Non-Classical Logic

Summary

An undergraduate course in non-classical logic, with an emphasis on modal logic.

Prerequisite

This course requires having taken a previous course in mathematical or symbolic logic.¹

Part 1 - Foundations

In this part of the course, we will study the foundations of modal logic. The perspective will be mainly proof-theoretic, and we will not spend much time on applications.

Section 1A - Introduction and Review of Classical Logic

Required

• Priest (2008, chaps. 0–1)

Supplementary

• Priest (2008, chap. 12)

Section 1B - Basic Modal Logic

Required

• Priest (2008, chap. 2)

Supplementary

• Benthem (2010, chaps. 1–5)

Section 1C - Normal Modal Logic

Required

• Priest (2008, chap. 3)
• Lewis (1986)
• R. C. Stalnaker (1976)

Supplementary

• Benthem (2010, chaps. 7–10)

Part 2 - Knowledge and Proof

We will look at two of the most accessible applications of modal logic: epistemic logic and intuitionist logic.

Section 2A - Epistemic Logic

Required

• Benthem (2010, chap. 13)
• R. Stalnaker (2006)

Supplementary

• Girle (2010, chap. 12)

Section 2B - Intuitionist Logic

Required

• Priest (2008, chap. 3)

Supplementary

Part 3 - Indeterminacy and Vagueness

Sometimes P is neither true nor false; it is indeterminate or vague whether it is true. This is where many-valued logics enter the picture.

Section 3A - Introduction to Truth Value Gaps

Required

• Priest (2008, chaps. 7–8)
• Sainsbury (2009, chap. 3)

Section 3B - Indeterminacy and Fuzziness

Required

• Priest (2008, chap. 11)

Part 4 - Contradictions and Inconsistency

Can a sentence be both true and false? The dialetheist says so, and gives you a logic to make sense of it.

Section 4A - Introduction to Truth Value Gluts

Required

• Priest (2008, chap. 8)
• Priest (1997)

Supplementary

• Weber (2022)

Section 4B - Dialetheism

Required

• Priest (2006)

Supplementary

• Weber (2021)

Part 5 - Additional Topics

This part of the course varies each year. I typically focus on one (but not all!) of the following subjects.

Temporal Logic

Required

• Priest (2008, chap. 3)
• Benthem (2010, chap. 18)

Supplementary

• Girle (2010, chap. 9)

Quantified Modal Logic

Required

• Priest (2006, chaps. 12–15)

Higher Order Logic and Type Theory

Higher-order logic can be hard-going. Type theory can also be tough. But I know you love a challenge! I give a very light introduction to higher-order logic.

Required

• Bacon (2023)
• Nederpelt and Geuvers (2014)
• Bell (2022)

Supplementary

• Skiba (2021)

References

Bacon, Andrew. 2023. A Philosophical Introduction to Higher-Order Logics. Taylor & Francis.

Bell, John L. 2022. Higher-Order Logic and Type Theory. Elements in Philosophy and Logic. Cambridge: Cambridge University Press. https://doi.org/10.1017/9781108981804.

Benthem, Johan van. 2010. Modal Logic for Open Minds -. Stanford, CA, USA: Center for the Study of Language; Inf.

Girle, Rod. 2010. Modal Logics and Philosophy: Second Edition. Mcgill-Queen’s University Press.

Lewis, David. 1986. On the Plurality of Worlds. Wiley-Blackwell.

Nederpelt, Rob, and Herman Geuvers. 2014. Type Theory and Formal Proof: An Introduction. Cambridge University Press.

Priest, Graham. 1997. “Sylvan’s Box: A Short Story and Ten Morals.” Notre Dame Journal of Formal Logic 38 (4): 573–82. https://doi.org/10.1305/ndjfl/1039540770.

———. 2006. In Contradiction: A Study of the Transconsistent. Second Edition, Second Edition. Oxford, New York: Oxford University Press.

———. 2008. An Introduction to Non-Classical Logic: From If to Is. 2nd ed. Cambridge Introductions to Philosophy. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511801174.

Sainsbury, R. M. 2009. Paradoxes. Cambridge University Press.

Skiba, Lukas. 2021. “Higher-Order Metaphysics.” Philosophy Compass 16 (10): e12756. https://doi.org/10.1111/phc3.12756.

Stalnaker, Robert. 2006. “On Logics of Knowledge and Belief.” Philosophical Studies 128 (1): 169–99. https://doi.org/10.1007/s11098-005-4062-y.

Stalnaker, Robert C. 1976. “Possible Worlds.” Noûs 10 (1): 65–75. https://doi.org/10.2307/2214477.

Weber, Zach. 2021. Paradoxes and Inconsistent Mathematics. Cambridge University Press.

———. 2022. “Paraconsistency in Mathematics.” Elements in the Philosophy of Mathematics, July. https://doi.org/10.1017/9781108993968.

Footnotes

1. Mathematical logic is not the logic used in the vast majority of math and computer sciences. (You can be a great math student while knowing nothing of substance about mathematical logic.) This means that the ability to prove mathematical theorems is not sufficient for taking this course. I do not allow students that have not taken a formal course in mathematical logic to take this course. (No, it is not enough to self-study the prerequisite material; no, I do not administer placement tests; no, your high performance in other math and computer science courses is not enough to persuade me to make an exception.)