Entering Categorical Propositions A categorical proposition is a statement which makes a claim about a subject's relation to a predicate. It is called 'categorical' because the claim that it makes is organized into the Categories listed by Aristotle in the Organon and the Metaphysics. A standard form for these propositions has emerged over the long years, and it is: A quantifier + a subject + the verb 'to be' + a qualifier + a predicate. The standard forms for the A, E, I, and O propositions are thus: A: All S are P E: No S are P I: Some S are P O: Some S are not P Because of this standard, and to ease parsing difficulties, certain conventions must be followed when you enter a categorical proposition. The proposition must begin with 'All,' 'No,' or 'Some.' It must have the present tense of the verb 'to be' as its operating verb, e.g.--'is' or 'are.' The subject and the predicate phrases may be several words long, but the first occurrence of 'is' or 'are' is treated as the verb, with all that follows as the predicate. This is a convention easier to learn than describe. Try entering a few propositions for immediate inference, and you'll get the trick. Entering Syllogisms If you're not sure of the form of a syllogism, you should look at the "Read" section of the "Aristotelian" menu. Both to follow this form, and to ease parsing, the syllogism you enter here must follow certain conventions: 1) Each of the 3 propositions must begin with "All," "No," or "Some," and must have "is" or "are" as its verb. The terms may be several words long, ("All fat geese are bona fide mammals" will parse just fine,) but the first occurrence of the verb "to be" will be treated as the operating verb. 2) The subject of the conclusion must occur in SECOND premise, and the predicate must occur in the FIRST premise. For example, though it seems natural, the following puts the two premises in the WRONG order: All Greeks are men. All men are mortal. (Therefore) All Greeks are men. 3) The two occurrences of each term, the major, minor and middle, must be identical. For example, the following cannot be parsed, and thus will returnthe Fallacy of Four Terms: All mortal beings are finite. All men are mortal. (Therefore) All men are finite. Truth Table Generator This program generates standard truth tables for any propositions entered. You may enter from one to five propositions. If you enter more than one, the final proposition is treated as a conclusion to the premises of the earlier propositions, and the resulting table is tested for validity, consistency, contradiction, and tautology. The valid symbols for constructing the propositions are: & for AND V for OR (Inclusive) > for IMPLICATION - for NOT { [ ( and ) ] } for BRACKETS You may have up to four variables, using the letters p, q, r, or s. Brackets must be nested in this order: { [ ( ) ] }. Simply press the return key on a blank line to generate the table, and the return key on the blank first line to exit. Translating Logical Notations The most commonly used systems of notation for the relations in propositional calculus are ones in which the dyadic operators come between the elements they relate. So, the system used by Bertrand Russell, and everyone who sat through high school math, would translate the proposition 'Either p or q, or both' as 'pVq', (The 'V' standing for the Latin 'vel.') 'If p, then q' as 'p>q', and so on. (Because not all the symbols commonly used are available in the IBM ASCII set, we've approximated them as best we can.) But such systems are not elegant, for they rely upon bracketing to avoid ambiguity. The proposition '-p&-q>r' is awkward, for it could mean two different things. Thus, we write it instead with bracketing, '-[p&(-q>r)]'. And this bracketing meshed well with Russell's work in set theory. But alternate systems have been developed to overcome the need for bracketing. These are usually of the form of 'Polish' notations (named for the great work done in Poland before the war.) 'Reverse-Polish' notation, used on Hewlett-Packard calculators, is a form of these. In Polish notation, the operator comes before the elements it relates. So, we would translate the proposition, 'If not p, then not q' as 'CNpNq', where 'C' stands for material implication, and 'N' for negation. From left to right, the 'C' means implication, and needs the next two full elements. The 'N' means negate the next element, and the 'p' means we have 'not p,' and still need a second element for the 'C.' The next 'N' waits for the next element, 'q.' Now we have 'Do 'C' to the 'N' of 'p' and the 'N' of 'q'. The complete list of symbols for standard operators is: 'C' for 'If..., then ...' (Implication) 'A' for 'Either..., or ..., or both.' (Inclusive Or) 'K' for '...and...' (And) 'E' for '...equivalent to...' (Equivalence: '...<>...') 'N' for 'not...' (Negation.) So, the proposition, '-{-pv[-q>(q>p)]}' (always false!) is translated 'NANpCNqCqp'. Try entering a few for translation. You'll find it easy enough to learn, and much easier to use for many purposes, especially computing. Truth Tables Truth tables, along with Truth Trees and Venn Diagrams, are mechanical devices for proving the validity of arguments. There are only four basic relations commonly expressed in truth tables: NEGATION, IMPLICATION, DISJUNCTION and CONJUNCTION. From these, any logical relation may be expressed. In fact, any logical relation may be expressed with negation plus any one functor. But it is standard to use at least these four. Truth tables show the relations expressed by these functors, and help, most of all, to express the curious consequences of material implication. The truth table for each of these looks like this: p q -p p>q pVq p&q ------------------------------------------------------------ T T F T T T T F F T T F F T T F T F F F T T F F Other logical functions, EQUIVALENCE, XOR, NOR, and NAND, often used in Math and Computing can all be expressed in terms of these relations. p q p=q pxq p NOR q p NAND q ------------------------------------------------------------ T T T F F F T F F T F T F T F T F T F F T F T T Here, 'p=q' is equivalent to the conjunct of the implications of p and q, '(p>q)&(q>p),' and equivalent to the disjunct of the conjuncts, '(p&q)V(-p&-q).' NOR (neither/nor) and NAND (neither/and) are even more easily generated, for they are simply the negation of their positive counterparts: (p NOR q)=-(pVq), and (p NAND q)=-(p&q). But the XOR relation is a little harder to express. It stands for the 'EXCLUSIVE OR' as distinct from the 'INCLUSIVE OR,' a distinction we lack in everyday English. It asserts that the two elements it relates are not both true, but at least one of them is. In our notation, that comes out as '-(p&q)&(pVq).' The truth table is used to test not only propositions, but whole arguments for validity. An argument is said to be valid when no truth values for its substitution variables can make the premises all true, and the conclusion false. Expressed another way, a valid argument means that a single statement will always be true when it expresses the conjunct of all the premises implying the conclusion of a valid argument. The statement '[(Premise 1) & (Premise 2) & ... ] > [Conclusion]' is a tautology for a valid argument. One of the implications of this is that inconsistent premises, premises which contradict each other and can never be all true at the same time, can prove anything. Remember that proof of invalidity is only found when the premises are all true, and the conclusion false. Premises which are never all true can never prove the invalidity of an argument. This result is echoed in the propositional calculus, where from inconsistent premises, any conclusion can be found. By the same reasoning, a tautological conclusion can be proved from any premises no matter what, for the conclusion will never be false. Enter a few arguments in the Truth Table Generator to get the feel for this simplest of the mechanical devices for proving validity. The Square of Opposition The Square of Opposition is a graphic device for recollecting the relations among the four standard propositions, A, E, I, & O: Contraries A ┌─────────────────────────────────┐ E │\C s/│ │ \o e/ │ │ \n i/ │ │ \t r/ │ A │ \r o/ │ A l │ \a t/ │ l t │ \d c/ │ t e │ \ i / │ e r │ / │ r A: All S is P n │ d/ \c │ n E: No S is P s │ a/ \t │ s I: Some S is P │ r/ \o │ O: Some S is not P │ t/ \r │ │ n/ \i │ │ o/ \e │ │ C/ \s │ I └─────────────────────────────────┘ O Subcontraries The Contrary relation states that an A proposition and its corresponding E proposition can't both be true at the same time. That all of something has a characteristic must mean that it is false that none of it does. "All stiff- necked mockers read Voltaire" and "No stiff-necked mockers read Voltaire" can't both be true. They can, however, both be false, e.g.--"All humans are children" and "No humans are children" are both false. The Subcontrary relation between I and O is exactly the reverse: I and O can both be true, but can't both be false. "Some cars are tangerine-colored" and "Some cars are not tangerine-colored" are both true. They cannot, however, both be false, for that would leave no color left for the cars to be. The Contradictories combine the Contraries and Subcontraries. A and O are exact opposites: if A is true, O is false; if O is true, A is false. The two can't both be true, and can't both be false. So with E and I: if E is true, I is false; if I is true, E is false. The alterns are the most disputed of the relations. The Subaltern relation is between the universal propositions A and E, and the particular I and O. If it is true that all of some class has a characteristic, then surely some of it must. And if none of the class has the characteristic, then some of it must not. If "All men are mortal," then "Some men are mortal." If "No horses are airborne," then "Some horses are not airborne." If the universal is true, then the particular is also true. The Superaltern states the corresponding truth: If the particular is false, the universal must also be false. If it's false that "Some cats are larger than a house," then it must be false that "All cats are larger than a house." If it's false that "Some men are not mortal," then it must be false that "No men are mortal." All these relations are easier to master when pictured on the square. Try the "Draw the Square" option on this menu. Immediate Inference There are three standard inferences you can make immediately upon knowing a true proposition. These inferences are called CONVERSION, OBVERSION, and CONTRAPOSITION. The CONVERSE of a proposition is formed simply by switching the predicate and subject, and leaving the rest of the proposition (its QUALITY and QUANTITY) alone. The converse of the E proposition "No dogs are reptiles" is "No reptiles are dogs." And this is perfectly true, and a valid inference. But not all conversions are valid. Although it's true that "All sodium phosphates are chemicals," it's not true that "All chemicals are sodium phosphates." (Notice that we can say that "Some chemicals are sodium phosphates." This is called conversion BY LIMITATION.) So too, the conversion of the O is invalid: although "Some dogs are not collies," it doesn't follow that "Some collies are not dogs." So the conversions possible are: A: All S is P converts to A: All P is S (invalid) or I: Some P is S (valid by limitation) E: No S is P converts to E: No P is S I: Some S is P converts to I: Some P is S O: Some S is not P converts to O: Some P is not S (invalid) The OBVERSE of a proposition is formed by replacing the predicate with its complement, and reversing the quality of the proposition. Now, what does that mean? The complement of a term is the class of everything else in the world besides the things referred to by that term. The complement of "cats" is "non-cats," of "hammers" is "non-hammers," of "P" is "non-P." Notice, though, that the complement of "lightfoot lads" is NOT "non-lightfoot lads." The complement means EVERYTHING else, and not just all other "lads." The proper complement of "lightfoot lads" is "non-(lightfoot lads)" or "everything in the world that isn't a lightfoot lad, including lads who aren't lightfoot, and things that aren't lads at all." We reverse the quality of a proposition by making it negative if it was affirmative, or affirmative if it was negative. We must leave its quantity alone, however. If it was universal, it's got to stay universal. If it was particular, it's got to stay particular. So A turns to E, E turns to A, I turns to O, and O turns to I. The possible obversions are thus: A: All S is P obverts to E: No S is non-P E: No S is P obverts to A: All S is non-P I: Some S is P obverts to O: Some S is not non-P O: Some S is not P obverts to I: Some S is non-P and these are all valid. The CONTRAPOSITIVE of a proposition is formed by taking the complement of the subject and the predicate, and then switching the two. If it's true that "All dogs are mammals," then it's going to be true that "All non-mammals are non-dogs." Again, if "Some dogs are not collies," then "Some non-collies are not non-dogs." Another way to look at contraposition is to notice that we can arrive at it by obverting a proposition, converting it, and then obverting it again. For instance: If we know that (1) "All sailors are workers," (A) then we know that (2) "No sailors are non-workers," (obverse of 1) and we know that (3) "No non-workers are sailors" (converse of 2) and we know that (4) "All non-workers are non-sailors" (obverse of 3). And this, number 4, is the contrapositive of number 1. Since conversion lies at the root of contrapositive, the contrapositive is going to fail in certain situations, just as conversion fails in certain situations. The valid contapositives are thus: A: All S is P contraposes to A: All non-P is non-S E: No S is P contraposes to E: No non-P is non-S (invalid) or O: Some non-P is not non-S (valid by limitation) I: Some S is P contraposes to I: Some non-P is non-S (invalid) O: Some S is not P contraposes to O: Some non-P is not non-S Bibliography This program is no replacement for a good introductory text on logic. It is intended rather as a supplement for such a text. Although every year sees the publication of many introductory books on logic, the king of the hill is still Irving M. Copi's INTRODUCTION TO LOGIC, published by MacMillan, and now in its seventh edition. This is one of bestselling philosophy books ever in America. That is not to say it's perfect. But it has these advantages: it combines Aristotelian and Symbolic logic, and EVERYBODY has read it, or used it, or taught it. You can't go far wrong using Copi's notation and Copi's rule sets for symbolic logic. Everyone will know what you mean. As an alternative, you would need to get a text on Aristotelian logic and a text on symbolic logic. Texts I have recently heard good reports about include E.J. Lemmon's SYMBOLIC LOGIC (Hackett: 1978 reprint of 1965 edition.) David Kelley's THE ART OF REASONING (Norton: 1988.) Often recommended books on more advanced topics include Lewis Carroll, SYMBOLIC LOGIC, GAME OF LOGIC (Dover: 1958 reprint of 1897 edition.) Irving Copi and J.A. Gould, READINGS ON LOGIC (Macmillian: 2nd edition, 1972.) William and Martha Kneale, THE DEVELOPMENT OF LOGIC (Clarendon: 1962.) E.J. Lemmon, AN INTRODUCTION TO MODAL LOGIC, (American Philosophical Quarterly Monograph Series, #11:1977.) Fernando Pereira, LOGIC FOR NATURAL LANGUAGE ANALYSIS (SRI International.) G.J. Satty, T.J. Blakeley, J.G. Colbert, COMPUTING AND LOGIC: MATHEMATICS AND LANGUAGE, (Philosophia Verlag Munchen Wien: 1988.) And the classics, difficult and brilliant, include Aristotle's ORGANON (Logical Works), especially the TOPICS and the PRIOR ANALYTICS, (Innumerable editions.) George Boole, AN INVESTIGATION OF THE LAWS OF THOUGHT, (Dover: 1960 reprint of 1854 edition.) Alfred North Whitehead and Bertrand Russell, PRINCIPIA MATHEMATICA, (Cambridge Univ.: 1957 reprint of 1927 edition.) Any good book on logic will include a much more extensive bibliography. Mood, Figure and the Syllogism A proposition can only have one of four forms: it may affirm something about an entire class, it may deny something about an entire class, it may affirm something about part of a class, or it may deny something about part of a class. From the first vowels in the Latin words to affirm and to deny, we label these four forms A, E, I, and O. Those propositions which speak of an entire class are called UNIVERSAL, and those which speak of only part of a class are called PARTICULAR. Affirmative Negative Universal: A: All S is P E: No S is P Particular: I: Some S is P O: Some S is not P A SYLLOGISM is an argument which consists of three of these propositions arranged in a specific order: the major premise, the minor premise, and a conclusion. The MOOD of a syllogism is simply the three vowels naming the form of each of the premises: "AAA" is the mood of all the syllogisms whose three propositions all have the form "All S is P." "EEE" is the mood of all the syllogisms whose three propositions all have the form "No S is P." But this doesn't explain the syllogism completely. There must be some relation between the propositions, or no conclusion can be drawn. In fact, the conclusion must have as its predicate a term which occurred in the premises. This is called the MAJOR TERM, and the premise in which it occurs is called the MAJOR PREMISE. By convention, we always name the major premise first. So, too, the subject of the conclusion must occur in the premises. This is called the MINOR TERM, and occurs in the second, or MINOR, PREMISE. A syllogism, thus, with the mood "AAA" might have the form: All S is the MAJOR TERM All the MINOR TERM is P Therefore, All the MINOR TERM is the MAJOR TERM. For example: All men are mortal. (MAJOR PREMISE) All Greeks are men. (MINOR PREMISE) Therefore, All Greeks are mortal. (CONCLUSION) But now we see yet another aspect of syllogisms. There must be a connection between the two premises. This connecting term is called the MIDDLE TERM, since it is the term through which the argument moves to its conclusion. Notice that the MIDDLE TERM is the term which doesn't occur in the conclusion. Now, since the middle term must occur twice in the premises, there are only four possible arrangements of the middle, major and minor terms. These four arrangements are called the FIGURE of the syllogism. Every mood, for example, "EIO," has four possible arrangements of its terms. These FIGURES look like this, (using EIO as the example mood): Figure 1: No Middle is Major M-->P Some Minor is Middle S-->M ------------------------------------------------------------ Therefore, some Minor is not Major S-->P Figure 2: No Major is Middle P-->M Some Minor is Middle S-->M ------------------------------------------------------------ Therefore, some Minor is not Major S-->P Figure 3: No Middle is Major M-->P Some Middle is Minor M-->S ------------------------------------------------------------ Therefore, some Minor is not Major S-->P Figure 4: No Middle is Major P-->M Some Minor is Middle M-->S ------------------------------------------------------------ Therefore, some Minor is not Major S-->P For all moods, the figures look like this: 1: M-->P 2: P-->M 3: M-->P 4: P-->M S-->M S-->M M-->S M-->S ------- ------- ------- ------- S-->P S-->P S-->P S-->P Every syllogism, thus, consists of a mood and figure, for example, AAA-1, EIO-3, IAI-2, OEO-4, etc. These moods and figures put together make 256 possible syllogisms. Unfortunately, very few of these syllogisms are actually valid arguments. Various rules have been stated by different logicians to express the invalidity of the bad syllogisms. These rules vary from writer to writer, and you should consult a good text to see the rules explained and justified. The rules for testing syllogisms that this program follows are drawn from Irving Copi's INTRODUCTION TO LOGIC, (but see the bibliography section of the Information Module of this program.) The most important of the rules for expressing the invalidity of the bad syllogisms are the RULES OF DISTRIBUTION. To be valid, the middle term of a syllogism must be DISTRIBUTED in one of the premises. DISTRIBUTION is usually said to refer to whether or not a term is making a claim about an entire class. This is not the best description of distribution, but it is the most graphic. In the A proposition "All men are mortal," "men" is DISTRIBUTED, because we are speaking about all men. But "mortal" is NOT DISTRIBUTED, because we have made no claim about everything that is mortal. We haven't said anything about rabbits, horses, dogs, fish, and all the other things that die. Distribution for the four forms of propositions looks like this: A: All S is P ----------------- S is distributed, but P is not. E: No S is P ----------------- S and P are both distributed. I: Some S is P ----------------- Neither S nor P are distributed. O: Some S is not P ----------------- P is distributed, but S is not. To be valid, a syllogism's middle term must be distributed in at least one premise. A syllogism which fails this test is said to commit the FALLACY OF THE UNDISTRIBUTED MIDDLE. An example is the AAA-2 syllogism: A: All men are mortal. 2: P-->M A: All cats are mortal. S-->M ------------------------------------------------------------ Therefore, A: All cats are men. S-->P Further, if a term is distributed in the conclusion, it must have been distributed in the premise which contained it. This surely is simple to see: you can't suddenly start talking about all of a class in the conclusion, if you weren't talking about all of it in the premises. A syllogism which fails this test is said to commit the FALLACY OF ILLICIT PROCESS. If the term distributed in the conclusion is the major term, the syllogism has an ILLICIT MAJOR, and if it is the minor term, the syllogism has an ILLICIT MINOR. An example of an ILLICIT MAJOR is the IEO-1 syllogism: I: Some men are mortal. 1: M-->P E: No rabbits are men. S-->M ------------------------------------------------------------ Therefore, O: Some rabbits are not mortal. S-->P An example of an ILLICIT MINOR is AIA-1: A: All cats are pets. 1: M-->P I: Some mice catchers are cats. S-->M ------------------------------------------------------------ Therefore, A: All mice catchers are pets. S-->P There are various other rules, the description of which varies from author to author. A syllogism shouldn't have two negative premises (FALLACY OF EXCLUSIVE PREMISES), ought not to have a negative premise without a negative conclusion (FALLACY OF NEGATIVE PREMISE), and according to some authors, ought not to draw a particular conclusion from universal premises (THE EXISTENTIAL FALLACY.) Symbolic Logic Symbolic logic is the result of an effort in the last century and a half to develop a new and universal phrasing of logical relations. In part, the need for such a new phrasing, particularly in symbols, came from the disappearance of Latin as a universal language for logic. In larger part, however, it arose from the decline of theistic philosophy. As the western world came less and less to accept God as a ground for metaphysics, the entire concept of CATEGORICAL propositions came under attack. That is to say, logicians felt they could no longer accept the real metaphysical claims of Aristotle's Categories. And, of course, the propositional logic of the Syllogism, which relies upon the metaphysical inter-relations of Aristotle's Categories, could not long survive the loss of its underpinnings. Logicians thus began to evaluate propositions in a more formal and artificial way, a way more influenced by the theory of classes. In part, this direction was thrust upon them by the increasing dilemmas of mathematics. When Western Civilization first rejected the traditional underpinnings of mathematics, it did so in the Enlightenment, in the confident assurance that Science would shortly find the new and real grounding for mathematics, knowledge, and all that previously had been grounded by understanding the world in relation to a God. And the development of a scientific psychology seemed to hold out a bright light for such an endeavor. Unfortunately, and at last, the work of Frege and Husserl showed what had long been suspected, that such a psychological grounding for mathematics was not to be found. And yet the need for some grounding of mathematics was only intensified by the implicit dilemmas shown to exist in mathematics by Gödel and others. Symbolic logic thus developed, pressured by this need to ground mathematics and the mathematical advances being made, a pressure felt keenly by Bertrand Russell, Frege, Boole, and others. These men developed the PROPOSITIONAL CALCULUS, a symbolized way of speaking about the relations between propositions, and a way of speaking about the logical possibilities of Truth and Falsity in a way divorced from considerations of actual being. (So, for instance, modern logicians will speak of possibility "in any possible universe, not just this one which happens to exist.") These relations of the propositional calculus are explained in detail in any good logic text. See the "Read" entries under "Translation" and "Truth Tables" for more information. This program uses only four of the standard operators: MATERIAL IMPLICATION, CONJUNCTION, DISJUNCTION, and NEGATION. The precise symbolization for these relations is difficult to learn, because it varies from author to author. This program suffers from an added burden: in order to be reasonably fast and reasonably transportable, it writes to the screen in text mode, and thus must draw its characters from the ASCII set of 256 characters. This means an approximation. This program uses the "greater than" sign (">") for implication ("if...then..."), a capital "V" for disjunction ("or"), the minus sign ("-") for negation, and the ampersand ("&") for conjunction ("and"). Perhaps future versions of this program will attempt to change the character fonts, or use the fast graphics modes now becoming available. Program Information This program was written by Joseph Bottum, a fellow in philosophy at Boston College, and an instructor of logic at the University of Lowell. It was written with the tools at hand, which is to say Microsoft's QuickBASIC, fleshed out by a few assembly video routines. This is not to decry QuickBASIC. I wrote it as well as I could with no experience, and more expensive tools would not have made my programming better than it is. Any comments, complaints, or suggestions would be most welcome. Please write me at 411 Marrett Road, Lexington, Massachusetts 02173. I began the program in answer to a challenge put to me by a friend and colleague, Kevin Connolly, and wrote the program in a month of work at the beginning of a semester. Should there be future additions, a matter of how much time I can find, I would like to add modules on the inferences of the propositional calculus, and on Venn Diagrams.
