274:
249:
These three themes are related. The categorical semantics of a logic consists in describing a category of structured categories that is related to the category of theories in that logic by an adjunction, where the two functors in the adjunction give the internal language of a structured category on
137:. One defines a suitable internal language naming relevant constituents of a category, and then applies categorical semantics to turn assertions in a logic over the internal language into corresponding categorical statements. This has been most successful in the theory of
148:
in a topos enables one to reason about the objects and morphisms of a topos as if they were sets and functions. This has been successful in dealing with toposes that have "sets" with properties incompatible with
769:
Fairly accessible introduction, but somewhat dated. The categorical approach to higher-order logics over polymorphic and dependent types was developed largely after this book was published.
791:
A comprehensive monograph written by a computer scientist; it covers both first-order and higher-order logics, and also polymorphic and dependent types. The focus is on
502:
302:
567:
97:
687:
Actes : Du
Congres International Des Mathematiciens Nice 1-10 Septembre 1970. Pub. Sous La Direction Du Comite D'organisation Du Congres
474:
300:
Goguen, Joseph; Mossakowski, Till; de Paiva, Valeria; Rabe, Florian; Schroder, Lutz (2007). "An
Institutional View on Categorical Logic".
906:
815:
762:
741:
541:
516:
488:
460:
413:
392:
238:
201:
in the logic and instances of an appropriate kind of category. A classic example is the correspondence between theories of
784:
434:
373:
847:
901:
16:
This article is about mathematical logic in the context of category theory. For
Aristotle's system of logic, see
197:
In many cases, the categorical semantics of a logic provide a basis for establishing a correspondence between
448:
795:
as universal tool in categorical logic, which is necessary in dealing with polymorphic and dependent types.
444:
230:
210:
39:
198:
896:
218:
214:
162:
316:
217:. Categories arising from theories via term model constructions can usually be characterized up to
558:
498:
259:
158:
779:. Studies in Logic and the Foundations of Mathematics. Vol. 141. North Holland, Elsevier.
311:
47:
43:
834:
803:
752:
531:
506:
478:
403:
637:
576:
242:
702:
8:
527:
641:
580:
655:
594:
279:
222:
145:
122:
114:
35:
757:. Cambridge studies in advanced mathematics. Vol. 7. Cambridge University Press.
668:
625:
607:
562:
483:. Cambridge studies in advanced mathematics. Vol. 7. Cambridge University Press.
811:
780:
758:
737:
690:
673:
612:
537:
512:
484:
456:
430:
409:
388:
369:
273:
723:
729:
698:
663:
645:
602:
584:
206:
178:
89:
58:
constructions. The subject has been recognisable in these terms since around 1970.
54:. The categorical framework provides a rich conceptual background for logical and
774:
792:
182:
150:
134:
118:
31:
874:
823:
799:
719:
470:
368:. Handbook of Logic in Computer Science. Vol. 5. Oxford University Press.
264:
234:
202:
166:
142:
42:. In broad terms, categorical logic represents both syntax and semantics by a
890:
694:
186:
170:
101:
93:
854:
677:
650:
616:
117:
of pre-categorical logic were more clearly understood using the concept of
81:
589:
870:
858:
55:
27:
141:, where the internal language of a topos together with the semantics of
733:
154:
66:
There are three important themes in the categorical approach to logic:
17:
659:
598:
226:
174:
133:
This can be seen as a formalization and generalization of proof by
105:
51:
827:
408:. Handbook of the History of Logic. Vol. 6. North-Holland.
387:(1st ed.). American Mathematical Society. pp. 18β20.
299:
138:
84:
notion of a structure appearing in the particular case where
810:. Vol. 12 (2nd ed.). Springer. pp. 279β361.
250:
the one hand, and the term model of a theory on the other.
108:, is an example of the usefulness of categorical semantics.
96:
notion of a model lacks generality and/or is inconvenient.
533:
Conceptual
Mathematics: A First Introduction to Categories
728:. Lecture Notes in Mathematics. Vol. 611. Springer.
229:
properties of some logics by means of an appropriate
402:
Gabbay, D.M.; Kanamori, A.; Woods, J., eds. (2012).
269:
839:
401:
303:International Journal of Software and Informatics
125:were also best understood using adjoint functors.
888:
497:
630:Proceedings of the National Academy of Sciences
568:Proceedings of the National Academy of Sciences
525:
427:Encyclopedia of Computer Science and Technology
754:Introduction to Higher Order Categorical Logic
480:Introduction to Higher Order Categorical Logic
363:
835:"The History of Categorical Logic 1963β1977"
806:. In Gabbay, D.M.; Guenthner, Franz (eds.).
563:"Functorial Semantics of Algebraic Theories"
536:(2nd ed.). Cambridge University Press.
424:
405:Sets and Extensions in the Twentieth Century
38:. It is also notable for its connections to
750:
718:
685:— (1971). "Quantifiers and Sheaves".
626:"Elementary Theory of the Category of Sets"
469:
73:Categorical logic introduces the notion of
667:
649:
606:
588:
443:
315:
92:. This notion has proven useful when the
833:Marquis, Jean-Pierre; Reyes, Gonzalo E.
832:
425:Kent, Allen; Williams, James G. (1990).
684:
623:
557:
330:
889:
853:
804:"The Development of Categorical Logic"
772:
689:. Gauthier-Villars. pp. 1506β11.
382:
364:Abramsky, Samson; Gabbay, Dov (2001).
342:
453:Category Theory for Computing Science
798:
239:disjunction and existence properties
13:
711:
14:
918:
869:
840:Gabbay, Kanamori & Woods 2012
776:Categorical Logic and Type Theory
30:in which tools and concepts from
751:Lambek, J.; Scott, P.J. (1988).
272:
808:Handbook of Philosophical Logic
455:(2nd ed.). Prentice Hall.
511:. Cambridge University Press.
336:
324:
293:
90:category of sets and functions
75:structure valued in a category
1:
725:First order categorical logic
351:
225:. This has enabled proofs of
907:Theoretical computer science
722:; Reyes, Gonzalo E. (1977).
211:simply typed lambda calculus
40:theoretical computer science
34:are applied to the study of
7:
366:Logic and algebraic methods
253:
215:Cartesian closed categories
61:
10:
923:
875:"Categorical Logic (278x)"
15:
624:— (December 1964).
333:, Quantifiers and Sheaves
161:in terms of objects that
286:
194:Term model constructions
902:Systems of formal logic
260:History of topos theory
159:untyped lambda calculus
100:'s modeling of various
651:10.1073/pnas.52.6.1506
383:Aluffi, Paolo (2009).
113:It was found that the
828:John Bell's homepage.
773:Jacobs, Bart (1999).
590:10.1073/pnas.50.5.869
153:. A prime example is
70:Categorical semantics
508:Sets for Mathematics
243:intuitionistic logic
237:gave a proof of the
859:"Categorical Logic"
843:. pp. 689β800.
642:1964PNAS...52.1506L
581:1963PNAS...50..869L
231:categorical algebra
80:with the classical
822:Version available
734:10.1007/BFb0066201
385:Algebra: Chapter 0
280:Mathematics portal
223:universal property
146:higher-order logic
130:Internal languages
104:theories, such as
36:mathematical logic
897:Categorical logic
817:978-1-4020-3091-8
764:978-0-521-35653-4
743:978-3-540-08439-6
561:(November 1963).
543:978-1-139-64396-2
518:978-0-521-01060-3
490:978-0-521-35653-4
462:978-0-13-323809-9
429:. Marcel Dekker.
415:978-0-444-51621-3
394:978-1-4704-1168-8
173:βHyland model of
169:. Another is the
26:is the branch of
24:Categorical logic
914:
882:
866:
857:(12 July 2024).
844:
821:
790:
768:
747:
706:
681:
671:
653:
620:
610:
592:
547:
522:
494:
466:
440:
419:
398:
379:
345:
340:
334:
328:
322:
321:
319:
297:
282:
277:
276:
233:. For instance,
227:meta-theoretical
207:equational logic
179:full subcategory
922:
921:
917:
916:
915:
913:
912:
911:
887:
886:
885:
845:
818:
800:Bell, John Lane
793:fibred category
787:
765:
744:
720:Makkai, Michael
714:
712:Further reading
709:
544:
526:Lawvere, F.W.;
519:
491:
463:
437:
416:
395:
376:
354:
349:
348:
341:
337:
329:
325:
317:10.1.1.126.2361
298:
294:
289:
278:
271:
256:
183:effective topos
177:by an internal
165:onto their own
151:classical logic
135:diagram chasing
121:, and that the
119:adjoint functor
82:model theoretic
64:
32:category theory
21:
12:
11:
5:
920:
910:
909:
904:
899:
884:
883:
867:
851:
846:A preliminary
830:
816:
796:
785:
770:
763:
748:
742:
715:
713:
710:
708:
707:
682:
636:(6): 1506β11.
621:
575:(5): 869β872.
552:Seminal papers
549:
548:
542:
528:Schanuel, S.H.
523:
517:
495:
489:
467:
461:
441:
435:
421:
420:
414:
399:
393:
380:
374:
360:
359:
358:
353:
350:
347:
346:
335:
323:
310:(1): 129β152.
291:
290:
288:
285:
284:
283:
268:
267:
265:Coherent topos
262:
255:
252:
247:
246:
221:by a suitable
195:
191:
190:
167:function space
143:intuitionistic
131:
127:
126:
110:
109:
71:
63:
60:
56:type-theoretic
48:interpretation
9:
6:
4:
3:
2:
919:
908:
905:
903:
900:
898:
895:
894:
892:
880:
879:lecture notes
876:
872:
868:
864:
863:lecture notes
860:
856:
855:Awodey, Steve
852:
849:
842:
841:
836:
831:
829:
825:
819:
813:
809:
805:
801:
797:
794:
788:
786:0-444-50170-3
782:
778:
777:
771:
766:
760:
756:
755:
749:
745:
739:
735:
731:
727:
726:
721:
717:
716:
704:
700:
696:
692:
688:
683:
679:
675:
670:
665:
661:
657:
652:
647:
643:
639:
635:
631:
627:
622:
618:
614:
609:
604:
600:
596:
591:
586:
582:
578:
574:
570:
569:
564:
560:
559:Lawvere, F.W.
556:
555:
554:
553:
545:
539:
535:
534:
529:
524:
520:
514:
510:
509:
504:
503:Rosebrugh, R.
500:
499:Lawvere, F.W.
496:
492:
486:
482:
481:
476:
472:
468:
464:
458:
454:
450:
446:
442:
438:
436:0-8247-2272-8
432:
428:
423:
422:
417:
411:
407:
406:
400:
396:
390:
386:
381:
377:
375:0-19-853781-6
371:
367:
362:
361:
356:
355:
344:
339:
332:
327:
318:
313:
309:
305:
304:
296:
292:
281:
275:
270:
266:
263:
261:
258:
257:
251:
244:
240:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
196:
193:
192:
188:
187:Martin Hyland
184:
180:
176:
172:
168:
164:
160:
156:
152:
147:
144:
140:
136:
132:
129:
128:
124:
120:
116:
112:
111:
107:
103:
102:impredicative
99:
95:
94:set-theoretic
91:
87:
83:
79:
76:
72:
69:
68:
67:
59:
57:
53:
49:
45:
41:
37:
33:
29:
25:
19:
878:
871:Lurie, Jacob
862:
838:
807:
775:
753:
724:
686:
633:
629:
572:
566:
551:
550:
532:
507:
479:
452:
426:
404:
384:
365:
338:
331:Lawvere 1971
326:
307:
301:
295:
248:
157:'s model of
98:R.A.G. Seely
85:
77:
74:
65:
23:
22:
475:Scott, P.J.
343:Aluffi 2009
219:equivalence
123:quantifiers
115:connectives
28:mathematics
891:Categories
703:0261.18010
471:Lambek, J.
352:References
155:Dana Scott
18:term logic
695:217031451
449:Wells, C.
312:CiteSeerX
245:this way.
46:, and an
802:(2001).
678:16591243
617:16591125
530:(2009).
505:(2003).
477:(1988).
451:(1996).
445:Barr, M.
254:See also
199:theories
175:system F
106:System F
62:Overview
44:category
848:version
638:Bibcode
577:Bibcode
181:of the
163:retract
139:toposes
88:is the
52:functor
824:online
814:
783:
761:
740:
701:
693:
676:
669:300477
666:
658:
615:
608:221940
605:
597:
540:
515:
487:
459:
433:
412:
391:
372:
314:
660:72513
656:JSTOR
599:71935
595:JSTOR
357:Books
287:Notes
235:Freyd
209:over
171:Moggi
50:by a
812:ISBN
781:ISBN
759:ISBN
738:ISBN
691:OCLC
674:PMID
613:PMID
538:ISBN
513:ISBN
485:ISBN
457:ISBN
431:ISBN
410:ISBN
389:ISBN
370:ISBN
213:and
826:at
730:doi
699:Zbl
664:PMC
646:doi
603:PMC
585:doi
241:of
185:of
893::
877:.
873:.
861:.
837:.
736:.
697:.
672:.
662:.
654:.
644:.
634:52
632:.
628:.
611:.
601:.
593:.
583:.
573:50
571:.
565:.
501:;
473:;
447:;
306:.
203:Ξ²Ξ·
881:.
865:.
850:.
820:.
789:.
767:.
746:.
732::
705:.
680:.
648::
640::
619:.
587::
579::
546:.
521:.
493:.
465:.
439:.
418:.
397:.
378:.
320:.
308:1
205:-
189:.
86:C
78:C
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.