Abstract
In this chapter, we discuss three flavours of non-classical logics. Intuitionistic logic is a weakened classical logic, whereas linear logic is stronger than classical logic. Linear temporal logic can express the future of paths one can take. We focus on the propositional fragment of these logics as we have seen the difficulty that comes with quantifiers. Our discussion of intuitionistic logic and linear logic is in the form of syntax and proof theory, and we only give an informal discussion of their semantics. On the other hand, linear temporal logic is widely used in model checking, which is different from proving valid formulae. We introduce the last logic by its syntax and semantics, from which we will change gear to automata theory of the next chapter.
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Notes
- 1.
Also called (pure) existence proof.
- 2.
e is a constant called Euler’s number, which is an irrational number that is approximately equal to 2.71828. \(\pi \) is also an irrational number, which is approximately equal to 3.14159.
- 3.
In fact, the law of contraposition \((\lnot B \rightarrow \lnot A) \rightarrow (A \rightarrow B)\) and Peirce’s law \(((A \rightarrow B) \rightarrow A) \rightarrow A\) do not hold in intuitionism, either. Accepting any of the four formulae as a valid formula makes the logic classical.
- 4.
Unsurprisingly, he was a student of Brouwer.
- 5.
An operation \(\circ \) is said to be idempotent over a set S of items if \(\forall x\in S. \ x \circ x = x\).
- 6.
We use [] to denote an empty list.
- 7.
Quantifiers are not considered modalities as they quantify over variables and are not logical connectives.
- 8.
Sometimes called a word or an \(\omega \)-word, hence the notation w.
- 9.
Do not confuse intension, which is the opposite to extension, with intention, which is about the will to act in a certain way.
- 10.
- 11.
Without the context of which logic it refers to, SAT usually means satisfiability checking for classical propositional logic. In this paragraph, we just assume that we are talking about the three problems for the same logic.
- 12.
US$475 million in 1994 is roughly US$838 million in 2020.
- 13.
Assuming that the model is correct, the model checking algorithm is correct, and their implementation is correct, etc. There will always be assumptions such as these, and we will not descend into this rabbit hole in this book.
- 14.
See John Harrison’s account of formal verification at Intel here: https://shemesh.larc.nasa.gov/NFM2010/talks/harrison.pdf.
- 15.
The original algorithm is invented by the Dutch computer scientist Edsger Dijsktra, and he used letters P and V for these two operations; they stand for Dutch words “passering” (passing), and “vrijgave” (release).
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Hou, Z. (2021). Non-classical Logics. In: Fundamentals of Logic and Computation. Texts in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-87882-5_3
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