*Alexander Broadie*

- Published in print:
- 1993
- Published Online:
- October 2011
- ISBN:
- 9780198240266
- eISBN:
- 9780191680137
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198240266.003.0007
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, History of Philosophy

This chapter considers the inferential power of categorical propositions where the structure of those propositions is taken into account. At the heart of the medieval theory of valid inference for ...
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This chapter considers the inferential power of categorical propositions where the structure of those propositions is taken into account. At the heart of the medieval theory of valid inference for analysed propositions lies an account of three ways in which two categorical propositions with the same categorematic terms may be related to each other. They may be related by opposition, equipollence, or conversion. Propositions related by opposition or equipollence have the same categorematic terms in the same order; where the relation is that of conversion the order is not the same. Equipollence is a relation of equivalence between two propositions structurally related to each other in a certain quite specific way. Opposition is not a relation of equivalence. Perhaps the best-known notion of medieval logic is that of the square of opposition, a square that medieval logicians liked and of which they drew a considerable variety.Less

This chapter considers the inferential power of categorical propositions where the structure of those propositions is taken into account. At the heart of the medieval theory of valid inference for analysed propositions lies an account of three ways in which two categorical propositions with the same categorematic terms may be related to each other. They may be related by opposition, equipollence, or conversion. Propositions related by opposition or equipollence have the same categorematic terms in the same order; where the relation is that of conversion the order is not the same. Equipollence is a relation of equivalence between two propositions structurally related to each other in a certain quite specific way. Opposition is not a relation of equivalence. Perhaps the best-known notion of medieval logic is that of the square of opposition, a square that medieval logicians liked and of which they drew a considerable variety.

*John Buridan*

- Published in print:
- 2014
- Published Online:
- May 2015
- ISBN:
- 9780823257188
- eISBN:
- 9780823261499
- Item type:
- chapter

- Publisher:
- Fordham University Press
- DOI:
- 10.5422/fordham/9780823257188.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Buridan first describes the causes of the truth and falsity of propositions, before defining consequence and distinguishing formal from material consequence. He describes the ampliation of the ...
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Buridan first describes the causes of the truth and falsity of propositions, before defining consequence and distinguishing formal from material consequence. He describes the ampliation of the supposition of terms in certain propositions, prior to drawing seventeen Conclusions. The first is that any proposition follows from an impossible one, and a necessary proposition from any other. He shows that what follows from the consequent follows from the antecedent, and what does not follow from the antecedent does not follow from the consequent. Consequences are necessarily truth-preserving, and necessary premises can be suppressed. The rules of distribution are set out and the rules of conversion. Finally, he describes in detail how universal, particular, affirmative and negative propositions are related.Less

Buridan first describes the causes of the truth and falsity of propositions, before defining consequence and distinguishing formal from material consequence. He describes the ampliation of the supposition of terms in certain propositions, prior to drawing seventeen Conclusions. The first is that any proposition follows from an impossible one, and a necessary proposition from any other. He shows that what follows from the consequent follows from the antecedent, and what does not follow from the antecedent does not follow from the consequent. Consequences are necessarily truth-preserving, and necessary premises can be suppressed. The rules of distribution are set out and the rules of conversion. Finally, he describes in detail how universal, particular, affirmative and negative propositions are related.

*Terence Parsons*

- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199688845
- eISBN:
- 9780191768002
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199688845.003.0002
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Ancient Philosophy

In Prior Analytics Aristotle proves the conversion principles, using three logical techniques: reductio (indirect derivation), exposition (a kind of existential instantiation: given ‘Some S is a P’, ...
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In Prior Analytics Aristotle proves the conversion principles, using three logical techniques: reductio (indirect derivation), exposition (a kind of existential instantiation: given ‘Some S is a P’, introduce a previously unused name m, and write ‘m is an S’ and ‘m is a P’), and expository syllogism (given ‘m is a P’ and ‘m is an S’ infer ‘Some S is a P’). He assumes four basic forms of syllogism, and uses them together with reductio, exposition, and expository syllogisms to prove the remaining forms. The four basic forms can be proved too, though this was unknown to him or to medieval logicians. Monotonicity properties of the determiners ‘every’, ‘no’, and ‘some’ are explained; Aristotle’s four basic argument forms allow easy proofs of their monotonicity properties. By the 13th century a memorizable verse was developed that encodes Aristotle’s validation of all syllogistic forms from the basic four.Less

In *Prior Analytics* Aristotle proves the conversion principles, using three logical techniques: reductio (indirect derivation), exposition (a kind of existential instantiation: given ‘*Some S is a P*’, introduce a previously unused name *m*, and write ‘*m is an S*’ and ‘*m is a P*’), and expository syllogism (given ‘*m is a P*’ and ‘*m is an S*’ infer ‘*Some S is a P*’). He assumes four basic forms of syllogism, and uses them together with reductio, exposition, and expository syllogisms to prove the remaining forms. The four basic forms can be proved too, though this was unknown to him or to medieval logicians. Monotonicity properties of the determiners ‘*every*’, ‘*no*’, and ‘*some*’ are explained; Aristotle’s four basic argument forms allow easy proofs of their monotonicity properties. By the 13^{th} century a memorizable verse was developed that encodes Aristotle’s validation of all syllogistic forms from the basic four.

*Terence Parsons*

- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199688845
- eISBN:
- 9780191768002
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199688845.003.0003
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, Ancient Philosophy

Medieval logicians expanded Aristotle’s notation; some rules naturally come along with the expansions. Predicates are quantified, as in ‘No donkey is every animal’, and negations are sprinkled ...
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Medieval logicians expanded Aristotle’s notation; some rules naturally come along with the expansions. Predicates are quantified, as in ‘No donkey is every animal’, and negations are sprinkled throughout sentences, so quantifier equipollences are introduced, such as ‘not some P’ is equivalent to ‘every P not’. One can now say that prefixing ‘not’ to any proposition produces its contradictory; this makes reductio proofs widely applicable. When singular terms occur, new rules are needed to let them permute with each other, with negations, and with denoting phrases. Conversion “by contraposition” is discussed; an example is converting ‘Every P is a Q’ to and from ‘Every non-Q is a non-P’; this is valid except for counterexamples involving empty terms. (‘Every chimera is an animal’ is false, but ‘Every non-animal is a non-chimera’ is true.) A set of rules of inference are given which are complete for the notation developed up to this point.Less

Medieval logicians expanded Aristotle’s notation; some rules naturally come along with the expansions. Predicates are quantified, as in ‘*No donkey is every animal*’, and negations are sprinkled throughout sentences, so quantifier equipollences are introduced, such as ‘*not some P*’ is equivalent to ‘*every P not*’. One can now say that prefixing ‘*not*’ to any proposition produces its contradictory; this makes reductio proofs widely applicable. When singular terms occur, new rules are needed to let them permute with each other, with negations, and with denoting phrases. Conversion “by contraposition” is discussed; an example is converting ‘*Every P is a Q*’ to and from ‘*Every non-Q is a non-P*’; this is valid except for counterexamples involving empty terms. (‘*Every chimera is an animal*’ is false, but ‘*Every non-animal is a non-chimera*’ is true.) A set of rules of inference are given which are complete for the notation developed up to this point.