Truth suffers from too much analysis

Posts Tagged ‘Aristotle’

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.

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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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Aristotle’s Sea Battle and Future Contingents

Posted by allzermalmer on December 18, 2012

This comes from Aristotle’s book “On Interpretation” and is Part 9. It deals with future contingents and statement either being true or not being true, i.e. law of excluded middle.

“In the case of that which is or which has taken place, propositions, whether positive or negative, must be true or false. Again, in the case of a pair of contradictories, either when the subject is universal and the propositions are of a universal character, or when it is individual, as has been said,’ one of the two must be true and the other false; whereas when the subject is universal, but the propositions are not of a universal character, there is no such necessity. We have discussed this type also in a previous chapter.

When the subject, however, is individual, and that which is predicated of it relates to the future, the case is altered. For if all propositions whether positive or negative are either true or false, then any given predicate must either belong to the subject or not, so that if one man affirms that an event of a given character will take place and another denies it, it is plain that the statement of the one will correspond with reality and that of the other will not. For the predicate cannot both belong and not belong to the subject at one and the same time with regard to the future.

Thus, if it is true to say that a thing is white, it must necessarily be white; if the reverse proposition is true, it will of necessity not be white. Again, if it is white, the proposition stating that it is white was true; if it is not white, the proposition to the opposite effect was true. And if it is not white, the man who states that it is making a false statement; and if the man who states that it is white is making a false statement, it follows that it is not white. It may therefore be argued that it is necessary that affirmations or denials must be either true or false.

Now if this be so, nothing is or takes place fortuitously, either in the present or in the future, and there are no real alternatives; everything takes place of necessity and is fixed. For either he that affirms that it will take place or he that denies this is in correspondence with fact, whereas if things did not take place of necessity, an event might just as easily not happen as happen; for the meaning of the word ‘fortuitous’ with regard to present or future events is that reality is so constituted that it may issue in either of two opposite directions. Again, if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place it was always true to say ‘it is’ or ‘it will be’. But if it was always true to say that a thing is or will be, it is not possible that it should not be or not be about to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be. All, then, that is about to be must of necessity take place. It results from this that nothing is uncertain or fortuitous, for if it were fortuitous it would not be necessary.

Again, to say that neither the affirmation nor the denial is true, maintaining, let us say, that an event neither will take place nor will not take place, is to take up a position impossible to defend. In the first place, though facts should prove the one proposition false, the opposite would still be untrue. Secondly, if it was true to say that a thing was both white and large, both these qualities must necessarily belong to it; and if they will belong to it the next day, they must necessarily belong to it the next day. But if an event is neither to take place nor not to take place the next day, the element of chance will be eliminated. For example, it would be necessary that a sea-fight should neither take place nor fail to take place on the next day.

These awkward results and others of the same kind follow, if it is an irrefragable law that of every pair of contradictory propositions, whether they have regard to universals and are stated as universally applicable, or whether they have regard to individuals, one must be true and the other false, and that there are no real alternatives, but that all that is or takes place is the outcome of necessity. There would be no need to deliberate or to take trouble, on the supposition that if we should adopt a certain course, a certain result would follow, while, if we did not, the result would not follow. For a man may predict an event ten thousand years beforehand, and another may predict the reverse; that which was truly predicted at the moment in the past will of necessity take place in the fullness of time.

Further, it makes no difference whether people have or have not actually made the contradictory statements. For it is manifest that the circumstances are not influenced by the fact of an affirmation or denial on the part of anyone. For events will not take place or fail to take place because it was stated that they would or would not take place, nor is this any more the case if the prediction dates back ten thousand years or any other space of time. Wherefore, if through all time the nature of things was so constituted that a prediction about an event was true, then through all time it was necessary that that should find fulfillment; and with regard to all events, circumstances have always been such that their occurrence is a matter of necessity. For that of which someone has said truly that it will be, cannot fail to take place; and of that which takes place, it was always true to say that it would be.

Yet this view leads to an impossible conclusion; for we see that both deliberation and action are causative with regard to the future, and that, to speak more generally, in those things which are not continuously actual there is potentiality in either direction. Such things may either be or not be; events also therefore may either take place or not take place. There are many obvious instances of this. It is possible that this coat may be cut in half, and yet it may not be cut in half, but wear out first. In the same way, it is possible that it should not be cut in half; unless this were so, it would not be possible that it should wear out first. So it is therefore with all other events which possess this kind of potentiality. It is therefore plain that it is not of necessity that everything is or takes place; but in some instances there are real alternatives, in which case the affirmation is no more true and no more false than the denial; while some exhibit a predisposition and general tendency in one direction or the other, and yet can issue in the opposite direction by exception.

Now that which is must needs be when it is, and that which is not must needs not be when it is not. Yet it cannot be said without qualification that all existence and non-existence is the outcome of necessity. For there is a difference between saying that that which is, when it is, must needs be, and simply saying that all that is must needs be, and similarly in the case of that which is not. In the case, also, of two contradictory propositions this holds good. Everything must either be or not be, whether in the present or in the future, but it is not always possible to distinguish and state determinately which of these alternatives must necessarily come about.

Let me illustrate. A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow. Since propositions correspond with facts, it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character.

This is the case with regard to that which is not always existent or not always nonexistent. One of the two propositions in such instances must be true and the other false, but we cannot say determinately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an affirmation and a denial one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good. The case is rather as we have indicated.”

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Avicenna’s Response to Meno’s Paradox

Posted by allzermalmer on December 18, 2012

Avicenna (Ibn-Sina) was an Arabic philosopher who wrote a book called The Healing. In Book 1, the sixth  chapter is called “On the Manner of Apprehending (iābah) Unknown [Things] (al-majhūlāt) from Known [Things] (al-malūmāt).” This is a response to Meno’s Paradox presented to Socrates about learning or coming to obtain new knew knowledge. The response of Avicenna is translated into English by Michael E. Marmura

“Every seeking (malab) of these [unknown things] is only achieved by means of existing, realized things. Here, however, there is a place for doubt, namely, that [with respect] to the thing whose entity (al-dhāt) is nonexistent [and] whose existence is impossible, how is one to conceive it so as to answer the question “What is it?” when asked about it, so that thereafter one could ask “Does it exist?” For if no meaning (manan) for it is realized in the soul, how can one judge it to be either realized or not realized, when the impossible has no form in existence? How, then, would a form of it be taken into the mind so that that conceived thing would constitute its meaning?

[To this] we say: this impossible [thing] is either (a) singular, having  neither composition nor separability [of parts] (wa lā tafīl), [or (b) composite]. If [(a), singular], it would not be at all conceivable except by some kind of analogy (muqāyasa) with the existent and in relation to it. This would be similar to our saying “vacuum” and “the contrary of God.” For vacuum is conceived as belonging to bodies as though a recipient, and the contrary of God would be conceived as belonging to God in the way the hot relates to the cold. Thus the impossible would be conceived in terms of a possible thing to which it relates, being conceived in relation to it and likened to it. In itself, however, it is neither conceivable nor intellectually apprehendable (lā maqūlan), nor would it have any being (wa lā dhāta lahu).

As for [(b)], which has some sort of composition and separable parts, such as the goatstag, the phoenix, or a human that flies, what is first conceived is their separable parts whose existence is not impossible. Thereafter there is conceived for the parts some kind of connections, analogous to the connections that obtain for the separable parts of existing composite entities. There would thus be three things, two of them consisting of parts, where each by itself [can] exist, and the third a composition of them. This in some respect is a conceived composition, because composition inasmuch as it is a composition is one of the things that exist. In this way it yields the meaning signifying the nonexistent’s name. Thus the nonexistent is only conceived because of the prior conception of existing things.

We now say: if a judgment about a universal is realized for us, its first  realization being self-evident, as, for example, “every human is an animal” and “the whole is greater than the part,” or evident by induction or experiment (tajriba) in the ways things are believed to be true (p.73) without the help of a [demonstrative] syllogism (qiyās), we would know potentially the judgment over every particular beneath it, but we would be ignorant of it in [its] actuality. Thus, for example, we would not know that Zayd, who is in India, is an animal. In actuality we would be ignorant of this because thus far we would have known [it] only potentially, since we have known that every human is an animal. In actuality we would be ignorant of it because one needs to add to this knowledge another knowledge, or two other [instances] of knowledge, for the potential to become actual. This is because we must know that Zayd exists and know that he exists as a human. If then through the senses we attain immediate cognition (marifa) that he exists and that he is a human, without this being sought after (malūban) through a [demonstrative] syllogism or [this being]taught, and if this is connected with [universal] knowledge (ilm), which is realized for us, also without a [demonstrative] syllogism, in [the] manner by which the connection that in itself brings about a third knowledge [takes place], we would then know that Zayd is an animal. It would thus be the case that from the combining of immediate cognition (marifa) and [universal] knowledge (ilm), knowledge [of the specific particular] comes about for us. Of the two, immediate cognition comes about through sensation, [universal] knowledge through the intellect. Immediate cognition would have come to us at a temporal moment, whereas [universal] knowledge is prior to it.

That which comes about from both of them may well have been sought by us, and we [may have] sought its principles, which bring us to it, or it could have been something to which we have been led by reason of the sufficiency of its causes without our seeking it. With all this, the conception of the thing sought after and its principles must precede every instance of it.

It may also happen that it is not like this, but that the judgment of the universal would have come about for us by a [demonstrative] syllogism, and the judgment of the particular would have come about by another [demonstrative] syllogism, such that if the two are combined, a third instance of knowledge is realized. But even if this is the case, the first syllogisms would have been through self-evident premises or else acquired by induction, experiment, or sensory perception without a syllogism, as we will make clear later.

Then, if a questioner should ask someone and say, “Do you know that every dyad is even?” it is known that the answer would be, “I know this.” [The questioner] would then come back and say, “Is what is in my hand even or odd, and is the number of people in such and such a city even or odd?” If it is answered, “We do not know this,” [the questioner] will come back and say, “You then do not know that every numerical dyad is even; for what is in my hand is a dyad, but you did not know that it is even.”

It is said in The Teaching [of Aristotle] that some people have answered this incorrectly, saying, “We only know that every dyad known to us is even.” But this answer is false; (p. 74) for we know that every existing dyad, whether known or not known, is even.

Rather, [our own] response to this would be, “We did not say that we know every dyad that is even, so that if we did not know that [the] two things [in the hands of the questioner] constitute a dyad our statement is contradicted”; we also did not say, “We know of everything that is a pair, that it is a pair, so that [consequently] we know that it is even.”

Rather, we said one of two things: either that every dyad that we know, we know it to be even; or every dyad in itself, whether we know it or not, is in itself even. As for the first disjunct, it is not contradicted by the doubt mentioned. As for the second [disjunct], it consists of general knowledge, not contradicted by our ignorance [of some aspect belonging to it]. This is because even though we do not know whether what is in the hand of so and so is even or not even, the meaning of our knowledge that every dyad is even stands established and is not false. But what we would be ignorant of is included in our potential, not actual, knowledge. Ignorance of it would not be an ignorance in actuality of [all of] what we have. If we come to know that what is in [the questioner’s] hand is a
dyad, and we remember the thing which was known to us, [namely, that every dyad is even,] we would immediately know that what is in his hand is even. What we are ignorant of is other than that which we know. It is not the case that if we do not know whether or not something is even because we do not know whether or not it is a dyad that this falsifies our knowledge that whatever is a dyad is even. We would then have also known that in one respect that thing is even. With this the doubt raised ceases.

It was mentioned that Meno, who addressed Socrates regarding the  nullification of teaching and learning, said to him, “The seeker of some kind of knowledge will either be seeking knowledge of what he [already] knows, in which case his quest would be inconsequential, or else he would be seeking knowledge of what he does not know—[In this case,] how would he know it, once he attains it? This is akin to one who seeks a runaway slave he does not know. If he finds him, he would not recognize him.”

Socrates then undertook in refuting him (p. 75) to present [Meno] with a geometrical figure, establishing for him how the unknown, after being unknown, is captured through the known. This, however, is not logical discourse. For he showed that this is possible, bringing about a syllogism that yielded as a conclusion the possibility of what Meno brought about by a syllogism that yielded other than its possibility. [As such,] he did not resolve the doubt.

Plato, on the other hand, undertook to resolve the doubt, saying that  learning is recollection. In this he attempted to convey that the thing being sought had been known prior to the quest and prior to [its] attainment, but it was only being sought because it had been forgotten. Once the investigation led to the thing sought, it was remembered and learned. Thus the seeker would have known something he had [previously] known. It is as though Plato had acquiesced to the doubt and sought toescape from it and fell into an impossibility. This is something we have treated in exhaustive detail in our solution to it in our summary of the book on the Syllogism.

Still with all this we say: if what is being sought after is known for us in every respect, we would not seek it; and if unknown for us in every respect we would not seek it. It is, [however,] known to us in two respects, and unknown in one respect. For it is known to us conceptually in actuality, and known to us potentially in terms of assent. It is unknown to us in being specified with actuality, even though it is also known to us [as] being not specified with actuality. If, then, we have had previous knowledge that every thing which is of a certain character is of that character,  without [that knowledge] having been sought after, but by an inborn rational act, or through sensation, or some other [similar] ways of knowing, we would attain in potentiality knowledge of numerous things. If through sensation we apprehend some of these particulars without our seeking them, they would immediately fall within [the category] of being actual, within the [category] of first knowledge. This, in some respect, would parallel step by step what Meno had brought about in the example of the runaway slave.

For we would know at first the thing sought after conceptually, as we would know the runaway slave at first conceptually. And we know prior to this what would bring us to knowing it by assent, just as we would know the road [taken by the slave] prior to our knowing the place of the runaway slave. For if we follow the path to what is being sought after, and we have had a previous conception of it and of the road leading to it, and [if] we reach it, we would have apprehended the thing sought after. [This is] similar to our travelling path to the runaway slave when we have a previous conception of his self, and of the road leading to him  Once we reach him, we would recognize him, even if we had never seen the runaway slave at all [before]; but we would have conceived for him a distinguishing mark (alāma), [which indicates] that everyone having this mark is a runaway.

If there were added to this some knowledge occurring not by acquisition but coincidentally by observation, or [some knowledge] occurring by acquisition, demonstration, testing, and cognizing, and we found that mark on a slave, we would know that he is a runway.

The distinguishing mark would thus be like the middle term in the  syllogism. Our grasping that distinguishing mark in a slave would be like the minor term. Our knowledge that whoever has this mark is a runaway is akin to the prior occurrence for us of the major premise; and our finding the runaway would be akin to the conclusion. This runaway, moreover, would not have been known to us in every respect; otherwise we would not have sought him. Rather, he was known to us by way of conception, unknown to us as regards [the physical] place [where he is]. Hence, we would be seeking him with respect to his being unknown, not with respect to his being known. Once we recognized him and caught him, there would have occurred through [our] demonstrative quest knowledge of him which [previously] did not exist. This only happened on account of the combination of two causes of knowledge, one being [knowledge of] the way and its leading to him, the second, his being the object of sense apprehension.

The unknown things sought after are similar to this: they are known by the combination of two things. The first is something which is prior for us, namely, that B is A, which parallels the first cause in the example of the runaway slave; the second is a thing which occurs immediately, namely, our sensory cognition that C is B, which equals the second cause in the example of the runaway slave. Just as the two causes there necessitate the apprehension of the runaway slave, similarly the two causes here necessitate the apprehension of what one is seeking. It is not the case that what he, [that is, Meno,] demanded as a condition, [namely,] that whatever is not known in every respect would not be known when attained, is admitted. Rather it is every thing unknown in every respect that is not recognized when attained. If, however, knowledge of a thing which had been known [before] had been pursued, [the prior knowledge of it] would constitute potential knowledge of a part of what is being sought after, it being like a distinguishing mark for it. Itwouldthen only require something to connect with it to change it into actuality. As soon as that which actualizes it is connected with it, the thing sought after is attained.

[Now] that it has been established how mental (dhihnī) instruction and  learning takes place and that this takes place only through previous knowledge, we must have first principles for conception and first principles for assent. If every instruction and learning was through previous knowledge, and every existing knowledge is through instruction and learning, the state of affairs would regress ad infinitum, and there would be neither instruction nor learning. It is hence necessary that we have matters believed to be true without mediation, and matters that are conceptualized without mediation, and that these would be the first principles for both assent and conceptualization.”

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