Beppe Brivec

Infinite sets and finite sets have different properties and follow different rules. C “All the elements of the set of ravens are elements of the set of black ravens”. CE “All the things that are not elements of the set of black ravens are elements of the set of the things that are not ravens”. It is assumed that the number of ravens is infinite, but it is not assumed that the number of black ravens is infinite: that the number of black ravens is infinite is only a hypothesis (for which confirmation is sought), it cannot be assumed as true that the number of black ravens is infinite (the set of ravens could contain an infinite number of red ravens and only a finite number of black ravens). Therefore, in the generalization C we refer to a set that is assumed to be an infinite set (the set of ravens) and to a set (the set of black ravens) that is not assumed to be an infinite set. By contrast, in the generalization CE we refer to two sets that are assumed to be infinite sets (the set of all objects that are not elements of the set of black ravens and the set of all objects that are not elements of the set of ravens). That CE has infinite instances is something we already know a priori (by virtue of the assumptions made); by contrast, we do not know a priori that C has infinite instances—that is merely an aspiration. Therefore, an instance of C carries weight, whereas an instance of CE tells us nothing more than what we already know a priori. To summarize the case of finite sets: ​1) The Disparity in Set Cardinality The set of ravens is vastly smaller than the set of non-black things that are non-ravens. In any plausible finite universe, the number of non-black objects is overwhelmingly larger than the number of ravens. Therefore, while investigating the restricted set of ravens significantly reduces the number of potential counterexamples, sampling from the near-infinite reservoir of non-black objects leaves the hypothesis virtually untested. ​2) The Problem of Indiscriminate Confirmation The observation of a green emerald is a positive instance of the contrapositive hypothesis CE ("All non-black things are non-ravens"). However, by the exact same logical token, that very same green emerald is also a positive instance of DE ("All non-yellow things are non-ravens", which is logically equivalentto D "All ravens are yellow"). If we accept that observing a non-black non-raven confirms that all ravens are black, we must also accept that it confirms that all ravens are yellow. Because a single piece of external evidence would simultaneously confirm two mutually exclusive hypotheses, looking outside the target set lacks any real discriminative power. ​3) Parallelism with the Infinite Case: Information vs. Redundancy There is a parallelism with the infinite sets scenario based on our background knowledge. That the universe contains an immense, virtually inexhaustible number of non-black things that are non-ravens is a structural fact already fully possessed by our background knowledge; it is already baked into our prior assumptions about the world. Consequently, observing a non-black thing that is a non-raven (for example, a green emerald) adds zero new information—it merely repeats an a priori platitude. Observing a black raven adds new information.

According to Aristotle, a true universal proposition must have the contrary proposition that is false. But there are universal propositions which have no contrary proposition. For example: X “All the propositions that contradict this proposition are true” does not have the contrary proposition (and does not have the contradictory proposition). In fact: FEX “All the propositions that contradict this proposition are not true” has a different subject: the subject of the proposition X is constituted by all the propositions that contradict the proposition X; by contrast, the subject of the proposition FEX is constituted by all the propositions that contradict the proposition FEX. According to Aristotle, a true singular proposition must have the negative proposition that is false. But there are singular propositions which have no negative proposition. For example: The proposition (or, rather, the pseudo-proposition) L “This same statement is not true” does not have the negative proposition because the proposition FNL “This same statement is true” has a different subject from the subject of the proposition L: the subject of the proposition L is the proposition L; by contrast, the subject of the proposition FNL is the proposition FNL. L and FNL cannot have the same subject. Aristotle in chapter 8 of “Peri Hermeneias” provides an example of something that has the form of a proposition but is actually a set of sentences. L has the form of a proposition, but is not a proposition because it lacks the corresponding negative proposition; L is actually a set of sentences containing an implicit sentence advising that, although L has the form of a proposition, L is not a proposition because it lacks the corresponding negative proposition. This is confirmed by the fact that L1 "The statement L1 is not true" and L2 “The statement L1 is not true” have different truth values (L1 has paradoxical truth values, L2 has non-paradoxical truth values) even though what is contained between the quotation marks is identical: in the case of L1 additional information is smuggled into the statement, the information that L1 is logically equivalent to L “This same statement is not true.” (which is not a proposition because it lacks the negative proposition). Thus, using the same procedure as above, the proposition G “This same sentence is written in English” would turn out to be a set of sentences (and would not turn out to be a proposition, because it is implicit the affirmation that G lacks the negative proposition, since FEG “This same sentence is not written in English” has a different subject), but it would be true that every word written in G (in the set of sentences G) is written in English. The second part of the paper is about Russell’s Paradox by introducing the method of "importing the subject into the predicate." Drawing a parallel with Hempel’s Paradox of the Ravens, the paper highlights hat the "set of all sets that do not contain themselves" is an impossible construct not because of an arbitrary axiomatic prohibition (as in ZF/ZFC), but because its predication results in a necessary falsehood. By importing the subject, the Russell Set reveals a self-contradictory predicate—analogous to a "black raven that is a non-raven"—which renders the existence of the set logically null. Following Aristotle’s Categories (13b), the author concludes that since the subject is impossible, affirmations regarding its existence are false by necessity, while their denials remain tautologically true.

This paper examines the structural limitations of Bayesian confirmation theory when applied to the justification of universal laws. In the analysis of the Raven Paradox and Goodman’s New Riddle of Induction Bayesian framework encounters systematic failures in infinite domains, where the prior probability of any non-tautological universal law collapses to zero, rendering incremental confirmation mathematically null or epistemologically irrelevant. New critical tools, such as the predicates "greeneverest" and "welement," are introduced to illustrate the "mechanical transmission of undecidability." These cases demonstrate that formal equivalence between interdefinable predicates does not imply epistemic equivalence: Bayesianism, unable to distinguish between decidable properties and those structurally blocked by global historical facts or mathematical undecidability, fails to represent the logic of universal laws. The paper concludes that the failure of Bayesianism occurs not in the face of extreme scientific complexity, but at the very foundation of inductive logic. The metaphor of "sinking in the harbor" suggests that the flaw lies in the structural integrity of the framework itself, which is unable to anchor the justification of universal laws to a solid logical base, thereby suggesting the need for a return to a rationality grounded in semantic-deductive principles.

Peter Godfrey-Smith writes in the section 3.3 “The Ravens Problem” of his book Theory and Reality [chapter “Induction and Confirmation”]: “First, the logical empiricists were concerned to deal with the case where generalizations cover an infinite number of instances. In that case, as we see each raven we are not reducing the number of ways in which the hypothesis might fail.” Infinite sets and finite sets have different properties and follow different rules. This distinction is fundamental when dealing with scientific laws. For a statement to qualify as a law of nature, it must be assumed to hold across an infinite number of instances. These instances are not merely a collection of physical objects existing at a single moment, but are more accurately understood as time-slices of existence. Consider a physical generalization such as M: "All metals expand when heated." To verify this as a universal law, we are not simply looking at a finite number of copper or iron rods. We are making a claim about every possible moment (time-slice) in which any metal exists, has existed, or will exist. Because time is continuous and the potential future is open-ended, the set of instances covered by M is necessarily assumed to be infinite. In such an infinite case, observing a finite number of instances (e.g., a thousand expanding metal slices) does not reduce the number of ways the hypothesis might fail. The remaining "pool" of instances stays infinite. There is a profound asymmetry when we move from the direct generalization to its contrapositive. Let us examine this through a finite example first, to isolate the logical mechanics. Suppose "bag X" contains exactly 90 marbles. The generalization A: All marbles in bag X are red" covers a finite number of instances. By contrast, the generalization AE: "All things that are not red are not marbles in bag X" refers to every non-red object in the universe. Even though A and AE are logically equivalent, there is an epistemic asymmetry: A can be fully confirmed (or disconfirmed) by checking a finite set, whereas AE refers to an effectively infinite set of non-red things. This asymmetry becomes critical in the context of infinite sets and scientific laws. Consider a barrique X (205 liters): B: "All minerals are in barrique X." BE: "All objects that are not in barrique X are not minerals." In generalization B, we refer to a set that is assumed to be an infinite set (the set of minerals) and to a finite set (the interior of barrique X). By contrast, in BE, we refer to two sets that are both assumed to be infinite: the set of all objects not in barrique X and the set of all non-minerals. Thus, an instance of BE (a non-mineral outside barrique X) cannot be epistemically equivalent to an instance of B. C “All the elements of the set of ravens are elements of the set of black ravens”. CE “All the things that are not elements of the set of black ravens are not elements of the set of ravens”. It is assumed that the number of ravens is infinite, but it is not assumed that the number of black ravens is infinite: that the number of black ravens is infinite is only a hypothesis (for which confirmation is sought), it cannot be assumed as true that the number of black ravens is infinite. Therefore, in the generalization C we refer to a set that is assumed to be an infinite set (the set of ravens) and to a set (the set of black ravens) that is not assumed to be an infinite set. By contrast, in the generalization CE we refer to two sets that are assumed to be infinite sets (the set of all objects that are not elements of the set of black ravens and the set of all objects that are not elements of the set of ravens). That CE has infinite instances is something we already know a priori (by virtue of the assumptions made); by contrast, we do not know a priori that C has infinite instances—that is merely an aspiration. Therefore, an instance of C carries weight, whereas an instance of CE tells us nothing more than what we already know a priori.

This article argues that the epistemic limitations famously associated with grue-like predicates in inductive contexts extend equally to deductive systems, thereby undermining the hope for a purely formal conception of deduction. Building on Nelson Goodman’s original insight concerning inductive projection, we show that any predicate—whether mathematically elementary such as "is even" or arbitrarily constructed—can be formally re-expressed as either an indecidable predicate K or its negation, thereby collapsing all extensional distinctions among predicates. In both first-order and second-order frameworks, this collapse renders every predicate truth-conditionally equivalent to an undecidable one. As a result, the aspiration for purely formal criteria of deductive validity becomes untenable. The article examines the conceptual consequences of this collapse across seven chapters, culminating in a discussion of the broader philosophical implications concerning meaning, analyticity, formalism, and the limits of syntactic approaches to logical theory.

This paper argues that mathematics cannot be regarded as either a mere invention or a purely hypothetico-deductive enterprise. By extending Goodman’s “grue” problem into the mathematical domain, it is shown that definitional interequivalence between predicates does not preserve epistemic properties such as decidability. Even when two predicates are formally interdefinable, only one may be epistemically admissible within a consistent formal system. This constraint reveals that mathematical language is not a free invention but is shaped by deeper semantic and computational necessities. Mathematics thus occupies a unique epistemological position: it is a discipline in which formal equivalence does not entail epistemic equivalence, and where the very possibility of truth and knowledge is conditioned by the structure of the system itself.

A - "All ravens are black." B - "All non-black things are non-ravens." C - "All the elements of the set of ravens are elements of the set of black ravens." D - "All the elements of the set of the things that are not black ravens are elements of the set of the things that are not ravens." The propositions A, B, C, D are logically equivalent: if one proposition is true, the other three propositions are also true; if one proposition is false, the other three propositions are also false. The generalizations A and C have the same instances: a black raven is an instance of A and is an instance of C. The generalizations B and D have some instances in common: a non-black object that is a non-raven is an instance of B and is an instance of D; in addition, the generalization D also has as its instances black objects that are non-ravens (for example, a black shoe is an instance of D but is not an instance of B). E - "All ravens are non-black." F - "All black things are non-ravens." G - "All the elements of the set of ravens are elements of the set of the things that are not black ravens". H - "All the elements of the set of black ravens are elements of the set of the things that are not ravens." The generalizations E and G have the same instances: a non-black raven is an instance of E and is an instance of G. By contrast, the generalizations F and H do not have the same instances: a black thing that is a non-raven is an instance of F; by contrast, H is a generalization that cannot have instances because a black raven that is a non-raven is a contradiction in terms. To summarize: A and C have the same instances; B and D have some instances in common; E and G have the same instances; by contrast, F and H do not have instances in common because H has no instances, H cannot have instances. To state that all black ravens are not ravens (as proposition H does) implies that the set of black ravens is not a subset of the set of ravens: hence proposition H is self-contradictory. Affirming that the set of black ravens is not a subset of the set of ravens contradicts the very definition of what a black raven is. Thus, even though the proposition H is the contrapositive of the proposition G, the propositions G and H are not logically equivalent since the proposition G is coherent and is a contingent proposition; the proposition H is incoherent and is a necessary falsehood. Furthermore: Peter Godfrey-Smith writes in the section 3.3 “The Ravens Problem” of his book “Theory and Reality” [chapter “Induction and Confirmation”]: “First, the logical empiricists were concerned to deal with the case where generalizations cover an infinite number of instances. In that case, as we see each raven we are not reducing the number of ways in which the hypothesis might fail”. In the raven paradox it is assumed that the set of ravens has infinite elements, therefore it is assumed that the set of ravens is not empty. The proposition G implies that the set of ravens contains an empty subset of black ravens: the proposition G is true in case there are not black ravens. The proposition H implies that the set of black ravens contains an empty subset of ravens: but in the raven paradox it is assumed that the set of ravens is not empty; hence the proposition H is false. The proposition H is not logically equivalent to the proposition G: in case there are ravens (as assumed in the raven paradox) but there are not black ravens, the proposition H is false but the proposition G is true. Thus, even though the proposition H is the contrapositive of the proposition G, the propositions G and H are not logically equivalent.

Let’s consider the skeptical example of the brain in a vat. It has been pointed out by Crispin Wright (1994) that Putnam's argument does not affect certain cases such as my brain being removed from my skull by a mad scientist and hooked up to a computer. Since Putnam's argument falls flat at least in cases where the brain is first removed from a human body and then hooked up to a computer, I consider the skeptical aspects of the brain-in-a-vat thought experiment below. Let’s consider the following hypotheses: C “I’m a brain in a vat” EC “I’m not a brain in a vat”. It is argued by skeptics that it cannot be proven that we are not brains in a vat. If we assume the skeptical thesis according to which it is impossible to prove that we are not brains in a vat, i.e. there is the impossibility of falsifying C, then the truthfulness of EC implies that both C and EC can not be falsified. On the other hand, the truthfulness of C does not imply the unfalsifiability of EC: for example, the mad scientist mentioned above could decide (perhaps for a sadistic pleasure) to inform us and prove to us that we are brains in a vat; in the latter case the EC hypothesis would be falsified. The truthfulness of EC implies the unfalsifiability of both EC and C. By contrast, the truthfulness of C does not imply the unfalsifiability of EC. The current evidence is that both C and EC are not falsified: the hypothesis EC necessarily implies that both EC and C result not falsified; the hypothesis C does not necessarily imply that both C and EC result not falsified And so, it is more rational to bet on EC rather than on C.

Peter Godfrey-Smith writes in the section 3.3 “The Ravens Problem” of his book “Theory and Reality” [chapter “Induction and Confirmation”]: “First, the logical empiricists were concerned to deal with the case where generalizations cover an infinite number of instances. In that case, as we see each raven we are not reducing the number of ways in which the hypothesis might fail”. Infinite sets and finite sets have different properties and follow different rules. For example: let’s call “bag X” a specific bag that contains 90 marbles; the generalization A “All marbles in bag X are red” covers a finite number of instances. By contrast, the generalization AE “All things that are not red are not marbles in bag X” covers an infinite number of instances. The propositions A and AE are logically equivalent, but there is an asymmetry due to the fact that A can only have a finite number of instances (since the set of the marbles contained in bag X is a finite set). Because of the aforementioned asymmetry, even though A and AE are logically equivalent propositions, it is wrong to equate an instance of A with an instance of AE. Another example involving a finite set: let’s call “barrique X” a certain 205-liter (54.1553 US gal) barrique. In the following generalization, the expression "all minerals" should not be understood as "all types of minerals", but should be understood literally as "all minerals". B “All minerals are in barrique X”. BE “All objects that are not in barrique X are not minerals”. Of course, a 205-litre barrique cannot contain an infinite number of minerals; the set of objects contained in barrique X is a finite set: therefore in the generalization B we refer to a set that is assumed to be an infinite set (the set of minerals) and to a set (the finite set of minerals contained in barrique X) that is not assumed to be an infinite set. By contrast, in the generalization BE we refer to two sets that are assumed to be infinite sets (the set of all objects that are not contained in barrique X and the set of all objects that are not minerals). Therefore, although B and BE are logically equivalent propositions, an instance of BE cannot be equivalent to an instance of B. Now, let’s analyze the following generalizations. C “All the elements of the set of ravens are elements of the set of black ravens”. CE “All the things that are not elements of the set of black ravens are not elements of the set of ravens”. It is assumed that the number of ravens is infinite, but it is not assumed that the number of black ravens is infinite: that the number of black ravens is infinite is only a hypothesis (for which confirmation is sought), it cannot be assumed as true that the number of black ravens is infinite. Therefore, in the generalization C we refer to a set that is assumed to be an infinite set (the set of ravens) and to a set (the set of black ravens) that is not assumed to be an infinite set. By contrast, in the generalization CE we refer to two sets that are assumed to be infinite sets (the set of all objects that are not elements of the set of black ravens and the set of all objects that are not elements of the set of ravens). Someone could argue that in the generalizations C “All the elements of the set of ravens are elements of the set of black ravens” and CE “All the things that are not elements of the set of black ravens are not elements of the set of ravens” only the set of ravens is assumed to be infinite: but even in this case an asymmetry would remain because C “ All the elements of the set of ravens are elements of the set of black ravens” refers to a set (the set of ravens) that is assumed to be an infinite set and to a set (the set of black ravens) that is not assumed to be infinite; by contrast, CE “All the things that are not elements of the set of black ravens are not elements of the set of ravens” in this case would refer to two sets that are not assumed to be infinite sets. Because of the aforementioned asymmetries, an instance of C cannot be equivalent to an instance of CE.

The pairs of predicates green/blue and grue/bleen are interdefinable: The predicate “grue” applies to an object if the object either is green and first examined before time t, or is blue and not examined before time t. The predicate “bleen” applies to an object if the object either is blue and first examined before time t, or is green and not examined before time t. The predicate “green” applies to an object if the object either is grue and first examined before time t, or is bleen and not examined before time t. The predicate “blue” applies to an object if the object either is bleen and first examined before time t, or is grue and not examined before time t. According to Goodman, if the green-speaker is able to apply the predicate "green" without needing temporal information, then also the grue-speaker is able to apply the predicate "grue" without needing temporal information; in other words, for the grue-speaker "grue" is a primitive predicate, exactly as for the green-speaker "green" is a primitive predicate. Let’s say that time t is 12:34 p.m.(New York time) on January 1, 2030. A committee of scientists knows that time t has been arbitrarily fixed and is 12:34 on January 1, 2030. The committee of scientists calls a meeting at 9 a.m. on January 1st, 2030 one person who takes “green” and “blue” as primitive predicates (a green-speaker) and another person, who takes “grue” and “bleen” as primitive predicates (a grue-speaker): these two people can talk to each other, exchange information, but are kept in the dark about what time t is (which corresponds to 12.34 on January 1, 2030) arbitrarily set by scientists. The committee of scientists asks the person who takes “green” and “blue” as primitive predicates (the green-speaker) to indicate which objects are green and which objects are blue. The committee of scientists asks the person who takes “grue” and “bleen” as primitive predicates (the grue-speaker) to indicate which objects are grue and which objects are bleen. Let’s imagine that at 11:00 a.m. (January 1, 2030) an object Z is classified as green by the green-speaker and as grue by the grue-speaker. At this point, the green-speaker and the grue-speaker can, by talking to each other, deduce that 11:00 a.m. is before the time t (12:34 p.m. on January 1, 2030) set by the scientists. In fact, according to the definition of "grue" we know that a grue object can only be: 1) either green (in case it is examined before time t ); 2) or blue (in case it is not examined before time t ); knowing that the object Z is green and is grue, it can be deduced that this is the case 1, the case of an object that is green and examined before time t. But this would contradict the initial assumptions: it was assumed that the green-speaker and the grue-speaker cannot know what time t is set by the scientists. Let’s imagine that at 12:00 (January 1, 2030) an object J is classified as blue by the green-speaker and as bleen by the grue-speaker. At this point, the green-speaker and the grue-speaker talking to each other can deduce that 12:00 is before the time t (12:34 p.m. on January 1, 2030) set by the scientists. In fact, according to the definition of "bleen" we know that a bleen object can only be: 1) either blue (in case it is examined before time t ); 2) or green (in the case it is not examined before time t); knowing that the object J is blue and is bleen, it can be deduced that this is the case 1, the case of an object that is blue and examined before time t. But this would contradict the initial assumptions: it was assumed that the green-speaker and the grue-speaker cannot know what time t is set by the scientists. Let’s imagine that at 13:00 (1:00 p.m. January 1, 2030) an object K is classified as green by the green-speaker and as bleen by the grue-speaker. At this point, the green-speaker and the grue-speaker, by talking to each other, can deduce that 13:00 (1:00 p.m.) is after the time t (12:34 on January 1, 2030) set by the scientists. In fact, according to the definition of "bleen" we know that a bleen object can only be: 1) either blue (in case it is examined before the time t); 2) or green (in case it is not examined before time t); knowing that the object K is green and is bleen, it can be deduced that this is the case 2, the case of an object that is green and not examined before time t. But this would contradict the initial assumptions: it was assumed that the green-speaker and the grue-speaker cannot know what time t is set by the scientists. Let’s imagine that at 14:00 (2:00 p.m. January 1, 2030) an object L is classified as blue by the green-speaker and as grue by the grue-speaker. At this point, the green-speaker and the grue-speaker, by talking to each other, can deduce that 2:00 p.m. is after the time t (12:34 p.m. on January 1, 2030) set by the scientists. In fact, according to the definition of "grue" we know that a grue object can only be: 1) either green (in case it is examined before time t ); 2) or blue (in case it is not examined before time t ); knowing that the object L is blue and is grue, it can be deduced that this is the case 2, the case of an object that is blue and not examined before time t. But this would contradict the initial assumptions: it was assumed that the green-speaker and the grue-speaker cannot know what the time t set by the scientists is.

According to Aristotle if a universal proposition (for example: “All men are white”) is true, its contrary proposition (“All men are not white”) must be false; and, according to Aristotle, if a universal proposition (for example: “All men are white”) is true, its contradictory proposition (“Not all men are white”) must be false. I agree with what Aristotle wrote about universal propositions, but there are universal propositions which have no contrary proposition and have no contradictory proposition. The proposition X “All the propositions that contradict this proposition are true” does not have the contrary proposition and does not have the contradictory proposition. In fact: FEX “All the propositions that contradict this proposition are not true” has a different subject: the subject of the proposition X is constituted by all the propositions that contradict the proposition X; by contrast, the subject of the proposition FEX is constituted by all the propositions that contradict the proposition FEX. And FOX “Not all the propositions that contradict this proposition are true” has a different subject: the subject of the proposition X is constituted by the propositions that contradict the proposition X; by contrast, the subject of the proposition FOX is constituted by the propositions that contradict the proposition FOX. According to Aristotle, a singular proposition (in his example: "Socrates is white") which is true must have its negative proposition ("Socrates is not white") which is false. I agree with Aristotle, but there are singular propositions which do not have the corresponding negative proposition. The proposition (or, rather, the pseudo-proposition) L “This same statement is not true” does not have the negative proposition because the proposition FNL “This same statement is true” has a different subject from the subject of the proposition L: the subject of the proposition L is the proposition L; by contrast, the subject of the proposition FNL is the proposition FNL. L and FNL cannot have the same subject. By contrast, the proposition M “This mount is entirely in Swiss territory” and the proposition NM “This mount is not entirely in Swiss territory” can have the same subject (for example, the Mount Eiger): in case the subject of the proposition M and the subject of the proposition NM is the same, M and NM are opposite propositions, NM is the negative proposition of M. By contrast, the proposition (or, rather, the pseudo-proposition) L "This same statement is not true" cannot have the corresponding negative proposition because FNL "This same statement is true" has a different subject: the subject of L is L ; by contrast, the subject of FNL is FNL. Then the paper continues by analyzing some variants of the liar’s paradox: L1 “The statement L1 is not true”; the so-called liar cycle; and the so-called Yablo’s paradox.

This paper reports (in section 1 “Introduction”) some quotes from Nelson Goodman which clarify that, contrary to a common misunderstanding, Goodman always denied that “grue” requires temporal information and “green” does not require temporal information; and, more in general, that Goodman always denied that grue-like predicates require additional information compared to what green-like predicates require. One of the quotations is the following, taken from the first page of the Foreword to chapter 8 “Induction” of the Goodman’s book “Problems and Projects”: “Nevertheless, we may by now confidently conclude that no general distinction between projectible and non- projectible predicates can be drawn on syntactic or even on semantic grounds. Attempts to distinguish projectible predicates as purely qualitative, or non-projectible ones as time-dependent, for example, have plainly failed”. Barker and Achinstein in their famous paper of 1960 tried to demonstrate that the grue-speaker (named Mr. Grue in their paper) needs temporal information to be able to determine whether an object is grue, but Goodman replied (in “Positionality and Pictures”, contained in his book “Problems and Projects”, chapter 8, section 6b) that they failed to prove that Mr. Grue needs temporal information to determine whether an object is grue. According to Goodman, since the predicates “blue” and “green” are interdefinable with the predicates “grue” and “bleen”, “if we can tell which objects are blue and which objects are green, we can tell which ones are grue and which ones are bleen” [pages 12-13 of “Reconceptions in Philosophy and Other Arts and Sciences”]. But this paper points out that an example of interdefinability is also that one about the predicate “gruet”, which is a predicate that applies to an object if the object either is green and examined before time t, or is non-green and not examined before time t. The three predicates “green”, “gruet”, “examined before time t” are interdefinable: and even though the predicates “green” and “examined before time t” are interdefinable, being able to tell if an object is green does not imply being able to tell if an object is examined before time t (the interdefinability among three elements is a type of interdefinability present, for example, also among the logical connectives). Thus, it is wrong the Goodman’s thesis according to which if it is possible to determine without having temporal information whether the predicate “green” has to be applied to an object, then it is also possible to determine without having temporal information whether a predicate interdefinable with “green” has to be applied to an object. Another example of interdefinability is that one about a decidable predicate PD, which is interdefinable with an undecidable predicate PU: therefore even though we can tell whether an object is PD and whether an object is non-PD, we cannot tell whether an object is PU (since PU is an undecidable predicate) and whether an object is non-PU. Although the predicates PD and PU are interdefinable, the possibility to determine whether an object is PD does not imply the possibility to determine whether an object is PU (since PU is an undecidable predicate). Similarly, although the predicates “green” and “grue” are interdefinable, the possibility to determine whether an object is “green” even in absence of temporal information does not imply the possibility to determine whether an object is “grue” even in absence of temporal information. These and other examples about “grue” and “bleen” point out that even in case two predicates are interdefinable, the possibility to apply a predicate P does not imply the possibility to apply a predicate interdefinable with P. And that the possibility to apply the predicate “green” without having temporal information does not imply the possibility to apply the predicate “grue” without having temporal information. Furthermore, knowing that an object is both green and grue implies temporal information: in fact, we know by definition that a grue object can only be: 1) either green (in case the object is examined before time t); 2) or blue (in case the object is not examined before time t). Thus, knowing that an object is both grue and green, we know that we are faced with case 1, the case of a grue object that is green and examined before time t. Then the paper points out why the Goodman-Kripke paradox is a paradox about meaning that cannot have repercussions on induction. Finally the paper points out why Hume’s problem is a problem different from Goodman’s paradox and requires a specific treatment.