Question

I tried Googling "demi-possibility demi-negation" and got nothing (just "demi-possibility" gave results mostly related to demisexuality). And my analysis of demi-negation didn't seem like it would directly carry over to modal logic as such anyway:

I differentiated between additive, multiplicative, and exponential demi-negation/demiproof, with conjunctions carrying demi-negation/demiproof for nonstandard truth values of ±1/2; disjunctions carrying {1, i, -1, -i}; and conditionals carrying solutions to xx = ±1 (including then 0 for "neither true nor false nor demitrue nor demifalse nor..."). The truth-table for conditionals was rather appealing in light of non-factivity issues in conditional semantics, but the truth-table for disjunction was puzzling at best, unacceptable at worst (a disjunction would be false if one disjunct was true and the other was false, for example).
Offhand, I was inclined to read possibility as involving conditionals which evaluate to 1 even though neither their antecedents nor consequents are true (or false). For example, if the truth table for A → B was (0, 0) then since there is a model(?) of arithmetic where 00 = 1, but there is also a model where the expression is undefined, then in some sense A → B would represent a mere/abstract possibility of truth, here.
At any rate, then, are there logic experiments with terms like √◊ or √OB, etc.? Or does the difference between a connective and a propositional attitude report(?) make such experiments prima facie unwieldy? (Note, however, that ¬A performs a syntactic role akin to various modalities MA.)

Answer 1

The student's question delves into modal logic and the normative aspects of logical reasoning, questioning the construction and interpretation of logical statements, particularly conditionals. It hints at the exploration of non-standard logical connectives or propositional attitudes, pondering the potential for expanding the logical framework to account for broader concepts of truth and possibility.

Step-by-step explanation:

The subject of this complex question touches upon the realms of modal logic, philosophy, and analytical reasoning. At the heart of the question is an exploration into the nuances of logical statements, particularly conditionals, and their relation to notions such as possibility and negation. In a broader sense, the query investigates the structure of logic itself and the normative implications of logical contradictions and possibilities.

Normativity in logic is essential because it suggests that logic prescribes a particular way of reasoning that people ought to follow. As an example, in ethical terms, logic, like ethics, has a normative character because it sets standards or norms for thinking processes. When Lulu states contradictorily that she is both 5 feet and 7 feet tall, she violates a logical standard, specifically the law of noncontradiction, which dictates that contradictory statements cannot both be true.

The connection between various logical forms such as disjunctive syllogism, modus ponens, and modus tollens is central to understanding conditionals and their logical implications. For instance, modus ponens and modus tollens are methods of reasoning which, respectively, affirm or deny a consequence to derive a conclusion. Similarly, logical constructs like conditionals and universal statements are used to articulate necessary and sufficient conditions within a logical framework. Philosophical examples involving the work of Moore and Descartes help illustrate how logical structures like modus tollens can be employed in arguments concerning external world skepticism and the certainty of mathematical truths even in the face of such skepticism.

Ultimately, the question seeks insights into whether there are experiments in logic that use novel terms like '√◊' or '√OB'. While the question acknowledges the syntactic role of logical negations akin to modalities, it appears to propose an exploration into the feasibility of extending traditional logic to accommodate alternative or expanded concepts of truth and possibility.