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m.pdf IntroductionCould we see logic? Which are some of the most relevant images in logic?What do we see when we understand logic? (Is it not perhaps true that when weunderstand logic we somehow see it?) Is it possible to make logic more evident?Logic is crucially concerned with correct reasonings or arguments, that is, withways of passing from true premises (or at least valid premises) to true conclusions(or at least valid conclusions). So, when are these reasonings or arguments correct?When do these reasonings or arguments work? They work all the times in which asubject of experiences is able to focus, and thus reflect, on different kinds of forms,or structures (or models, or patterns, etc.). If one is able phenomenologically tofocus and describe these forms or structures, then one has also the opportunity toshow these forms or structures to other persons: one can detect and display the relationsamong the elements of these forms or structures. A real proof in logic hasthus always the features of a literal demonstration: a reasoning that one could inprinciple always reconstruct visually. Logic, it is maintained here, is not very differentin spirit from geometry, though logic deals with more abstract ideas thangeometry (the importance of geometry for philosophy was wholly recognized byPlato). Since the ideas of logic are abstract it is of the utmost importance to drawsuch ideas. On the contrary, strict symbolic logic, that is, mathematical logic in thestyle of written algebra, has never faced the problem of making its notations, andthus its representations, more impressionistic and explicit. But one cannot preciselygrasp the meaning of a logical symbol if one is not able to say which images aregenerated by the use of such symbol (and thus which images could fall under suchsymbol). For example: one is not able to grasp the meaning of '⌐ A' if one does notknow that it generates a figure of negation ('⌐' means 'not'): a figure in whichsomething (or someone) A is imagined to be absent, or cancelled, and thus it isimagined to designate only, at most, what is external with respect to A itself, etc.Now, if one examines the first developed logic of the ancient times, the logic ofAristotle, one sees it is based on certain basic figures too: the four famous Aristoteliansyllogistic figures 'A, E, O, I', with each of such letters referring to a paradigmaticstructure of categorical propositions (here, precisely, the letter 'A' designatesa structure of 'universal affirmatives' ('All X are Y'); the letter 'E' designates astructure of 'universal negatives' ('No X are Y'); the letter 'O' designates a structureof 'particulars affirmatives' ('Some X are Y'); and finally, the letter 'I' designatesa structure of 'particular negatives' ('Some X are not Y')). Figures, in short,are crucial for Aristotle's logic too. However: is Aristotle's logic wholly coherentwith a philosophy centered on images? No, at least not if one takes into account thefollowing decisive fact: that the four main figures of Aristotle's logic are not builtby analyzing different perceivable images, but by analyzing different linguisticterms and sentences (this fact about Aristotle's logic is for example lucidly stressedby Descartes in his 'Replies', in particular when he observes that his 'cogito ergosum' argument has not the features of a terminological syllogism but of a beliefgrounded on an immediate act of mental intuition or vision). Aristotle's idea ofdeveloping logic on a strict linguistic basis is then embraced, more than twentycenturies later, by Gottlob Frege (indeed Frege's prominence in logic is due,among the other things, to his ability to extend Aristotle's term logic - this by conceivingwhat is nowadays called predicate or quantificational logic). Frege's maindevice for representing some basic logical notions (identity, implication, etc.) is theBegriffsschrift, a German expression to be translated as 'concept writing' or 'conceptscript': a notational system modeled on formulas used in algebra and arithmetic,to be applied to linguistic propositions. In Frege, once again, one thus findsthis: that logic is not conceived as a way of shading light on perceivable forms orstructures at the basis of our valid reasonings, but as a way of packing words andsentences into formulas. In short: first Aristotle and then, above all, Frege somehowset the path that brings logic to its present indirect shape and symbolization(the attempt by the American logician Charles Sanders Peirce to make logic moreexplicit and iconic is not in the end successful for Peirce does not anchor his logicon a robust philosophy of images - he anchors it on a mere semiotic or 'semeiotic'reflection, a mere philosophy of signs).But one could now ask: would not it be possible to invert the main trend inlogic that wants it more and more departed from our impressions and ideas? Wouldnot it be possible to reconcile logic with our experiences, so to make logic closer,to begin, to our visual impressions and ideas? It is possible: it is possible if onerealizes that one's activity of reflexive imagination plays an indispensable role inlogic; such activity, however, should then also be accompanied by one's capacitypublicly to reorganize and display one's internal images.One could begin here by noticing this: that if one depicts the most importantlogical reasonings, one sees that they take two basic iconic forms: i) a form orstructure to be described as (literal) containment - as, for example, with the inference'If I have two books then I have one book'; and ii) a form or structure to bedescribed as (literal) entailment (or implicature or involvement) - as, for example,with the inference 'If I have a book then I have an object' (while an inference as 'IfI have a book then I have a blue object' would be incorrect). Logical reasoningsthat depend on the form of i) containment are inferences that start from somethingbigger (in this sense external) and move towards something smaller (in this senseinternal). Logical reasonings that depend on the form of ii) entailment (in the strictsense) are inferences that start from something smaller (in this sense internal) andmove towards something bigger (in this sense external).Cases of logical inference expressing the very same idea of identity ('a = a','a = b', etc.) could then also be seen to point to boundary forms or structures ofcontainment: they are an extreme variant of the containment kind (one central ideaof logical identity is the idea of something '(being) in itself'). Indeed, the basicform of these 'identity inferences' is that of a geometric figure containing itself oranother geometrically congruent figure - as, for example, with the inference 'If Ihave a book then I have a book'.(To recapitulate about the previous examples: as for the inference (1) 'If I havetwo books then I have one book' one sees this: that the image of two books necessarilycontains, as its own part, the image of one book. As for the inference (2) 'If Ihave a book then I have a book' one sees this: that the image of a book necessarilycontains itself, that is, a congruent image of the book. As for the inference (3) 'If Ihave a book then I have an object' one sees this: that the image of a book necessarilyentails or involves (or ingenerates, etc.) the image of an object.)The form of i) containment makes then possible to see the most intuitive inferencesassociated with the so called propositional logical connectives: negation ('Ifnot a then logical absence of a'); conjunction ('If a and b then a, b'); (inclusive)disjunction ('If a or b then a, b'. Etc.); conditional ('If a then b then a, b'. Etc.);biconditional ('If and only if a then b then a, b'. Etc.). In these cases, one is able toevaluate the correct inferences also because one is able to see the specific imagesthat are associated with each specific logical connective: one is able to see that theimage of negation of one thing contains the image of absence of such thing (negation);one is able to see that the image of two things together contains the image ofeach one of such things (conjunction). Etc.What about the relevance of the form of ii) entailment for reasonings exploitingthe five propositional connectives? One could reflect here on an inference likethis: 'If a, b then a and b'. In this case, if one accepts this inference as correct, onedoes so because one sees that compositionally the image of 'one thing, anotherthing' brings about, and thus entails, the image of 'one thing and another thing'(not surprisingly, the truth-tables for the logical connectives have historically beenbuilt, at the beginning of the twentieth century, in a compositional way: they relyon a philosophy of logical atomism).The forms or structures of i) containment and ii) entailment are also at the basisof the central reasonings in what is called 'predicate logic' or 'quantificationallogic'. One could begin to consider, here, certain inferences in the style of Aristotle:if one sees, for example, that 'If all books are interesting then some books areinteresting' one just observes that the image of 'all' (expression of a universalquantifier) contains the image of 'some' (expression of an existential quantifier).Similarly: if one sees that 'If no books are interesting then some books are not interesting'one just observes that the image of 'no one thing' (expression of a negationplus an existential quantifier) contains the image of 'something does not' (expressionof an existential quantifier plus a negation). Moreover: the rule of inferenceof universal instantiation, that allows one to eliminate the universal quantifierin a logical demonstration, depends on the form or structure of containment: 'Ifeverything is a book then one concrete thing is a book'. The rule of inference ofuniversal generalization, that allows one to introduce the universal quantifier in alogical demonstration, depends instead on the form or structure of entailment: 'Ifevery concrete thing is a book then all things are a book'. Similarly: the rule ofinference of existential instantiation, that allows one to eliminate the existentialquantifier in a logical demonstration, depends on the form or structure of containment:'If something is a book then a concrete thing is a book'. The rule of inferenceof existential generalization, that allows one to introduce the existential quantifier,depends instead on the form or structure of entailment: 'If a concrete thing isa book then something is a book'. (In other terms, about these rules of inference inquantificational logic: one sees that a bound variable contains all the constants inthe domain of the bound variable; and one sees that all the constants in the domainof the bound variable entail the bound variable.)Let us consider now the so called 'modal logic', the logic built on the ideas ofnecessity and possibility. One could observe here, for example, that the inference'If it is necessary that I have a book then I have a book' rests on a form or structureof containment (one here imagines this: if the logical space must host a book then ithosts a book). Or one could observe, for example, that the inference 'If I have abook then it is possible that I have a book' rests on a form or structure of entailment(one here imagines this: if there is a book in the logical space then there isavailable logical space for the book in the logical space). Or one could observe, forexample, that the inference 'If it is possible that I have a book then it is not necessarythat I have a book' rests on a form or structure of containment (one here imaginesthis: if there is further available space in the logical space that hosts a bookthen such further space is not null). Again: one could observe, for example, that 'Ifit is necessary that I have a book then it is necessary that it is possible that I have abook' rests on a form or structure of entailment (one here imagines this: if the logicalspace must host a book then it must be possible for the logical space to host abook). Etcetera.The iconic forms or structures of i) containment and ii) entailment are thus atthe centre of logic - if one scrutinizes the most relevant images behind our reasonings.A further question is now this: is there also a way of more directly or intuitivelyshowing - that is, demonstrating - such relevant images behind our reasonings?Yes, a possibility is this: one should here simply begin to substitute the linguistic,alphabetical letters used to represent propositions or sentences in standardsymbolic logic - i.e., 'p', 'q', etc. - with geometrical points or dots referring tovisualizable things: '' (one should keep in mind here that a point/dot is the smallestimage - a point/dot is the highest expression of iconic parsimony in the sense ofOckham's razor). So, for example, one could represent the image 'The book isheavy' with two points: 'Book', as subject/object: ''; 'Being heavy', as property:''. For greater simplicity, one could represent a subject/object with its property (asin a subject-predicate sentence) by means of a single point (thus, 'The book isheavy' will be pictured like this: ''). If the idea or image of a (true) 'something' isthis, '', then the idea or image of negation (the negation of such something) willbe this: '¬ '; the conjunction of two things will be this: ' & ' (or ' '); the disjunctionwill be this: ' ¬ '; the conditional: ' ' (as containment: ' C ' (thuswith ≥ ); as entailment ' ' (thus with < )); the biconditional: ' ↔ '.Etcetera. In order to name this technique for representing things by means ofvisual points (or dots) I shall coin here the term 'pointography' (or 'dottography').One could notice that a pointography (in the sense just specified) is, among theother things, a more genuine instance of 'ideography' than Frege's Begriffsschriftfor it is sensitive to the nature of the 'idea' (the Ancient Greek 'ἰέ') as somethingvisual. By representing things as points/dots, and not (or not immediately) as symbolicletters or numbers, one is able to pass from verbal, indirect proofs to visualproofs - that is, to explicit demonstrations. Let us consider, for example, the followingstatement: 'It is not the case that a book is blue and a book is red, and that abook is not red'. If one draws such statement and simplifies it one sees this:¬((&)&¬)¬(()&¬)¬(&¬)¬(¬)¬¬(Here thus one sees an image of conjunction, 'a book is blue and a book is red':'(&)'; and then an 'image' of contradiction, 'a book is red and a book is not red':'(&¬)'; then one sees an image of negation of the contradiction, thus an image ofnegation of something false: '¬(¬)'; finally one sees an image of something true:''.) By reconstructing a certain logical situation by means of drawn points onemakes it possible both for the eye and the mind's eye more realistically to observesuch situation and calmly evaluate it (this of course makes it also possible for otherpeople publicly to observe one's reasonings; again, it allows one to make a visiblecomparison of different logical situations; etc.). An important consideration to behighlighted here is however this: the possibility of arriving at a more iconic andeven geometric representation of a logical state of affairs by employing a pointography(i.e. a dottography) should not be taken to imply the possibility of testingthe truth or falsehood of such state of affairs without the mind eye. The belief thatone could evaluate a logical case without intuition and philosophical reflection isjust illusion - as Gödel has shown, the idea that one could close logic under a finitelist of axioms, however long such finite list of axioms is taken to be, is mere chimera.Hence there is perhaps just one axiom that one could somehow put at thebeginning of logic, what one could call the 'mind eye's axiom': logic should beevaluated, first of all, on the basis of insight and philosophical speculation. Indeed:the 'mind eye's axiom' would clearly state that a logical system cannot but be anopen system. This suggests, among the other things, that one should be open to seewhether there are other possibilities of interpreting a given logical case - oneshould be ready to take into consideration counterfactuals (i.e. counterexamples). Itis also in this particular sense that the mind eye's axiom keeps the logical systemopen. On the other hand, the fact that one wants logic to be based on direct, reliableevidence - as, for example, when one is dealing with visual demonstrations - suggeststhat one thinks that logic should be put on a firm ground. The idea of a logicalspace, imagined as a visualizable space, and of something positive contained init looks as a firm, reliable ground - this is so because one just realizes here that ifone is in front of a positive thing then one cannot be in front of its opposite, that is,the negation of such positive thing (as we have said, the absence of such positivething, etc.). And one's ability, in general, clearly to see the logical space and all thenon contradictory possibilities contained in it is just one's capacity of logical vision.1Logic, from the written and spoken Greek word 'logos'. It has been translatedas 'word', 'speech', 'thought', 'reasoning', 'action' ('deed'), etc.Perhaps the best translation of 'logos', as conceived in Greek early philosophy,is 'mental kosmos'. One should see here that 'logos' (or 'Logos') is associatedwith the Greek verb 'legein': 'to recollect' (this is the reason why 'logos' could betaken to refer to the idea of 'mental kosmos' or 'mental all' - one should also noticehere that the Greek word 'kosmos' refers to some intuition of 'ordered universe').Here is a possible picture of logos as mental kosmos: (Image 1)2The idea of mental cosmos suggests this: that one cannot do logic - one cannotunderstand logic - without a theory of mind. More in general: one cannot do logicwithout a metaphysics.3Before Aristotle, logic - as discourse about logos - is conceived as a reflectionon human beings' experiences about the principles of thought and world: beforeAristotle, logic is experiential, not linguistic - with Aristotle it becomes a 'termlogic'. (Aristotle's syllogistic logic is thus not built on simple intuitions. It is based,for example, on two premises and a conclusion, not on a single premise and a conclusion.Etc.)4At the end of the 19th century-beginning of the 20th's, the mathematician andphilosopher Edmund Husserl tries to move back to a conception of logic as a generalmeditation on the principles of (immediate) phenomenal thought and experience.(Image 4)5At the end of the 19th century-beginning of the 20th's, Charles Sanders Peircetries to develop a more iconic approach to logic. Peirce's more iconic approach tologic is not, however, based on a substantive interest on images. Peirce's main ideais not that of putting images at the basis of his philosophical reflections, but that ofputting signs at the basis of his philosophical reflections. This is the reason whyPeirce is nowadays recognized, correctly, as one of the fathers of semiotics (or'semeiotic'), not as a contributor to the development of a philosophy of images.Logic is conceived by Peirce as a formal branch of his pragmatic (or 'pragmaticist')theory of signs. Moreover, his writings on icons and logic are wholly basedon non ordinary images: they are based on diagrams and graphs, specifically on'existential graphs'. Since Peirce does not pay much attention to human beings'common experiences, many of his existential graphs come out to be counterintuitive- in his texts concerned with the existential graphs, Peirce has thus often to addtechnical 'conventions', or 'permissions', or assumptions, etc., in order to explainsuch graphs. Here is an example: when Peirce depicts the idea of negation, hecounterintuitively puts a circle, i.e. an oval enclosure, around a propositional letter.For instance: 'There is not a car' is drawn as '- car' inside an oval enclosure, or ashaded oval enclosure (the sign '-', a dash, is used by Peirce to signal an assertion).Peirce does not discuss, then, other possible ways in which people usuallyimagine or visualize a negation. (About this point one could also reflect onJ.F. Sowa's commentaries on Peirce's manuscripts.) (Image 5)6Nikolaj Vasilev's (in Russian, Николай Александрович Васильев's) 'imaginarylogic' is not an iconic logic - it is, historically, one among the very first instancesof non classic or (so called) 'paraconsistent' logic.7What is erroneous in David Hilbert's 'formalistic' approach to logic is this: tothink that the main problem of logic is syntactically to derive or prove theoremsfrom self-evident axioms. The main, or primary, problem is instead this: to showthe self-evident form of the axioms or first principles. Hilbert's attempt to fix theaxioms is then vitiated by the fact that his symbols and representations concerninglogic and mathematics are incommensurable: they are not homogenous (they sometimesare alphanumerical symbols; sometimes geometrical elements; etc.). Moreover,Hilbert does not make it explicit that someone - a subject of experience - hasto see or intuit such forms or representations (Hilbert's main idea is just that ofclosing a logical or mathematical object by bringing it under a finite system).From a more general point of view, the limit of Hilbert's approach to logic andmathematics is the following: trying to be objective on logic and mathematics justby ruling on the connection of certain possible objects of logic and mathematics(e.g. finitary numerals such as 1 ('1'), 11 ('2'), 111 ('3') and so on, or Euclideanelements, etc.), without paying much attention to the conscious subjects of logicand mathematics. In one of his first works, Foundations of Geometry, Hilbert neverpresents a geometrical figure together with an explicit image of a subject perceivingor intuiting such figure. Indeed, it is only in the latter years of his studies thatHilbert begins to take into consideration representational and heuristic problems atthe basis of logic and mathematics). (Image 7)8Intuitionistic and constructivist logics - like those developed by L. E. J. Brouwer- could be seen as attempts to put logic closer to a person's experiences. However:even these conceptions of logic have not culpably been developed on iconicbases. (Brouwer also maintains that time is the most primitive and crucial elementfor logic. In accordance with what I have till now tried to show, I think it is howeverspace, that is space-time, in particular logical or experiential space-time, to bethe most primitive and crucial element for logic.)9The space of logic, or logical space, is the space of the conceivable - one couldmore precisely say it is the space of the conceivable as experienceable.A possible drawing of such logical space is this (an open circle or an opensphere): (Image 9)10Logic is concerned with basic structures or forms or models (even dynamicalstructures or forms or models) inside the logical space.An image of a structure is, for example, the image of a simple net or reticulate- one could then also think about such things as the structure of a temple, etc.:(Image 10)11Developing logic: trying to show the simplest structures or forms underlyingthis or that phenomenon or thing (for example a truth-preserving reasoning).12A net or reticulate is made, for example, of vertical and horizontal lines - thatis, of columns and rows. (Image 12)13The Cartesian axes, which give rise to a Cartesian plane, are a kind of structure- here one could just observe the two perpendicular lines, without immediatelypaying attention to the fact that such lines could then also be ordered, etcetera. (Image13)14Two perpendicular (or quasi perpendicular) lines meet in a point. This point -in particular if one has in mind a net, or a mesh, etc. - is sometimes called 'knot'.(The meeting-point of two Cartesian axes is usually called 'origin'.) (Image14)15The main principle of logic is the principle of identity, the idea of something orsomeone being itself or oneself. Here one usually writes this: a = a.The idea of identity is the most extreme idea of relation - it is an internal, orself-reflexive, relation. (Image 15)16Again on the idea of identity, in particular when it is represented as 'a = b':Gottlob Frege, for example, tries to explain the case in which two differentthoughts point to the same object. Frege discusses the case of the 'Evening Star' (inGerman 'der Abendstern') and of the 'Morning Star' (in German 'der Morgenstern').Frege claims that the 'sense' (in German 'Sinn') 'Evening Star' has thesame 'reference' (in German 'Bedeutung') of the 'sense' 'Morning Star' for bothsuch 'senses' point to one object: the planet Venus.Let's now however observe this: that we could draw the fact that 'Hesperus isPhosphorus' ('The Evening Star is the Morning Star') in the following way: (Image16)'Hesperus is Phosphorus' should thus not be regarded as a famous sentence ofa philosophy based on language but, first of all, as a famous imaginational experience- the 'sense' of a thought should not be regarded first of all as a linguistic'mode of presentation', but instead as a mode or circumstance of vision (as a visualinterpretation - an interpretation that also relies, for example, upon the perceptionof a given environment or context or background).17On the idea of identity written as 'a = a' or as 'a = b': according to Leibniz'sprinciple of the 'indiscernibility of identicals' two things are identical if they haveall their properties in common. Hence, two copies of the same book are not one andthe same, for they display at least one different property: they occupy two differentportions of space. Of course the two copies of the same book should be seen toshare the same form - or experienced form (Image 17).18One could see that certain numbers, or certain numerical expressions, etc. havethe same logical, abstract form. For example: the numeric expression '1 + 1' hasthe same logical form as the numeric expression '2' (indeed: '1 + 1 = 2'). Onecould clearly observe this if one draws the numbers or the numeric expressions aspoints: (Image 18)(The points ' = ' have different spatial positions, though one is assuminghere that numbers could be held to be abstract entities - entities that (taken singularly)do not have to be characterized as displaying a contingent position in spaceand time.)19Geometric congruence is two figures having the same i) shape and ii) size: twofigures are intuitively congruent if and only if the distance between two points inthe first figure is the same as the distance of the two corresponding points in thesecond figure (geometric congruence is analogous to equality or equivalence fornumbers). Isomorphism is two figures having the same shape. Geometric congruenceand isomorphism could be seen to express degrees of identity. (Image 19)20We have seen that the main condition of identity, conceived in the logicalsense, is the condition of isomorphism, and even more strictly of congruence. Onecould observe such isomorphism or congruence even by paying attention to theform or shape of certain words, as, for example, in the following linguistic proposition(i.e. a tautology): 'water is water' (that is, 'water = water').21A relevant idea for logic is symmetry. When one is concerned with logic oneshould remind oneself that one's bodily eyes are symmetrical and that one's inneror mental eye is symmetrical too - one should remind oneself that one's visual fieldand imaginative field appear symmetrical. (Image 21)22Here are some images of i) vertical symmetry, ii) horizontal symmetry, and iii)centered symmetry (of course the expression 'centered symmetry' should not betaken to mean that only centered symmetries have a centre - indeed every symmetryimplies, just qua symmetry, a centre.) (Image 22)23One of the first difficulties in seeing some logical forms inside our ordinary orcommon phrases - written propositions or statements - is this: the fact that manywritten languages are asymmetrical: they are written, for example, from left toright, etc. For instance: this proposition, 'the house is black', is asymmetricallywritten from left to right. (Image 23)24Symbolic logic - i.e. the development of symbolical notations in the history oflogic (in particular in the 19th and 20th centuries) - has not faced the problem ofmaking its signs and representations more impressionistic. In other words, symboliclogic has not faced the problem of making its representational conventionsmore natural - that is, closer to our everyday experience of the world.(Some cases and commentaries in support of this latter claim are offered below).25Classic symbolic notation has for example 'Fa' to refer to a certain subject orobject 'a' that has the property or relational property 'F' ('F' stands for 'function').Here one immediately sees this: that the letter 'F' is written as a big (capital) letterand the letter 'a' is written as a small (low) letter. As a matter of fact, however, itshould be the other way round: if it is true that the letter 'a' represents a subject orobject and the letter 'F' a property, then it is the letter 'a' that should be written asa big letter and it is the letter 'F' that should be written as a small letter. Moreover:assuming a left-right order of writing, the letter 'a' should have precedence withrespect to the letter 'F', and thus one should find the letter 'a' on the left and theletter 'F' on the right (indeed, in our common, non-technical way of thinking andwriting, the subject is usually, realistically, put on the left, at the very beginning ofthe phrase). An impressionistic, or more impressionistic, notation should thus havethis: 'Af' (for argument's sake I am here of course making tabula rasa of certainconventions in symbolic logic). (Image 25)26When one writes the subject-predicate as 'Fa' one should also keep in mindthis: that the predicate (or the function, etc.) 'F' is not, in reality, on the left of thesubject 'a'. For example, if 'F' stands for the property 'being white' and 'a' standsfor the subject 'Paul's skin', then it is not of course the case that Paul's skin iswhite on the left! If a property belongs to a subject, one has to assume that thisproperty is symmetrically distributed in the subject. (Image 26)27In the following points (from point 28 to point 36) I will try to put into focusthe ideas of i) subject and predicate; ii) relation; iii) negation; iv) conjunction; v)disjunction; vi) material implication (conditional); vii) strictly logical implication(biconditional).28Classic logic is seen to be based on the ideas of subject and predicate. Onecould draw a logical figure of a subject and its property as follows:--(For example: if the idea is that 'Socrates is mortal', then the point '' drawsthe subject 'Socrates'; and the little line '--' draws the property 'being mortal'.)For reasons of simplicity one could draw a subject with a given property just asa single point:(This latter solution might especially suit those philosophers that think that thedistinction object/property should be abandoned.)29An image that seems to capture the idea of a (basic) logical relation could bethe following: a line or segment connecting two subjects, or two objects, or twoindividuals, etc. (for example; if the idea is that 'Romeo and Juliet love each other'then one point '' will refer to the subject or individual 'Romeo'; the other point ''will refer to the subject or individual 'Juliet'; the line or segment connecting them'---' will picture their relation of love.)Here is a possible drawing of a basic logical relation:---In some cases, one could also find useful to show the direction of a certain relation.If 'Don Quixote loves (in an uncorresponded way) Lady Dulcinea', onecould, for example, draw this (asymmetric relation):Don Quixote --- Dulcinea'To love'30The idea of negation is usually expressed by the word 'not'.A possible image that refers to the negation of a certain thing is an image ofabsence of such thing: not A is the absence of A (here one somehow keeps A in the'corner' of one's mind eye and sees its absence).Another image of negation of a given thing (a negation that first of all negatesthe thing) is an image of a cross or of a slash (or of a sign in general) erasing suchthing: not A is the cancellation (or the elimination, etc.) of A.Again, thirdly, a negation of a certain thing could be visualized as anotherthing that is external with respect to that first thing: not A is external with respectto A (in this third case one can see why the idea of negation is sometimes called'logical complement'.)The ideas of negation as i) absence of something and ii) erasing of somethingcould be captured by the containment model. The idea of negation ('not A') as iii)external thing (with respect to A) could be captured by the entailment model.Here are the three images of negation that we have just mentioned: (Image 30)31About the idea of negation, in particular about the idea of absolute or independentnegation, one should observe this: that if one takes i) something (A) and ii)a negation (N) to refer to absolute things, one has i) being and ii) nothingness (herean absolute property would also become an absolute thing or subject).One sees here that being and nothingness could exist together at the same timeand in the same respect. This is so because the image of nothingness, as somethingindependent, does not corrode the image of being (similarly: '0 + 1 = 1').Here is a possible drawing of being and nothingness: (Image 31)32An image of conjunction between two (or more) things is the image of two (ormore) things taken together: one could also see the idea of conjunction as, for example,two things that meet: for example as two segments (two streets, etc.) thatmeet. This latter image could also be used intuitively to explain the shape of thelogical symbol ''.If, for example, one draws the proposition 'There are an orange and an apple'one has this: (Image 32)33One could visualize a logical disjunction by imaging two things that diverge,for example by imaging two segments that diverge, or two streets that bifurcate,etc. (this would intuitively explain the logical symbol '', which however comesfrom the Latin word 'vel': 'or'). The idea of inclusive disjunction refers to the casein which one could also take both the diverging segments, or both the divergingstreets, etc. The idea of an exclusive disjunction refers to the case in which one hasto make a real, hard choice - if one chooses, for example, the left segment, or theleft street, etc., one cannot then also choose the right segment, or the right street,etc.If one draws the proposition 'There are an orange or an apple' one has this:(Image 33)(An intuitive symbol that has also been employed to represent a logical disjunctionis this:|If one puts two objects-points on the left and right sides of such sign of disjunctionone sees this: | )34A 'material implication' (or a simple conditional) is captured, for example, bythe image of a ball causing the movement of another ball - this second ball is assumed,at the beginning, to be at rest. The idea behind the notion of material implicationis the idea of something - sometimes called antecedent - causing somethingelse: an effect, etc. - sometimes called consequent.Another image that captures the idea behind the simple conditional is the imageof containment. For example: one could visualize a material implication as abigger circle that contains a smaller circle. When the smaller and bigger circles areboth true, one sees that the bigger circle indeed contains, and thus implies, thesmaller circle; when the bigger circle is true and the smaller circle false one seesthat it cannot be the case that the bigger circle contains, and thus implies, thesmaller circle; when the bigger circle is false one could imagine, about the smallercircle, whatever one likes in Latin, ex falso quodlibet.Here is an image of an orange that hits and pushes an apple, and thus causes itto move. Here one says this: 'If the orange moves, then the apple moves'. (Image34)35The material implication has sometimes been expressed by means of the followingsymbol of entailment: '' (e.g. 'A B', to be read as 'A implies B'). If onesees the material implication as containment, one could however more preciselyrepresent it by means of a 'C' symbol: 'A C B', to be read as 'A contains B'. Therelevant point is now this: one should clearly distinguish a form of inference asentailment ('A entails or implies B': 'A B') from a form of inference as containment('A contains B': 'A C B').(About this point also see the notes in 52, 53 and 68.)36The idea behind the notion of (strict) logical implication, also called doubleconditional (biconditional), is the idea of two things, for example two objects, kepttogether by some kind of indissoluble link or tie. Since the link or tie between thetwo things is imagined to be indissoluble, when the first thing is true also the secondthing is true; and when the first thing is false also the second thing is false - ifthe first thing were true and the second false or vice versa, one could not see thatsuch things are kept together by an indissoluble link or tie.Another appropriate image for the strict logical implication is that of twowholly overlapping or congruent circles.The biconditional 'p if and only if q' is logically equivalent to the expression'p implies q and q implies p'.If, for example, one draws the proposition 'There is an orange if and only ifthere is an apple' one has this: (Image 36)37The idea of necessity in modal logic could be captured by showing that there isonly one logical space (or one 'possible world', etc.) inside which something is thecase. If one wants to draw the sentence 'It is necessary that it is sunny', one coulddraw a single circle that perfectly contains a point - the point, in this example,represents the sunny weather. Instead: the idea of possibility in modal logic couldbe captured by showing that there is more logical space than the space strictly containingwhat is the case (one usually searches this further logical space by usingwhat is sometimes called 'counterfactual imagination'). If one wants to draw thesentence 'It is possible that it is sunny', one can draw at least another circle close tothe circle that strictly contains a point - i.e. that strictly contains the sunny weather.In fact: if it is possible that it is sunny then there must be a logical space that cancontain the sunny weather, but also at least another logical space (that is, somemore logical space) that can for instance contain a rainy weather, or a foggyweather, etc.Another way of drawing 'necessity' and 'possibility' would be to show a closelogical space ('it is necessary that p'); and an open logical space ('it is possible thatp'). (Image 37)(The standard signs used to represent necessity and possibility in modal logicare the following: necessity, ''; possibility, '◊'. One might say that, sticking to thenecessity sign, a more intuitive or impressionistic way of representing the ideas ofnecessity and possibility is this: necessity, ''; possibility, '◊'.38Saul Kripke claims that rigid designators should be thought as proper names.Here I propose this: to think of rigid designators as genuinely distinctive forms, forexample as essential or identificational images (fingerprints, etc.).39Each little space in the graph of a 'truth-table' could be seen as a 'possibleworld', that is, as a possible state or stage of the world.40A classic way to signal, in an argument or reasoning (e.g. in a syllogism), thedistinction between premise or premises and conclusion is drawing a horizontalline between them. The line serves also to signal that the conclusion (the 'result')'follows' from the premises.An image that could adequately express the relationship between premises andconclusion is a picture according to which the premises contain or entail the conclusion.(Image 40)41When one uses different parentheses in a logical scheme (or formula, etc.) onewants usually to show, among the other things, that the truth of what is externalrests on the truth of what is internal. For example, the truth of '{ [ ()]}' containsthe truth of '{}'; and the truth of '{}' contains the truth of '[]'; and the truth of'[]' contains the truth of '()'. With a metaphor: the truth about a whole onion containsthe truth of the content of the first most external layer of the onion; and thetruth of the content of the first most external layer of the onion contains the truth ofthe content of the second most external layer of the onion; etc. With anothermethapor: the truth about a whole matrioska contains the truth of the first most externalmatrioska; and the truth of the first most external matrioska contains thetruth of the second most external matrioska; etc. (Image 41)42At the vertex of his logic Aristotle puts the principle of non contradiction(PNC, as discussed in particular in Metaphysics IV):"It is impossible for the same thing to belong and not to belong at the sametime to the same thing and in the same respect." (Metaph IV, 3, 1005, b19-20.)This is how modern symbolic logic writes the principle of non contradiction:⌐ (A ⌐ A)(or: ⌐ (A & ⌐ A)(Etc.).(Formulas to be read as: 'Not (A and not A)'.)If one draws A and not A by means of some sort of Euler-Venn diagram, as inset theory, one aims to express this: firstly, that if something is in the 'A zone',then it cannot be in the 'not-A zone' (there cannot be cases that are (in) A and (in)not A at the same time (thus even the line or segment that divides the zone A fromthe zone not A in the diagram cannot be seen as a 'mixed place' (A and not A)).(Image 42)43If one would like to escape from the contradiction '1 = 2' one should be able todraw a (unit-) point '' as completely congruent with two (unit-) points '' (orviceversa). Here a person will in the end say: 'I see that I cannot make one pointcongruent with two points!' (Image 43)44An attempt to arrive at an image of contradiction is, for example, a person's attemptto draw a squared circle - indeed one cannot imagine to put a circumferenceexactly inside a squared line or vice versa (indeed one cannot imagine to draw acircular square).Here is another case concerning an attempt to get close to a contradiction: aperson trying, for example, to write different alphabetical letters ('a', 'b', 'c', 'd','f', etc.) inside the very same space. The person sees, in this case, that the attemptto get close to a contradiction produces a messy scribble - the messy scribble isbrought about by some sort of clash of incompatible or inconsistent images.It has been said that a drunk (or drugged) person could for example have theexperience of seeing a room as still and moving at the very same time. This experienceshould, in fact, be interpreted as a person's getting close to the idea of contradiction- a contradiction being, in fact, an experiential impossibility. (Image 44)45Perhaps the simplest way to display a contradiction is to show the impossibilityof putting together the image of an object a that contains an object b with the imageof an object a that is contained by an object b. (Image 45)46About the so called 'quantum logic': one could observe that the quantum phenomenonin physics (double-slit experiment, etc.) does not force one to abandonthe principle of non contradiction. One could assume here that when a particle issaid to be in two places at one time, it is in fact in a 'unitary' wider place: it is inthe 'single' place corresponding to the absolute value of the sum of the two areasthat are believed to contain it (so, for example: 'The particle moves to the left(momentum) and is in the interval [0, 1] or in the interval [-1, 1]' becomes 'Theparticle moves to the left (momentum) and is in the interval [-1, 1]'. This way oflogically describing the quantum phenomenon is justified by the fact that a particlecould be seen as a wave, or as an induced field, etc. (as, for example, in the 'wave'or 'electro-magnetic field' interpretations of the microscopic behavior of light).(Image 46.)47On sintax (syntactic structures) and semantics: a syntactically consistent or noncontradictory formal system could be (or become) semantically inconsistent orcontradictory - hence it could in some cases become undecidable.(Syntactically comes from the word 'syntax', from the Greek 'ς' ('sintaxis'),which refers to the arrangement or ordering of certain signs; semanticallycomes from the Greek word 'ς' ('semantikos'), and thus from the word'' ('sema'), which refers to a significant sign or meaning.)Now suppose to have a formal system like this:'a, c, d, e, f, h, i, l, m, n, o, r, s, t, y, *'Suppose, then, to derive from the above signs - that is, from the above systemof letters as sign-types - this:'This statement is not contained in the formal system'.One can here see this: that syntactically (or structurally), the sign-types in thephrase 'this statement is not contained in the formal system' have been derivedfrom the sign-types 'a, c, d, e, f, h, i, l, m, n, o, r, s, t, y, *': the formal system thusactually contains 'this statement is not contained in the formal system': the formalsystem clearly contains one sign-type more than 'this statement is not contained inthe formal system': it contains all the types of letters of 'this statement is not containedin the formal system' plus the sign '*'. From a semantic point of view, however,'this statement is not contained in the formal system' suggests something opposite:a meaning that denies the fact that the statement is contained in the formalsystem - a meaning that thus generates some kind of inconsistency or contradiction.What does this case suggest? That syntactic consistency or syntactic non contradictionis not sufficient for semantic consistency or semantic non contradiction -meaning is something higher than syntax.(Kurt Gödel's incompleteness theorems should be seen to support similar conclusions.)(Image 47)48I take Gödel's incompleteness theorems to demonstrate this: that a given formalsystem cannot wholly display itself. In other terms: one point cannot representmore than one point - thus it cannot represent one point and the action needed forthe point to see or be aware of itself as a point (this suggests that the mind's eyecannot be closed inside a system if it has to be possible for the mind's eye to watchthe system).49One could notice that linguistic prepositions such as 'of', 'in', 'on', 'through',etc. usually function, inside a language, to generate visual forms or visual structures.One can thus see that not only there is a 'propositional logic' (with 'o') butalso a 'prepositional logic' (with 'e'). By saying, for example, that 'a man is sleepingon the floor', one communicates, first of all, the following idea: that a sleepingman is placed in a higher - i.e. more northern - position than the floor.Here one should then also take into account certain things such as spatial andtemporal adverbs, etc. (temporal adverbs that would for example be relevant for atemporal logic, etc.). (Image 49)50Some written symbols employed in the history of symbolic logic, and sometimesin the history of mathematics, are the following: '=': identity; '': non identity;'⌐' (or '~', etc.): negation; '' (or '&', '.', etc.): conjunction; '': disjunction(or '|'); '⊻': exclusive disjunction; '' (or ''): conditional (or implication); '↔'(or '', etc.): double conditional (or material equivalence); '': universal quantifier('for every x'); '': existential quantifier ('for some x'); '': necessity; '◊': possibility;'', 'therefore', usually pointing to a conclusion; ' ': 'yields' or 'proves',in proof theory; '': intersection, in set theory; '+': addition; '÷': division; '':infinite; etc.Of all these symbols, perhaps only a few could be held to be impressionistic (or atleast somehow intuitive): '='; ''; '⇐'; ''; '↔'; etc.51If one draws René Descartes' famous argument 'Cogito ergo sum', one seesthis: that the conclusion follows from the premise, for the premise visually entailsthe conclusion (this is so because the idea of 'res cogitans' ('thinking thing') or'cogitare' ('to think') is wholly crossed by the transcendental idea of 'ens' ('being')or 'esse' ('to be')): (Image 51)52A premise i) could contain the conclusion, and thus give rise to a containmentmodel; or a premise ii) could be crossed by a conclusion, and thus literally give riseto an entailment model.Here are two examples.First example: the reasoning 'If it is water, then it is H20' refers to a figure ofcontainment: the image in the premise 'it is water' contains the image in the conclusion'it is H20' (here I take this reasoning just to be expression of some kind ofidentity).Second example: the reasoning 'If it is Moscow, then it is Russia' refers to afigure of (literal) entailment: the image in the premise 'it is Moscow' entails orimplies the image in the conclusion 'it is Russia'. (Image 52)53Theoretical logic is thus not only deductive, as in the containment model(where one deduces the conclusion which is contained in the premises); theoreticallogic is also extractive, as in the entailment model (where one extrapolates the conclusionthat crosses the premises)). Thus: one not only could 'derive' conclusionsfrom premises, but also 'extract' conclusions from premises. Linguistic arguments- for example arguments based on a definition ('If he is a bachelor, then heis unmarried') - have usually the form of a containment. Transcendental arguments(as the 'cogito ergo sum' argument, etc.) have usually the form of an entailment.54In philosophy of logic and mathematics, and indeed in logic and mathematics,it is important to keep in mind the image of something being discrete; and the imageof something being continuous. Here are two possible drawings of such ideas:(Image 54)55How would it be something that is discretely continuous?Here are two possible pictures of something that is discretely continuous: (Image55)56A logic based on common images is the logic of a person who anchors her reasoningsfirst of all on her reflections on her concrete, everyday visual experiences.57When one represents a logical case (a logical reasoning, etc.) by means of dots(points) - '' - one more explicitly brings to light the form or structure underlyingsuch case. In other words: if one uses dots to represent subjects/objects and propertiesone more explicitly sees those subjects/objects and properties.58A subject and a predicate - a subject that has a certain property, or attribute, orquality, etc. - could be drawn like this: Here one has however the problem that the dot representing the property is notsymmetrically distributed with respect to the dot representing the subject. A betterimage would thus be this:For example: if one would like to draw the proposition 'The dog is brown' onecould draw this: ('Dog') ('Brownness')If one would like even more explicitly to show that the property of brownnesssymmetrically applies to the dog, one could draw this: ('Brownness') ('Dog') ('Brownness')For reasons of simplicity (and also again of symmetry) one could draw 'Thedog is brown' (that is, 'The dog has the property of brownness') by means of a singlepoint:('Dog with brownness')59If one combines dots and basic logical symbols one has the following images(here one also takes the idea of a thing, that is of 'something', as basic - it meansthat this idea could contain the notions of subject, object and property):Image of one thing:Image of two things: Image of negation of a thing:⌐ (not )Image of conjunction of two things: ( and )Image of disjunction of two things: ( or )Image of conditional - as (necessary) relation that links two things: (if then )Image of biconditional - as (necessary) stricter relation that links two things: ↔ (if and only if then )Image of identity (as isomorphism) ('a = b'): = Another, stricter, image of identity (as numerical sameness) ('a = a'):= =Image of contradiction:( & ⌐ ) ( and not )Image of non contradiction:⌐ ( & ⌐ )Image of equivalence between a thing and its double negation: = ⌐ ⌐ (or: ↔ ⌐ ⌐ )(…)60Let's now observe a possible image corresponding to a famous syllogism: 'Allmen are mortal. Socrates is a man. Thus Socrates is mortal' ('All men are mortal'is the first premise. 'Socrates is a man' is the second premise. 'Thus Socrates ismortal' is the conclusion.): (Image 60)61About quantification in logic: in his theory of syllogism, Aristotle introducesfour basic ideas concerning quantification in logic. These main four ideas are associatedwith what are known as 'A, E, I, O' subject-predicate propositions. Theyare:'A': 'All'. As in: 'All X are Y' (universal affirmative proposition).'E': 'None'. As in: 'No X are Y' (universal negative proposition).'I': 'Some'. As in: 'Some X are Y' (particular affirmative proposition).'O': 'Negative Some'. As in: 'Some X are not Y' (particular negative proposition).One could draw Aristotle's four ideas or 'forms' of quantification as follows:(Image 61)62Quantificational logic: one could derive the idea of 'some' from the idea of'all' - the idea of 'all' contains the idea of 'some'; similarly, one could derive theidea of 'some do not' from the idea of 'none' - the idea of 'none' contains the ideaof 'some do not' (in Aristotle's terms, the idea of 'some' is subaltern to the idea of'all'; and the idea of 'some do not' is subaltern to the idea of 'none').About the main rules of inference in quantificational logic: one can eliminate auniversal quantifier and keep a concrete constant because the universal quantifier atleast contains a concrete constant. One can introduce a universal quantifier becausethe sum of all the concrete constants entails the universal quantifier. Similarly, onecan eliminate an existential quantifier and keep a concrete constant because theexistential quantifier at least contains a concrete constant. One can introduce anexistential quantifier because one concrete constant entails the existential quantifier.(Image 62.)63Let's now come back to the so called propositional logic. If one conceives thetruth-tables in a more visual way one moves, for example, from the negation table(1) to the negation table (3):(1)P Not pT FF T(2) Not T FF T(3)T F Not Not 64Here are all the main truth-tables (for negation, conjunction, negative conjunction,disjunction, negative disjunction, conditional and biconditional) if one conceivesthem in a visual or more visual way:If one substitutes the word 'not' with the symbol '⌐' in the above table one hasthis:65Now an example: is the proposition 'Andrea is a boy and Hilary is a girl' true?Here one has to keep in mind the image corresponding to the (positive) true conjunction:' And 'If it is true that 'Andrea is a boy' one has, to begin, this: If it is true that 'Hilary is a girl' one has, to begin, this: Here one has ' And ', and thus the proposition 'Andrea is a boy and Hilary isa girl' is true.Let's now consider this other example: is the proposition 'Andrea is not a boyand Hilary is not a girl' true?Here one has to keep in mind the image corresponding to the (negative) trueconjunction:'⌐ And ⌐ 'If it is true that 'Andrea is not a boy' one has, to begin, this: ⌐ If it is true that 'Hilary is not a girl' one has, to begin, this: ⌐ Here one actually has '⌐ And ⌐ ', and thus the proposition 'Andrea is not aboy and Hilary is not a girl' is also true.Let's now notice this: if one employs the traditional, non visual, truth-table forthe conjunction (as in 63 (1)) for evaluating the propositions 'Andrea is a boy andHilary is a girl' and 'Andrea is not a boy and Hilary is not a girl', one tends immediatelyto write this (since they are both true, in different cases):'T And T' ('Andrea is a boy and Hilary is a girl')'T And T' ('Andrea is not a boy and Hilary is not a girl')So here one has, for example, that 'Andrea is a boy' and 'Andrea is not a boy'are immediately represented in the very same way: one has here that 'p' is immediatelyrepresented just as T (true) and 'not p' is also immediately represented just asT (true). This is somehow irrespectful even of the traditional truth-table for the negation,for the traditional truth-table for the negation at least formally displays a 'p'(T) distinct from a 'not p' (T). The problem with the traditional truth-table for thenegation (as in figure 63 (1)) is thus this: that it induces one to conflate differentcases together and makes one think that 'not p' is always automatically dependenton 'p' - it somehow induces one to think of 'not p' just functionally or algorithmically.(Of course: if one aims visually to demonstrate that a given logical situationcontains (or does not contain) a contradiction one should in the end come to workwith a drawing where the truth is always represented as truth presence ('') and thefalsity as truth absence ('⌐ ')).66One main idea suggested in the positive or constructive part of these notes isthe following: it seems to be important to draw a variable or its instantiation (hereseen as an empirical 'something' or quidditas) as a point/dot:By representing things in terms of points/dots one uncovers their minimalform.67About the idea of logical atomism - in Russell, etc. - one can observe this: thatan 'atomic' proposition (or 'basic' singular proposition) could even be conceivedas a mere subject proposition - not as a subject-predicate proposition, as the classiclogical atomist would maintain. The case of a mere subject proposition is of coursean extreme or limit case, though it seems to be important for logic to recognizesuch limit or basic case.An example of a subject proposition, a one argument meaningful proposition,is the single name that one finds as entry in a dictionary; or it is the single namethat one sometimes finds displayed as label of a certain thing: a good, a person, etc.(the singular proposition accompanies in this latter case an image (the propositionplays in this case a role similar to that one of a contracted demonstrative sentencesuch as 'This (is this)', or 'I (am I)', etc.)). For instance, the word 'banana' displayedas label on a banana is normally seen or interpreted to held a true value (andthe word 'banana' displayed as label on an orange is instead normally seen or interpretedto hold a false value). (Image 67).68If one sees the truth tables from a genuinely deductive, that is, subtractive ordivisional, perspective one sees, for example, that 'p' is true and 'q' is true because'p and q' is true - here the truth of 'p and q' contains the truth of 'p' and the truthof 'q' (likewise: it is true that I have one parent and another parent because it istrue that I have two parents). On the other hand, one could see the truth tables in acompositional way: one could see, for example, that 'p and q' is true because onehas defined this to be true when 'p' is true and 'q' is true - here the truth of 'p' andthe truth of 'q' do not contain but literally entail the truth of 'p and q'. (Image 68)69According to Leibniz, logic requires a 'lingua generalis'. Here I have insteadsuggested this: that logic requires a 'pictura generalis'.According to Boole, logic requires an algebraic system. Here I have insteadsuggested this: that logic requires a (human) geometrical view.According to Frege, logic requires a 'concept-script' or a 'concept-writing' (a'Begriffsschrift', in the original German). Here I have instead suggested this: thatlogic requires an iconography, that is, in its minimal form, a 'pointography' (or'dottography').70What is truth? What is falsity? The ideas of truth and falsity play a crucial rolein philosophy and logic. Here I will simply say this: truth is seeing (or 'seeing')that something is the case (i.e. that something is a fact); and falsity is seeing (or'seeing') that something is not the case (i.e. that something is not a fact).So for example: i) it is true that 'it rains' presupposes that one can actually seethat it rains; ii) it is true that 'it does not rain' presupposes that one can actually seethat it does not rain; iii) it is false that 'it rains' presupposes that one can actuallysee that it does not rain; iv) it is false that 'it does not rain' presupposes that onecan actually see that it rains.Let's now consider three ideas of truth by three logicians-philosophers: Aristotle,Tarski and Kripke.71In his Metaphysics (1011b25), Aristotle famously claims that "to say of what isthat it is not, or of what is not that it is, is false, while to say of what is that it is,and of what is not that it is not, is true". Now the difficulty with this idea of truth isthat it does not put into focus how we know what is and what is not. A more substantialidea of truth and falsity would thus make explicit this: that truth is seeingthat what is is and falsity is seeing that what is not is not (in other works - e.g.Categories - Aristotle seems to come closer to such idea of truth and falsity).72For Tarski, truth is captured by the following formula: (s) if and only if .Such formula should be understood in this way: all the time in which one has aname s (e.g. 'snow is white') for a sentence S in a Language L, s is true () if andonly if one has a copy of S (e.g. snow is white) in a Metalanguage M. In otherwords: s is true if and only if is the object that 'satisfies' s. Given what we havejust said, Tarski's account of truth is sometimes described as 'relation of satisfaction',a relation at the centre of a 'semantic conception of truth'.A problem with such interpretation of truth is that Tarski in not capturing herethe idea of truth but instead a formal relation of satisfaction. Tarski's reading oftruth is too abstract - it is, at most, an indirect conception of truth. The sentence'snow is white' is true if and only if snow is white just implies an act of disquotation.But one does not know if a sentence is true just by removing from it the signsof quotation (i.e. " "). In order for one to be able to know whether a sentence is trueone has to watch the world referred to by such sentence.73Kripke's account of truth has some points of similarity and difference with respectto Tarski's, though in the end Kripke's idea of truth signals that he would liketo take the opposite direction than that suggested by Tarski. A point of similaritybetween Kripke and Tarski is for example that both their conceptions of truth arebased on the idea of relation of satisfaction (or saturation, etc.). Differences:Kripke's account of truth is linguistic in spirit, at least much more linguistic thanTarski's. Why? Because Kripke's main intention is that of trying to bring the truepredicate, to be applied to a sentence, inside the Language containing the sentence- it is this that signals that Kripke's view of logic moves in the opposite directionthan Tarski's (who, as we have said, opens his view of what is true by exploitingthe idea of an explicit upper order, a Metalanguage).Kripke's linguistic move is of course coherent with his general ambition to reinvigoratea classic philosophy of language. Such move, however, makes truth, thatis the source of truth, once again wholly implicit. Indeed: by bringing truth insidethe Object Language, Kripke makes the source of truth - the knowing subject -completely hidden inside the object (from this it follows that Kripke's idea of the'fixed point' is unnecessarily artificial). Evidence of Kripke's doubts about theplausibility of his linguistic proposal can be found in a claim by Kripke himself.Indeed, towards the end of his Outline of a Theory of Truth, he writes: "The ghostof the Tarski's hierarchy is still with us". (Image 73)74Significant tests for the comprehension of logic are the so called logical paradoxes.One main family among the logical paradoxes is that of the semantic paradoxes.Perhaps the most discussed semantic paradox is the Liar's Paradox, theclearest version of which is the proposition or sentence 'This statement is false'.Now what are the difficulties with such proposition? The statement seems toimply a paradox because if one says that it is false, then it is true.But let us now observe better this proposition.The first point that one should notice here is that one does not know who - whichconscious subject - is concerned with the proposition 'This statement is false'. If onetakes the proposition at its face value it seems that it is the statement itself that is sayingof itself that it is false (let's of course observe here that we do not have 'This statement'is false). A proposition - a statement -, however, cannot decide about its truth or falsity!A proposition does not have experiences, nor think, etc.The second point that one should notice about this case is this: that one cannotsee the content of 'This statement is false'. Indeed: if one takes |This statement isfalse| to be the fact corresponding to 'This statement is false' one should claim that'This statement is false' is true. But now: if one takes |This statement is true| to bethe fact corresponding to 'This statement is true' one should claim that 'Thisstatement is true' is also true. Here one would thus have this impossible fact: |Thisstatement is both false and true|. (Notice here that the problem is not with

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