ~Q)
K 9:  P—> (~P—>~Q)

Proof of how K 6 – K 9 are derived are ignored for here. All that needs to be known is that K3 and applications of Modus Ponens is equal to If K4 then K 6. K 6 and applications of Modus Ponens is equal to If K 1 then K 7. K 7 and applications of Modus Ponens is equal to If K 5 then K 8. K 6 and applications of Modus Ponens is equal to If K 8 then K 7 implies K 9.

This forms Kolmogorov’s System.

Lewis System of Strict Implication

Strict Implication is defined by modal operator of “it is possible that P” or <>P. So “P strictly implies Q” is equal to “It is not possible that both P and not Q”. But taking the conditional statement —> as strict implication means that the implicational law of overfilling is not a theorem.

Material Implication as a conditional is usually defined by the logical relationship of a conjunction.

Material Conditional: P—>Q = ~(P & ~Q)
“P implies Q” is equal to “Not both P and not Q”
Strict Conditional: P—» Q = ~<>(P & ~Q)
“P strictly implies Q” is equal to “It is not possible that both P and not Q”

Under Strict Implication, Law of Overfilling is not a theorem. Under Material Implication, Law of Overfilling is a theorem. And set of theorems which include only strict implication and not material implication is very limited.

Many Valued Logics

Based on a certain Three Value logical matrix, which shall be ignored, the Law of Overfilling is not a theorem. But there is another theorem in the Three Value logic which has some similarity to the Law of Overfilling.

L 1: P—> (~P—> (~~P—> Q))

Based on the theorem (stated above) of this specific three valued logic, it holds the overfilling of a system when it includes the inconsistent thesis of P, ~P, and ~~P. And the implicational theses of two valued calculus remains valued in the three valued logic. But the three valued logic also holds other theorems that are not in two valued logic, which are as follows.

L 2: P—> ~~P
L 3: ~~~P—> P
L 4: ~P—> ~(P—> P)

So in the three valued logic, which is ignoring the logical matrix of this three valued logic, we cannot obtain the Law of Overfilling. The Law of Overfilling will thus be a theorem in two valued logic but not a theorem in this three valued logic. But the three valued logic has a theorem that is similar to the Law of Overfilling but is not equivalent to the Law of Overfilling. This three value logic also has some theorems that are not theorems in two valued logic. Besides the Law of Overfilling not being a theorem in the three valued logic, the rest of implicational theorems in two valued logic are theorems in the three valued logic.

Calculus of Modal Sentences (M2)

The Modal Sentences of (M2) will assume that modal assertions are either true or false, or simply that the Modal sentences are two valued. But now suppose that there are factors that do not allow for the assertion P to be determined strictly to be either true or false.

For example: Suppose that you are flipping a coin. Suppose that you make the assertion that “During the game heads will turn up more times than tails will” and this is represented by the variable of P. There will be certain sequences that turn up so that P is true, and there will be certain sequences that turn up so that P is false. So P may take on both true and false.

“It is necessary that P” = []P

Taking the example above, we can say that “P occurs for all possible events”.

Q is any formula that includes (1) operators —>, V, &, <—>, ~ and [], (2) and variables p,q,r,s..etc. R is any formula that is already a Q formula and is replacements of variables in Q by interpreting them as P(x), Q(x), R(x), S(x)…etc, and interpreting [] by universal quantifiers “for every x”. Every Q satisfies (1) and (2) and every R satisfies (1) and (2), and additionally satisfying (3).

The operators are implication, disjunction, conjunction, equivalence, and necessity. These are applied to variables or connects variables. When those conditions are met, then it is a formula of Q. The replacement of the variables and [] are formula of R. (1) and (2) can be recognized as P–>Q, or []P—>Q, or []P—>P. We can replace those variables to formulas in R: P(x) —> Q(x), or For every x, P(x) —> Q(x), or For every x, P(x) —> P(x).

“It is possible that P” = <>P

<>P can be taken as “it is not necessary that not P”.

<>P = ~[]~P

Like we could change []P into “for every x”, we may also change <>P into “for some x”.

Definition of Discussive Implication and Discussive Equivalence

As is known, even sets of those inscriptions which have no intuitive meaning at all can be turned into a formalized deductive system. In spite of this theoretical possibility, logical researches so far have been taking into consideration such deductive systems which are symbolic interpretations of consistent theories, so that theses in each such system are theorems in a theory formulated in a single symbolic language free from terms whose meanings are vague.

But suppose that theses which do not satisfy those conditions are included into a deductive system. It suffices, for instance, to deduce consequences from several hypotheses that are inconsistent with one another in order to change the nature of the theses, which thus shall no longer reflect a uniform opinion. The same happens if the theses advanced by several participants in a discourse are combined into a single system, or if one person’s opinions are so pooled into one system although that person is not sure whether the terms occurring in his various theses are not slightly differentiated in their meanings. Let such a system which cannot be said to include theses that express opinions in agreement with one another, be termed a discussive system. (Italics is authors and Bold is mine)

Each the theses in discussive logic are preceded so that each thesis has the speaker has the reservation such that each assertion means  “in accordance with the opinion of one of the participants in the discussion” or “for a certain admissible meaning of the terms used”. So when you add an assertion to a discussive system, that assertion will have a different intuitive meaning. Discussive assertions have the implicit condition of the equivalence to <>P.

King Solomon having to decide between two harlots claiming to be the mother of a baby. Woman A claimed to be the mother of the baby and not the mother of the dead baby, and Woman B claimed to be the mother of the baby and not the mother of the dead baby. King Solomon being the arbitrator, under Discussive assertions, would have taken each Woman’s claim as having the prefix of possibility, or “it is possible that Woman A is the mother” or “it is possible that Woman A is not the mother”.

Discussive logic is not based on ordinary two valued logic. Discussive logic would not hold Modus Ponens in all cases if it did.

Take the statement P—>Q is asserted in a discussion. It would be understood to mean “It is possible that If P, then Q”. P is asserted in the same discussion. It would be understood to mean “It is possible that P”. Q would not follow from the two assertions in the discussion. For by Q would not follow in the discussion because Q stands for “It is possible that Q”. So it is invalid to infer from “It is possible that if P, then Q” and “It is possible P” that “It is possible that Q”. But people might assume the normal two value logic in which Modus Ponens holds in all cases.

For Discussive Logic, Discussive Implication is defined as such:

Definition of Discussive Implication: P—>Q = <>P—>Q

There is a theorem of M2 based on Discussive Implication.

M2 Theorem 1: <>(<>P—>Q) —> (<>P—><>Q)

So Modus Ponens may be used in Discussrive Logic when we understand that from (<>P—><>Q) and <>P, we may infer that <>Q by Modus Ponens.

For Discussive Logic, Discussive Equivalence is defined as such:

Definition of Discussive Equivalence: P <—> Q = (<>P—>Q) & (<>Q—><>P)

There is a theorem of M2 based on Discussive Implication.

M2 Theorem 2: <> (P<—>Q) —> (<>P—> <>Q)
M2 Theorem 3: <> (P <—> Q) —> (<>Q —> <>P)

Two valued Discussive System of Sentential Calculus: D2

The system of D2 (i.e. Discussive Logic) of two valued discussive sentential calculus is marked by the formula T, and are marked by the following properties: (1) Sentential variables and functors of Discussive Implication, Discussive Equivalence, Disjunction, Conjunction, and Negation. (2)  precedening T with the symbol of <> yields a theorem in two valued sentential calculus of modal sentences M2.

As the author says, “The system defined in this way is discussive, i.e., its theses are provided with discussive assertion which implicitly includes the functor <>/ This is an essential fact, since even such a simple law as P—>P, on replacement of —> with —-> (i.e. discussive implication leads) to a new theorem.”

D2 Theorem 1: P—>P

D2 is not a theorem in M2, specifically because M2 did not have discussive implication. But in order to make D2 theorem 1 into a theorem in M2, you have to add <> to D2 theorem 1 like this:

M2 Theorem 4: <>(P—>P)

System M2 is decidable, so the discussive sentential calculus D2, defined by an interpretation in M2 is decidable too.

Methodological Theorem 1: “Every thesis T in two valued sentential calculus which does not include constant symbols of —>, <—>, V, becomes a thesis in T(d) in discusive sentential calculus D2 when in T the implication symbols is replaced by the [discurssive implication], and the equivalence symbols are replaced by [discusrive equivalence]. “

“Proof. Consider a formula T(d) constructed so as the theorem to be proved describes. It is to be demonstrated that <>T(d) is a thesis in M2. It is claimed that <>T(d) is equivalent to some other formulae; the equivalences will be proved gradually.”

Here are a couple more M2 Theorems.

M2 Theorem 5: <>(P—>Q) <—> (<>P—><>Q)
M2 Theorem 6: <>(P <—> Q) <—> (<>P <—> <>Q)
M2 Theorem 7: <>(P v Q) <—> (<>P v <>Q)

These theorems are about the distribution of <> over the variables. For example, M2 Theorem 5 distributes <> over implication, and M2 Theorem 6 distributes <> over equivalence, and M2 Theorem 7 distributes over Disjunction. M2 Theorem 5 and M2 Theorem 6 have Discussive Implication and Discussive Implication as the antecedents, respectively.

This shows how we can replace Discurssive Implication with regular implication and how we can replace Discurssive Equivalence with regular equivalence. So from <>(P—>Q), which contains Discurssive Implication, can be replaced with regular implication as <>P—><>Q. The form <>(P<—>Q), which contains Discurssive Equivalence, can be replaced with regular implication as <>P,—><>Q. Discurssive assertion like <>(PvQ) has the equivalence in M2, or Modal Logic, as <>P v <>Q.

The procedure yields the formula W, which is equivalent to <>T(d) and includes (1) only the symbols —>, <—>, and V, (2) variables, and (3) symbols <> in certain special positions, like each variable is directly preceded by <> and each symbol <> directly precedes a variable. Forming T(d) from the thesis T belonging to two value logic is possibly be seen that W can be obtained from T by preceding each variable by <>. For example, precede the variable P by <>P or precede the variable Q by <>Q. This procedure would yield the following theorems in M2.

(a) W is a result of the substitution in T
(b) <>T(d) is equivalent to W.
Hence T(d) is a thesis of D2

The theorems just listed above, immediately yields these theorems in Discussive Logic:

D2 Theorem 2: (P<—>Q) <—> (Q<—>P)
D2 Theorem 3: (P—>Q) —> ((Q—>P) —> (P<—>Q))

Each of the connectives in the D2 theorem just listed are Discurssive Equivalence for D2 Theorem 2 and Discurssive Implication for D2 Theorem 3.

Methodological Theorem 2: If T is a thesis in the two valued sentential calculus and includes variables and at the most the functors V, &, ~, then (1) T and (2) ~T —> q, are thesis in D2. The implication of (2) is Discurssive Implication.

Proof: The symbols V, &, and ~ retain respective meanings in M2 and D2, and that (3) []T is a thesis in M2. The symbols V, &, and ~ retain respective meanings in M2 and D2 and that (3) []T is a thesis in M2. Hence (1) by M2 Theorem 8 []P—><>P and (2) by M2 Theorem 9 []P—><>(<>~P—>Q).

M2 Theorem 8: []P—><>P
M2 Theorem 9: []P—> <>(<>~P—>Q)

We may apply the Methodological Theorem 2 to the Two Valued Logic Theorem of ~(Pv~P), i.e. Aristotle’s Principle of Contradiction.

L2 Theorem 3: ~(P&~P)

“Methodological Theorem 2 and Law of Contradiction in Two valued logic yieleds – in view of the law of double negation- the following theorem of Discussive logic.”

D2 Theorem 4: ~(P&~P) [Law of Contradiction]
D2 Theorem 5: (P&~P) —> Q [Conjunctional Law of Overfilling]

What these two theorems are basically stating is this: Suppose that we have an individual in a discussion, and this individual holds to the Discussive assertion of (P&~P), this individual would hold inconsistent opinions. And in Discussive logic, when an individual holds to inconsistent opinion, the persons opinion implies any and all discussive assertions. This basically forbids an individual from holding to discussive assertions that are contradictory to one another by D2 Theorem 4, and if we do hold to contradictory discussive assertions then any discussive assertion follows from the conjunction of contradictory discussive assertions. This is similar to Law of Overfilling in two value logic but not exactly the same.

We also have the following theorems in Discussive Logic.

D2 Theorem 6: (P&Q) —> P
D2 Theorem 7: P—> (P&P)
D2 Theorem 8: (P&Q) <—> (Q&P)
D2 Theorem 9: (P& (Q&R)) <—> ((P&Q) & R)
D2 Theorem 10: (P—> (Q—>R)) —> ((P&Q) —>R) [Law of Importation]
D2 Theorem 11:((P–>Q) & (P—>R)) <—> (P—>(Q & R))
D2 Theorem 12: ((P—>R) & (Q—>R)) <—> ((PvQ) —> R)
D2 Theorem 13:P <—> ~~P
D2 Theorem 14:(~P—>P) —>P
D2 Theorem 15:(P—>~P)>~P
D2 Theorem 16:(P<—>~P) —>P
D2 Theorem 17: (P<—>~P) —>~P
D2 Theorem 18: ((P—>Q) & ~Q) —> ~P

There are laws of inference by reductio ad absurdum that remain valid in Discurssive Logic.

D2 Theorem 19:((P—>Q) & (P—>~Q)) —> ~P
D2 Theorem 20:((~P—>Q) & (~P—>~Q)) > P
D2 Theorem 21:(P> (Q&~Q)) —> ~P
D2 Theorem 22: (~P —> (Q&~Q)) —>P

Here are some other theorems of Discurssive Logic.

D2 Theorem 23:~(P <—> ~P)
D2 Theorem 24:~(P—>Q) —>P
D2 Theorem 25: ~(P—>Q) —>~Q
D2 Theorem 26: P —> (~Q—>~(P—>Q))

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Whatever Is Conceivable Is Possible

Posted by allzermalmer on September 27, 2012

I am going to quote one little section in a book called Hume’s First Principles by Robert Fendel Anderson. This first part of the book is on Perceptions, and the first principle gone over on Perceptions is “Whatever is Conceivable is Possible”.

“The principle of the possible existence of whatever is conceivable is one which Hume finds both an evident principle and already an established maxim in metaphysics[1]. The application of the principle is frequently restricted to that which is clearly and distinctly conceivable: “…nothing of which we can form a clear and distinct idea is absurd and impossible.”[2] Again: “To form a clear idea of anything, is an undeniable argument for its possibility…”[3]. The possibility of existence, therefore, is of the essence of whatever is clearly and distinctly conceived; that is, its possibility is included or implied within it: “ ‘Tis an establish’d maxim in metaphysics, That whatever the mind clearly conceives includes the idea of possible existence…”[4] and: “Whatever can be conceiv’d by a clear and distinct idea necessarily implies the possibility of existence….”[5]

A clear and distinct idea, according to Hume’s doctrine, is one which neither contains nor implies a contradiction: “Now whatever is intelligible, and can be distinctly conceived, implies no contradiction…”[6] Again: “How any clear, and distinct idea can contain circumstances, contradictory to itself, or to any other clear, distinct idea, is absolutely incomprehensible….”[7] In saying that whatever is clearly and distinctly conceived is possible, therefore, it appears to be Hume’s intention also that whatever is self-consistent and noncontradictory is possible:

“Whatever can be conceiv’d by a clear and distinct idea necessarily implies the possibility of existence; and he who pretends to prove the impossibility of its existence by any argument deriv’d from the clear idea, in reality asserts, that we have no clear idea of it, because we have a clear idea. ‘Tis in vain to search for a contradiction in any thing that is distinctly conceiv’d by the mind.”[8]

The expression employed in the remarks thus far examined may lead the reader to suppose that there are some things clearly and distinctly conceived and some not- that some of our ideas are clear and distinct and some of them unclear and indistinct. Were this true, then it would follow that we have ideas of things the existence of which we must regard as impossible. There is evidence, however, that Hume considers all our ideas to be clear and distinct. He offers an argument to this conclusion, based on his doctrine that ideas are derived from impressions:

“…we need but reflect on that principle so oft insisted on, that all our ideas are copy’d from our impressions. For from thence we may immediately conclude, that since all impressions are clear and precise, the ideas, which are copy’d from them, must be of the same nature…”[9]

Since all perceptions are either impressions or ideas[10], we must conclude that there are no perceptions of any kind that are not clear and precise.

From the clarity and preciseness of all ideas, we may infer, moreover, that we possess no ideas of those things whose existence we must regard as impossible, but that any idea we may have is the idea of something the existence of which is possible. We find, indeed, that Hume does not always restrict the possibility of existence to that which is clearly and distinctly conceived, but extends it as well to everything that is conceived or imagined at all: “…whatever we conceive is possible.”[11] And: “…whatever we can imagine, is possible.”[12]Hume appears, indeed, to make no firm distinction between what is clearly and distinctly conceived and what is conceived or imagined merely, as is evidenced in his full statement of the metaphysical maxim: “ ‘Tis an establish’d maxim in metaphysics, That whatever the mind clearly conceives includes the idea of possible existence, or in other words, that nothing we imagine is absolutely impossible.”[13] We are thus again justified, apparently, in supposing that all our ideas are equally clear and distinct, and that all things conceived are possible. Things which are contradictory and therefore impossible, on the other hand, cannot be conceived or imagined at all: “We can form the idea of a golden mountain, and from thence conclude that such a mountain may actually exist. We can form no idea of a mountain without a valley, and therefore regard it as impossible.”[14] Again: “ ‘Tis in vain to search for a contradiction in any thing that is distinctly conceiv’d by the mind. Did it imply any contradiction, ‘tis impossible it cou’d ever be coneiv’d.”[15]

Knowing then that self-contradictory things are neither conceivable nor possible, and knowing that whatever is conceived or imagined is possible, we may next inquire what things are in fact conceived or imagined and hence possible. From certain of Hume’s remarks one might infer that we conceive only perceptions; for it is only perceptions that are “present to” the mind: “…nothing is ever really present with the mind but its perceptions or impressions and ideas…”[16] If this be true, then it is reasonable to suppose that we have clear and distinct ideas only of perceptions, as Hume sometimes appears to agree: “We have no perfect idea of any thing but of a perception.”[17] Now if we can conceive only of perceptions, then according to Hume’s principle it is only perceptions whose existence we may regard as possible. We may observe, moreover, that the remarks we have thus far examined do not imply that perceptions, as such, exist, but only that their existence is possible. Were there no further texts available to us from among Hume’s writings, we might justifiably conclude that what he calls “perceptions” are to be understood as a realm of mere essences which, taken together, comprehend all possibility, but which are not, of themselves, existence.”

[1] David Hume, A Treatise of Human Nature, ed. by L.A. Selby-Bigge (Oxford: Clarendon Press, 1888), pp. 32, 250, Hereafter cited as Treatise.

[2] Treatise, pp.19-20. Cf> David Hume, Dialogues Concerning Natural Religion, ed. and with an introduction by Henry D. Aiken(New York: Hafner Library of Classic, Hafner Publishing Company, 1948), p. 19, Philo speaking. Hereafter cited as Dialogues.

[3] Treatise, p. 89

[4] Treatise, p.32

[5] Treatise, p. 43

[6] David Hume, “An Enquiry Concerning the Human Understanding,” in An Enquiry Concerning the Human Understanding and an Enquiry Concerning the Principles of Morals, ed. and with an introduction by L.A. Selby-Bigge (2d ed.; Oxford: Clarendon Press, 1902), p. 35. Hereafter cited as “Understanding.” Cf. Dialogues, p. 58, Cleanthes speaking.

[7] “Understanding.” P. 157

[8] Treatise p. 43.

[9] Treatise, p. 72; cf. p. 366.

[10] Treatise, pp. 1, 96.

[11] Treatise, p. 236.

[12] Treatise, p. 250

[13] Treatise, p. 32.

[14] Treatise, p. 32.

[15] Treatise, p.43. Cf. “Understanding,” p. 164.

[16] Treatise, p. 67; cf. pp.197,212.

[17] Treatise, p. 234.

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Possibility and Necessity

Posted by allzermalmer on January 14, 2012

This blog is going to deal with logically necessary and logically possible. This is slightly different from Avicenna, but mostly based on Modal Logic, or some of the basic ideas of Modal Logic.

Because it is a form of logic, it also deals with one of the foundations of logic. Logic is concerned with statements, and the inferences that we draw from these statements. It is about having a couple of statements, and seeing if we can draw another statement with those statements that we accept. Logic helps give us some rules to follow in order to say that we drew a statement from the other statements that we held to, in a correct manner.

Law of Identity is “every individual thing is identical to itself”. Law of Excluded Middle is “every statement is either true or false”. Law of Non-Contradiction is “given any statement and its opposite, one is true and the other false”. And with possibility and necessary, they are mostly based on the Law of Non-Contradiction.

Possible means not self-contradictory. For example, “The sun won’t rise tomorrow” or “I ran a 2 minute mile” are possible. There’s nothing logically self-contradictory there. Necessary means self-contradictory to deny, which are based on logic, meaning of concepts, or necessary connection between properties. For example, “2+2=4” or “a bachelor is an unmarried male”.

A possible world is “a consistent and complete description of how things might have been or might in fact be.” A possible world is a consistent world, and this means that the statements that describe a possible world don’t entail self-contradiction. We can’t have statement X of possible world N and statement ~X of possible world N, being both affirmed at the same time like “statement X of possible world N and statement ~X of possible world N”. To do this would be to affirm a contradiction, or show that that possible world couldn’t exist because it’s a contradictory world. But the actual world is a description of how things in fact are. Yet, it would seem, that the actual world has to be consistent as well, which means there are no self-contradictions in the world. So the actual world is a possible world itself.

One of the difference between necessary and possible is that necessary statements are known to be true or false without experience. This means that information obtained from observation or sense-perception play no rule in determining if the necessary statement is true. This means we can know that a statement is true without recourse to evidence supplied by observation. But possible statements are known to be true or false with experience. This means that information from observation or sense-perception plays a part in determining if the possible statement is true. This means we know that a statement is true with recourse to evidence supplied by observation.

(A side note is that a necessary statement is a possible statement as well. This is because a necessary statement is a statement that it would be self-contradictory to deny, but the statement itself shouldn’t be self-contradictory and that’s what a possible statement is as well.)

From this idea of possible and necessary, we can say there are three types of statements. There are necessary statements, impossible statements, and contingent statements. A necessary statement is a statement that couldn’t be false. A impossible statement is a statement that couldn’t be true. A contingent statement is a statement that could be true or could be false, or could have been true or could have been false, or could be true in the future or could be false in the future.

So take “a bachelor is an unmarried male”. This is a necessary statement and means it’s necessarily true. But now say that I say “a bachelor isn’t an unmarried male”. This is a impossible statement, and means it’s necessarily false. To actually affirm the second statement is to affirm something that is false. Now say that I affirm “Justin Bieber is a bachelor”. That statement is a contingent statement. This means that Justin Bieber is a married male or isn’t a married male. The only way we could tell which of the two propositions is true is through experience.

So take the statement “Justin Bieber is a bachelor“, and accept it’s a contingent statement. So when we say “‘Justin Bieber is a bachelor‘ is a contingent statement”, we are also saying “‘Justin Bieber is a bachelor‘ is possible and not ‘Justin Bieber is a bachelor‘ is possible.”But take the same statement, and accept it’s a contingent truth. So when we say “‘Justin Bieber is a bachelor‘ is a contingent truth”, we are also saying “‘Justin Bieber is a bachelor’ is true but could have been false.”

As Raymond D. Bradley said, “Our own world – the real world, the actual one – is just one of many possible worlds. Indeed, it is just one of infinitely many, since for any possible world containing say n atoms there is another logically possible world containing n+1 atoms, and so on ad infinitum.” So there are an infinity of possible worlds, or possible ways of describing what might have been or what might be. Say we have the possible world of N, and within this world it contains M and M contains 140 particular things that are M. That is one possible world, but anther possible world, which is logically contradictory from N, and we can call it N*, contains 141 particulars in M. But 140 is logically contradictory from 141. So N is one possible world and N* is a different possible world, but they are logically contradictory from one another.We can continue on doing this infinitely, and so there’s an infinity of possible worlds.

Now science is concerned with contingent statements. This is because science is said to be empirical. The quotes, in order of the authors, are from Richard Feynman, Pierre Duhem, Stephen Hawking, and Henir Poincare: “The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific “truth.””; “Agreement with experiment is the sole criterion of truth for a physical theory.”; “[the] scientific method…w[as]…developed with goal of experimental verification.”; “Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty. These are two points that cannot be questioned.” Now experiment is based on coming to human sense-perception. This can be anything from looking at the squirrel climbing a tree to reading the numbers off of a volt-meter.

The reason that science wants to deal with contingent matters, besides dealing with sense-perception, is that science likes to try to have the ability to be shown that the theories are false. But if science only dealt with necessary statements, then scientific statements could never be shown to be false. For if a statement was presented that said a necessary statement was false, that statement would be an impossible statement and necessarily be false. But a contingent statement can be shown to be false.

But science creates models of how the actual world could possibly be. Take this example of what Richard Dawkins says of science and what science does: “There is a less familiar way in which a scientist can work out what is real when our five senses cannot detect it directly. This is through the use of a ‘model’ of what might be going on, which can be tested. We imagine- you might say we guess- what might be there. That is called the model. We then work out (often by doing a mathematical calculation) if the model were true. We then check whether that is what we see…We look carefully at the model and predict what we ought to see (hear, etc.) with our sense (with the aid of instruments, perhaps) if the model were correct. Then we look to see whether the predictions are right or wrong.  ”

So science creates a model of what is possible, and what happens in this possible world. And we can deduce what should be observed if this possible world is the actual world. We have the possible world of N. But N is our model. So, “If N then O. O. Therefore N.” This is the fallacy of affirming the consequent. Just because the model has a true prediction in the actual world, that doesn’t mean that the model is how the actual world is. In other words, just because this possible world (model) had one right prediction of how the actual world is (which is also another possible world), doesn’t mean that this actual world is that possible world. It also wouldn’t matter if the model has made nothing but correct predictions up till now, because the problem is still there.

If I robbed a bank, then I have 100 million dollars. I have 100 million dollars. Therefore, I robbed a bank. But me having 100 million dollars is also consistent with me winning the lottery, it is also consistent with me investing my money in a way where I got a lot of return in my investments, me starting up a business and my business got me 100 million dollars, or me getting the money from the death of a family member as a part of inheritance. In other words, there are many other possibilities that are consistent with the actual results or observations. As W.V. Quine once said, “Whatever observation would be counted for or against the one theory counts equally for or against another.”

This seems to raise a skeptical problem.

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Epistemological Possibility

Posted by allzermalmer on June 5, 2011

This blog comes from W.T. Stace’s book Theory of Existence and Knowledge. It comes from his chapter on categorical knowledge. The possibility that is being talked about is not logical possibility, which is anything is possible that is not self-contradictory. This possibility is based on epistemology, or theory of knowledge.

Meaning of Possibility: Possibility is part of a cluster or group of categories, that deal with the category of existence. The other categories that deal with existence are identity and substance. Possibility is a category, i.e. character of the external world, and clearly applies to the world and not just our knowledge of it.

A possibility is something which is part of the real world, or at least in some sense. For example, it could rain tomorrow or it may not rain tomorrow. We, typically, say that either option is a possibility. However, possible in such a sense is a concept that does not quality as part of the world, but only our knowledge of it. All it says is that we do not know whether it will rain or not. But tomorrow’s weather, when it comes, will be actual. So one could think that possibility has to deal with our uncertainty about the world, but that is not the meaning of the category to be mentioned.

Possibility (not based on uncertainty) is opposed to actuality. Possibility is said to be part of the world which is never actual, which means that it is something that is non-existent. Possibility is saying that there is something part of the world that does not exist, which makes it into some sort of mystery.

Let us look at some examples of a possibility that has no actuality, no part of the world. “If the British fleet did not defeat the Spanish Armada, then they would have invaded Britain.” Another one is, “If the horse I bet on in the Kentucky Derby had won, then I would have won 5,000 dollars.” However, the British fleet did defeat the Spanish Armada, and the Spanish Armada did not invade Britain. The horse I bet on in the Kentucky Derby did not win, and I did not win 5,000 dollars. Thus, we find that these propositions do not express any facts or ever did, or will ever exist in the world. But, they do express possibilities which might have happened, but did not.

With the image above, we are walking one path. This one path is the actuality. However, as we are walking, we come to a fork in the road, and there are many ways we could go. There are many possibilities. However, we can only walk one of those paths. Thus, there is only one actuality. What we do take is the actuality, and all the others that we do not take are possibilities.

Now I gave propositions of possibility of the past, but this also holds with propositions of the present and of the future. “If I put my hand out in front of me, then the wall will feel hard.”, “If I bite the Pear, then it will taste sweet”, “If I look through a telescope at Saturn, then I will see its rings”. I am looking at the wall, but I am not touching it. I have the visual sensation of the wall, which is actual, but I don’t have any tactile sense of the wall. Thus, the tactile sense would be possible and is not actual. The feel of the wall is a possible tactile experience which I might have if I stretched out my hand.

The essential notion of possibility is  the antecedent of the hypothetical proposition is not meet. Thus,  with “If I bite the Pear, then it will taste sweet”, but I do not bite down on the Pear. The actuality is that I never did bite down on the Pear, and thus there is no actuality of me biting down on the Pear. For if I do that, then the experience ceases to be possible and becomes actual, with the assumption that my prediction is correct.

Possibility is only expressible by the means of a hypothetical proposition, which is a conditional statement of X→Y. Now we could say, “X is a possibility”, but this categorical propositions is really a hypothetical. It means if certain conditions were meet, then X would actually exist.

The question of necessity: There is no necessity in possibility, as should be obvious. It is, however, essential to rational prediction and control of experience. Without possibility, our thinking would be confined to only what is present to our consciousness by the senses. Our thinking would be at a rudimentary stage. Possibility, then, is of a practical necessity. But it is not necessity of thought. We could, theoretically, confine our attention to what is actually given/existent.  There is nothing self-contradictory in such a course.

Epistemological Type: Possibility is a constructive category. It is a construction that is existential, since it creates in the imagination an existence that is not actual. It sets up hypothesis which cannot be proved by experience, and which posits an existence which is not part of the actual world.

Suppose we are in a totally dark room. I say, “If I had switched on the lights, then I should now see the walls of the room”. That proposition is alleged to be now a possible experience. It is not a prediction of future experience, since it does not assert that I shall turn on the light or that I shall see the walls. It does not assert anything, whatsoever, about what will happen in the future. It only makes an assertion of the present. The proposition asserts that if the room were now light, then I should see the walls. But this can never be proved or shown. For If I do turn the lights on, then the visual experience of the wall will exist at a time future to when the proposition was spoken. Also, the experience will have ceased to be possible and would have become actual, so we can never prove the existence of the possibility.

So we see we can never prove possibility, but it is even contrary to the facts, which makes it clear that it is a construction or a fiction. Take the proposition, “If I bite this Pear, then I shall taste that it is sweet.” This proposition does not state anything is the case, but only that something might be the case. But what does ‘might be’ even mean?A fact, an existence, a reality, either is or is not. There is no half-way in the universe for ‘might be’. ‘Might be’ is simply an ‘is not’. So there is no possibility has no part of the actually existing universe, and there is no such thing as possible experience. This makes it clear that possibility is a fiction.

The importance of such a category, though, is very great. It is involved in our existential constructions, and is involved in every scientific hypothesis that asserts the existence of something that we can never perceive (i.e. atoms, gas molecules, DNA, or viruses). These concepts depend on the category of possible experience. This construction lies at the foundation of the construction of the external world, and renders it possible.

The notion of possible experience is an assumption that things exist when no one is aware of them, like the wall when the lights are out, or the hardness of the wall when no one is touching it. At an early stage of the mind, it is only aware of its presentations and nothing else, which is just what actually exists. Esse was identical with percipi. All existence has to be conceived in terms of perception, and even unperceived existence is thought of, and only thought of, as if it were a perceived existence. To exist does not mean simply to be perceived, since the mind has determined it to be otherwise, and has come to project existence beyond its own perceptions, beyond the actual existing, and invented an unperceived world. However, all thought and all knowledge, has its foundation in perception.  The mind simply takes the materials given to it, i.e. what is actually perceived, and builds them up into fictitious worlds. The wall exists when no one is aware of it, and is supposed to be purple, shiny, hard, rectangular, and just like the wall that we see. So the everything that the mind constructs has roots in perceptions, and goes back to perception, and has to be understood in terms of perception.

What do we mean when we say things exists when no one is aware of it? Well, we simply mean that although we are not now looking at it, or perceiving it, yet if any one looked that he would see it. But this is the formula by which the category of possibility is expressed. Thus, the notion of the category of possibility, is based on unperceived existence, is really a construction of the mind.

What do I mean when saying that Beijing exists on the other side of the planet? It must ultimately be explained in the terms of perception. The statement means that some minds (like the inhabitants of Beijing) are actually perceiving Beijing. However, if there were no other minds there to perceive it, then my statement can only mean that if I traveled around the world to Beijing, then I should perceive Beijing, or if any other mind were to do it.

What about the other side of the moon, the dark side of the moon? It means that if any could look around the back of the moon, then he would see the other side.

The minds invention of possibility was its greatest creations, since it helped advance its knowledge. By inventing this imaginary realm of the possible, which is distinguished from the actual, from existence, it opened up all the future existential constructions. It helped us render the idea of permanence, existence, and a public independent world, into existence.

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