Philosophy Dictionary of ArgumentsHome
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| Conjunction: In logic, a conjunction is an operator that takes two propositions as input and produces a single proposition as output. The output proposition is true if and only if both of the input propositions are true. The symbol for conjunction is usually "∧" (or "and" in natural language). See also Disjunction._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
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Aristotle on Conjunction - Dictionary of Arguments
Geach I 16/17 Conjunction/Aristotle/Geach: In his early work, Aristotle considered conjunctions to be either true or false, but he later changed his mind. He considers what we might call a “merging of predicates”: e.g., “is white” and “is a man” to “is a white man.” This works. But: Example: “is good” and “is a cobbler” cannot be combined in this way to form “is a good cobbler.” Example: A compound noun such as “morse” (man and horse) cannot appear as the subject of a predication. This is because it does not designate anything, even or especially because it is required that “man horse” should combine all corresponding predicates. I 18 Geach: 1. That is actually questionable, but in any case it is not a conjunction if one attributes predicates to such a subject: Example: “S” is a “lawyer politician.” Then neither: “Every S is a scoundrel” nor “Some S are honest” can be considered a conjunction of predications that would be obtained by first substituting “lawyer” and then ‘politician’ for “S.” GeachVsAristotle: So his argument against the composite subject is irrelevant. 2. If the composite subject “morse” is regarded as the intended equivalent of a combination of two predications, then the result of the juxtaposition (antiphasis?) is not necessarily true and false. This shows that “morse” cannot be understood as a predication. Example: “Some morse is white” will be true if some man and some horse are white. “No morse is white” is true if no man and no horse are white. Problem: if no man is white but some horse is white, then we get true for one side and false for the other: “Some horse is white” “No horse is white”. (nsert, unity). So “Some horse is white” is not a well-formed sentence. (s) Unambiguity could be achieved if, at the same time, no man and no horse, or all men and all horses share the predicate in question. Example: In the case of four-leggedness, again, no sentence could be formed. Conjunction/GeachVsAristotle: However, this does not show that the conjunction Example: “Some men are white and some horses are white” would not be a sentence! Conjunction /Aristotle: (late, Sophistici elenchi): denies that conjunctions can be w/f. It would be the “ruin of discourse” to answer “Is it the case that p and q and r...?” with ‘yes’ or “no.” Even if it seems harmless because all terms might be true or false. GeachVsAristotle: Modern logic has no problem with this at all: the conjunction is true if all terms are true, otherwise false. Only a confused person would conclude from “no” that all terms must be false. Example Aristotle: “Are Koriscus and Kallias at home?” as if it were the same as “Is it the case that p and q?” I 19 GeachVsAristotle: but that is not exactly the same sentence as “Koriskus is at home and Kallias is at home.” (Everyone at home?). ((s) Are the subjects “summarized” or the predicates? It is impossible to separate them here.) Geach: “d and b are P's” or “d is (a) P and b is (a) P” (as Aristotle thought). But there are cases where the attribution in the plural becomes illegitimate, even though it is permissible in the singular. Example (see above). Parmenides/The Third Man Argument: Solution: if we concede that the predicate “large” can be said of itself and at the same time of many large things. However, this presupposes that we do not allow this form of “great” in the plural (ta polla megala/tanta megala) to be assumed to apply to itself at the same time. “Analogy”/Middle Ages/GeachVs: from the example “God is wise and Plato is wise,” one should not be able to conclude: “God and Plato are two wise men.” (sapiens/sapientes, plural, nominalization of the predicate). Structure: if d is P and b is P and a is a class of Ps, then we cannot conclude that “P” can be stated (predicated) in the plural by a class that has exactly a and b or only d as elements. (predication/singular/plural). Whether such a set is permissible at all depends on the accepted set theory. Aristotle: uses a devilish example to show that the plural form of the predicate cannot be attributed if it is in the singular form. Example Aristotle: Two animals “d” and “b” are blind. Is this equivalent to: “d is blind and b is blind”? Aristotle: (Sophistici elenchi): Even that is not permissible! (GeachVsAristotle). 1. “Blind” means: by nature capable of seeing, but without the ability to see. 2. If d and b are by nature capable of seeing, then they either have the ability to do so or they do not. I 20 3. If d and b are by nature sighted but do not have the ability, they are blind. 4. Therefore, if d and b are by nature sighted, either both have the ability or both do not. 5. If d has the ability and b is blind, then d and b are by nature sighted. 6. Therefore, if d has the ability and b is blind, either both have the ability or both are blind. Which is absurd. (Aristotle). Solution/Aristotle: Plural questions such as “Are they by nature sighted?” or “Are they blind?” should be banned. GeachVsAristotle: that is drastic and unnecessary. “Having the ability to see” can be constructed grammatically in two different ways: a) that no one in a class has the ability b) that not everyone in a class has the ability. Geach: To do step 3 correctly, it must say: every element of the class does not have the ability. I 25 Conjunction/Aristotle/Geach: A. Own proof of his metatheorem: Premises: should be “A is white” (of a valid syllogism) Conclusion: “B is large.” Then the premises of a presumed syllogism cannot be true: (14) If A is not white, then B is large. The syllogism itself should be represented by: (15) If A is white, then B is large. (15) leads to the contraposition (16) If B is not large, then A is not white. Then (1) and (14) lead to the conclusion in what Aristotle calls the “hypothetical syllogism”: (17) If B is not large, then B is large. Aristotle calls this “absurd.” VsAristotle: some authors: the form “If not p then p” does not have to be absurd! (Geach pro). Example: It can be used to achieve “p” itself (?). In geometry: “If AB and CD are not parallel, then they are parallel, so they are parallel.”(?). VsVs: but in this case, that overlooks the fact that “B is large” is not a proposition (statement, sentence) in the sense of a traditional syllogism: such as “Every X is Y.” GeachVsAristotle. He claims to have shown here that if we have two valid syllogistic schemata with a conjunction of the premises “p and q” and “not p and not q,” then if both yield the conclusion “Every X is Y,” we should be obliged to recognize the general validity of the formula: “If not every X is Y, then every X is Y,” and that is indeed absurd. ((s) Difference from above: If B is not large, then B is large. Does not contain “every” or “not every”: difference between contrary and contradictory). GeachVsAristotle: the error lies in his incorrect understanding of contradiction. He is right in saying that a conjunction is a proposition. Punchline: if “A is white” is supposed to represent a premise conjunction “p and q,” then its negation, “A is not white,” cannot represent “not p and not q.” Rather, the correct negation is “not both p and q.” Negation: A conjunction: ~(p u q) = (plq) not (~p u ~q). Not both, not “neither.” Example (s) “A is white”: Negation: “A is not white” not “A is not white and also not an object.” I 26 GeachVsAristotle: because he only implicitly assumed that conjunctions are sentences (Geach pro), he did not properly consider the question of what the contradiction of a conjunction actually is. (see below) In his later work, Aristotle explicitly recognized conjunctions as propositions with the ingenious invention of “A” and “B” etc. as sentence letters. (Sentence variables). >Modal logic/Geach, >Facts/Geach._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Gea I P.T. Geach Logic Matters Oxford 1972 |
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> Counter arguments against Aristotle
> Counter arguments in relation to Conjunction
