Singularities are now realizations of that original possibility tending to a general cohesiveness Peirce vol. We can understand change as a perfect continuity of possibilities of which any actual occurrence is a realization. We can define a Peircean perfect continuum by four characters: modality plasticity , transitivity, generality, and reflexivity, each underlying one aspect of the relationship between the parts and the whole of continuity, as seen in Zalamea Generality is the law of cohesiveness among parts beyond any individual and any possibility of metrically measuring it; modality means plasticity, namely the fact that a continuum is not tied to actualities but involves both possibility and necessity; transitivity is the internal passage between modalities possibility, actuality, and general necessity ; reflexivity means that any part shall have the same properties of the whole to which it belongs.
Mathematical diagrams work because they act synthetically, namely — according to an old Kantian definition — in mathematics we are dealing with universals in particulars, while in philosophy we have to deal with universals abstracted from particulars. The great power of generalization of mathematics is due to these contracted universals. It is the approach through logical modalities as such. Possibility, actuality, necessity, became the way in which Peirce explained transition within the continuum itself.
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But in what do these modalities consist? Possibility is the mode of reality in which the principle of contradiction does not hold. Actuality or existence is the mode of reality in which both the principle of contradiction and excluded third hold. Necessity is the mode of reality in which the principle of excluded third does not hold, that is, both alternatives can be false. Reality actualizes possibilities and tries to develop them as generalities.
According to Peirce, it is impossible to describe reality without modalities. A term can be vague, determinate, and general Peirce according to the characters inhering in it. If any character is predicated universally and affirmatively or recognized as inherent the term is determinate.
If it requires further determination by the utterer, it is vague; if it requires further determination by the interpretant it is general. Vagueness, determination, and generality fall under the same description of the logical modalities.
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What we have here is a transition in determination which is based on the ontological reality of the logical modalities. A term or a concept can be vague and become determinate and even general and vice versa. Possibility as firstness can also be understood as a not yet actualized border from where actions secondness and ideas thirdness eventually emerge.
Moreover, it is precisely in the horotic transposition between particulars seconds and generals thirds where many logical forces enter the panorama. Every element has the same properties of general modalities reflexivity ; the passage among vagueness, determination, and generality transitivity is the law of the development of meaning through categories generality ; this latter depends on the possible inchoative status of vague meaning possibility.
EG are the iconic formalization of logic of propositions Alpha , first order Beta , and modalities Gamma. In comparison to formalization by symbols that happened in the same years, EG display a greater simplicity and uniformity same rules for the three fragments, propositional, first order, modal. EG were still analytic, but their overall project was a synthetic one Zalamea If the generalization is the analytic result of the diagrams, diagrams are the synthetic happening of generals.
Identities are always passing through changes. In order to answer this question we have to represent both the continuity of change and the identity we want to inquire within this change. The first kind of continuity, the continuity of reality, is represented by the sheet of assertion. The sheet represents the universe of discourse we are dealing with in Alpha and Beta parts logic of propositions and first order.
In this continuum we are really scribing not the propositions as they actually are, but as they might be. In the Beta Graphs we have a graphical tool for identity: the line of identity, that can solve the problem of quantifiers. The line represents the existential quantifier when it is evenly enclosed and the universal quantifier through an appropriate nested double cut. But the definition becomes more complex when we do not have to deal with the two dimensional sheet of assertion, but with the multidimensional plastic continuum. When we scribe a line of identity on the multidimensional plastic continuum we are representing the identity into the continuum of possibilities.
The line of identity continues to assert the identity of the individuals denoted by its extremities, but here it means the continuity of possibilities of an individual considered as a changing object in its becoming, i. This switch from using a line of identity on a sheet of assertion to using it on a multidimensional continuum implies the passage from an Assertoric view of Truth to a broader conception of truth and identity based on possibility.
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Teridentity represents two relations of co-identity and means that there are two different relations of identity that have in common one end and part of the line. The loose end of the line of teridentity means that there is or was also a different possibility even if it is or was not being actualized. It means that we know identity as a continuity but we do not know at which point it would stop and the point might be or have been different from the one we have. We have to allow the possibility that brute existence will be different since we are not representing actuality but potentiality.
Finally, the line of identity has a direction or an aim because we draw it starting from one point and ceasing at another. The loose end guarantees the fullness of reflexivity: any part of the line, included the ends, are possibilities that might be realized according to a general law. Besides, every single dot actualized on the multidimensional continuum realizes a possibility according to the general law, and might have a different branch — the loose end — as a different actualization of the same possibility.
So there is a transition from possibility to necessity through actuality. The drawing itself is our recognizing; in EG there is no recognition without the actual drawing of the line. It can be a mental or a virtual act, but it has to be a kind of diagram within a general interpretation. It is what we call a mathematical gesture. We can say we really understood something when we are ready to act according to our ideas; so, it is the performance that synthetically involves our reasoning.
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But if every gesture carries meaning, not every gesture serves to recognize an identity. So, any gesture is meaningful, but not every gesture embodies a synthetic reasoning. Now, for a gesture to be perfect it has to be a general law symbol that generates replicas; it is actual when it indicates its particular object index ; and it expresses different possibilities of aspects of the object which it refers to icon.
The three semiotic characteristics describe what a perfect gesture is: creative because of possible aspects, singular in its individuality, recognizable for its conformity to an established pattern. Moreover, gestures carry a possible and vague meaning that progressively gets determinate till a new habit is established. So we can easily find in this movement generality, modality, and transitivity.
How can we perform this reasoning? We do that through perfect gestures, where creativity, particularity and generality overlap. Perfect gestures — like the line of identity in EG — save identity through changes because of their continuous semiotic nature. Analytic reasoning loses the identity through changes because it is concerned with breaking up the identity, as its etymology suggests from analyo , discomposing, breaking up. Categorization and the Moral Order Routledge Revivals 5. Introduction to Phenomenology 3.
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