How would I translate the following statement into higher order logic? Or what book would you recommend to teach myself higher order logic?

Question

"For every liar Sentence, there exists some person for whom the sentence is either self-referential or (purely) negative."

I am a behavior analyst with an undergraduate degree in philosophy and some training in first-order logic. I am interested in learning more about logic both (a) to apply it to behavior analysis in order to further advance the field of behavior analysis and (b) to apply behavior analysis to logic, namely, in explaining the behavior of logicians. One way I hope to do this is by attempting a behavior-analytic approach to the Liar Sentence/Paradox, but I want to inform it with the language of formal logic.

I wish to demonstrate that the problem with the liar Sentence/Paradox is not merely one of syntax or semantics, but also a problem of pragmatics, e.g., the behavior of logicians in their choosing to stipulate that the Liar Sentence be both self-referential and negative thus leading to the Liar Paradox, but which seems like an arbitrary stipulation. It seems that as long as a person does not make that stipulation, no paradox exists. Just because that stipulation can be made, and the paradox thus be constructed, does not mean it should be.

Informally, in English, the Liar Sentence is "This sentence is false," from which the Liar Paradox is generated. In Tarski's approach to the Liar Sentence/Paradox, he took the route of arguing that the concept of truth is ill-defined, specifically ill-defined within the object language in which the Liar Sentence is constructed, and that one must venture outside the object language into a meta-language in order to evaluate the truth of the Liar Sentence. Others have pursued similar routes of approaching the Liar Sentence/Paradox by questioning the nature of truth, such as arguing that the truth-value of the sentence is indeterminate. But, of course, there are problems with both of these lines of reasoning. To my understanding, both lines of reasoning rely on a truth-gap.

I want to take the opposite route of arguing that it is not the concept of truth that is necessarily ill-defined or the truth-value of the Liar sentence that is necessarily indeterminate, but rather the concepts of "This" and “negation.” If the word "This," which traditionally is stipulated as referring to the Liar Sentence itself, is ambiguous, that is, could refer to the Liar Sentence itself or to some other sentence, then it's debatable whether the Liar Sentence is self-referential. And if the concept of negation is impossible without the possibility of assertion, then it’s debatable whether the liar sentence is even negative.

Similar sentences to the Liar sentence can be constructed without invoking the concept of truth. For example:

This sentence does not mean what you think it means.

This sentence does not refer to itself.

This sentence does not refer.

These sentences demonstrate that it is language itself—and not the concept of truth—that is indeterminate and that language only becomes determinate in the act of reading and I want to argue that this is true no less for formal languages than for informal languages. What I want to argue is that the bearers of truth are not the usual candidates of sentences and propositions but rather pragmatics (behaviors) and people.

Thus, whichever side a person comes down on--whether the Liar Sentence is both self-referential and purely negative or either self-referential or purely negative--is a matter of choice, i.e., behavior. And thus, the Liar Sentence/Paradox is arguably a behavioral problem. And I believe the above statement is key in being able to demonstrate this. For people who insist the liar sentence be both self-referential and purely negative, the paradox persists. For those who don’t, there is no paradox.

The above, of course, is all informal. Out of respect for the literature on the Liar Paradox, I want to turn the above into a formal argument, starting with translating the above statement into formal logic.

My thinking is influenced by C. S. Peirce. While Peirce does not make the above argument (at least, not that I know of), he does insist on the importance of pragmatics and the necessity of people ("interpretants") in any theory of language and truth. Tarski's T Convention of truth (e.g., "Snow is white" is true iff snow is white) seems devoid of the pragmatic element. By including the perspectives of different people/interpretants/readers in my statement above, I am attempting to develop an argument that the Liar sentence is read qua a Liar sentence to some people and not qua a Liar sentence to other people, depending on whether one insists on the stipulation that the Liar sentence be both self-referential and purely negative.

In other words, if the Liar sentence is taken to be either syntactically correct or semantically correct, then it is pragmatically correct, e.g., useful. But if the Liar sentence is taken to be both syntactically correct and semantically correct, then it is pragmatically incorrect, e.g., useless, and as such, in behavior-analytic terms, it should be neither reinforced (affirmed) nor punished (denied) but extinguished (ignored without further argument, mentioned but not used).

In arguing this, I am essentially seeking to challenge Tarski’s T Convention. What I want to argue is that the issues with the Liar sentence / paradox are resolved if we modify his T Convention to something like: “Snow is white” is true iff, for some members of the verbal community, “Snow is white” is taken to mean snow is white and at the same time snow is white. For example:

The Liar sentence as purely negative but not self-referential:

“This sentence is false” is true iff, for some members of the verbal community, “This sentence is false” is taken to mean “This sentence, ‘Snow is black,’ is false,” and at the same time this sentence, “Snow is black,” is false.

Or

The Liar sentence as self-referential but not purely negative:

“This sentence is false” is true iff, for some members of the verbal community, “This sentence is false” is taken to mean “This sentence is false or this sentence is true” and at the same time this sentence is false or this sentence is true.

Answer 1

I'll give my go at the translation into higher-order logic:

∀ Sentence (Liar(Sentence) → ∃ Person (Grammatical(Sentence, Person) ∨ SelfReferential(Sentence, Person)))

In this translation, "Liar(Sentence)" represents the predicate that specifies whether Sentence is a liar sentence. "Grammatical(Sentence, Person)" represents the predicate that specifies whether Sentence is grammatical for Person. "SelfReferential(Sentence, Person)" represents the predicate that specifies whether Sentence is self-referential for Person.

The statement is saying that for every liar sentence, there exists some person for whom the sentence is either grammatical or self-referential. In other words, there is at least one person for whom every liar sentence is either grammatical or self-referential..

Further expounding on the formalization you are seeking

The higher-order sentence that I gave you earlier can be understood as follows:

∀ Sentence (Liar(Sentence) → ∃ Person (Grammatical(Sentence, Person) ∨ SelfReferential(Sentence, Person)))

This sentence contains two quantifiers, ∀ and ∃, which are used to range over higher-order entities. The first quantifier, ∀, stands for "for all," and it ranges over the variable "Sentence." This means that the sentence is making a claim about every possible sentence.

The second quantifier, ∃, stands for "there exists," and it ranges over the variable "Person." This means that the sentence is making a claim about at least one person.

The sentence also contains three predicates: Liar(Sentence), Grammatical(Sentence, Person), and SelfReferential(Sentence, Person). The predicate Liar(Sentence) specifies that Sentence is a liar sentence. The predicate Grammatical(Sentence, Person) specifies that Sentence is grammatical for Person. The predicate SelfReferential(Sentence, Person) specifies that Sentence is self-referential for Person.

The sentence can be read as follows: "For every sentence, if the sentence is a liar sentence, then there exists at least one person for whom the sentence is either grammatical or self-referential." In other words, the sentence is saying that for every liar sentence, there is at least one person for whom the sentence is either grammatical or self-referential.

Books

There are many books available that can help you learn higher-order logic. Here are a few books that may be helpful:

"Higher Order Logic" by Jon Barwise and John Etchemendy is a classic textbook on higher order logic. It provides a comprehensive introduction to the subject, covering topics such as higher order syntax, semantics, and proof theory.

"Logic, Language, and Computation" by Volodymyr Klonowski is a comprehensive textbook on logic and its applications to computer science. It includes a chapter on higher order logic, which covers topics such as higher order syntax, semantics, and type theory.

"The Logic of Provability" by George S. Boolos is a detailed and technical introduction to higher order logic and its applications to the foundations of mathematics. It covers topics such as the λ-calculus, Gödel's incompleteness theorems, and the consistency of set theory.

"A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity" by Shawn Hedman is a more accessible introduction to logic that covers topics such as first order logic, set theory, and model theory, as well as a brief introduction to higher order logic.

I hope these suggestions are helpful!

Answer 2

I'd look into quantified propositional logic first, since I read your opening premise as universally quantifying over propositions.

Not to put it formally, yet, but: you bringing up how the appearance of paradox can depend on our background assumptions is very interesting. For example, how is the liar sentence itself taken? How can we entertain it as a hypothesis? When we attribute truth or falsity to the liar sentence "from the outside," are we using the same truth- or falsity-operator as we built into the interior of the sentence? Do we conflate truth as a property of (or operation on) sentences with truth as a property/operation within a sentence?

And so on. C.f. dialogical logic, then. In other words, I'd look into erotetic logic, here, too: how do different people represent the liar sentence as a question? Can the liar sentence correspond to a question, as a logically well-formed possible answer? If not, then it seems as if the liar sentence itself could be interpreted as axiomatic. I'm not saying that that's a legitimate (or illegitimate!) interpretation, but it seems like a possible one, and differences between logicians on the level of said interpretation might correspond to variations on the background carried into deriving the liar paradox from the liar sentence.

Answer 3

My attempt (it's been a while)

For every liar Sentence, there exists some person for whom the sentence is either grammatical or self-referential.

Ax(Lx implies Ey(Py & (Gxy v Sxy)))

Ax = for all x

Lx = x is a liar sentence

Ey = there exists a y

Py = y is a person

Gxy = x is grammatical for y

Fxy = x is self-referential for y

Answer 4

What does it mean to say that in the practice of the language certain elements correspond the signs? (S53)... There are characteristic signs of it in the player's behaviour (S54)... F.P. Ramsey once emphasized in conversation with me that logic as a 'normative science'... But if someone says that our languages only approximate to such calculi, he is standing on the very brink of misunderstanding. For then it may look as if what we were talking about in logic were an ideal language... But here the world "ideal" is liable to mislead, for it sounds as if these languages were better. (S81)-- Ludwig Wittgenstein, Philosophical Investigations

Your following Tarskian notions of truth of reducing language to physical facts is already heavily embraced in some quarters:

[A]lmost all knowledge is implicit in the structure of the device that carries out the task, rather than the explicit in the states of units themselves. Knowledge is not directly accessible to interpretation by some separate processor, but it is built in the process itself and directly determines the course of processing. -- D.E. Rumelhart

This is typical for philosophers who reject that meaning is external, and argue instead that meaning is internal, that it is constructed by the agent. Experimental epistemology presumes these sorts of metaphysical statements when building models to observe their behavior, and measure it against reality. Seems you're in this frame of mind, looking to connect behavior to actual language practice and put the formal language of logic and its obsession of truth in a proper place. I would argue that what you're looking to do already has an active program outside of math and logic philosophy in the philosophy of psychology, economics, linguistics, and computer science. My goal then is to introduce some references tangential to your direct aims to challenge that your goal is sufficiently novel.

I find it interesting that you want to explain paradoxes from a behaviorist perspective. Radical behaviorism doesn't commit to them ontologically. It seems to me that action and logical contradiction are more fruitfully explored from examining logic from a psychological perspective, which could be done from a softer behavoristic view. For instance, you're absolutely right that any individual is free to interpret the laws of thought, even using all of them inconsistently with what might be called the received interpretation. Doesn't the work in behavioral economics on cognitive biases seem to account for irrationality already? In computer science, paradoxes don't seem very mysterious at all. They simply are mathematical objects that result from a permutation of negation, self-reference, and truth-conditional appraisal, and nothing more mysterious than that. I can write a function that recurses and never terminates by calling itself with the current parameter and writing a termination criterion that compares the argument passed with the negation. That's all a paradox is from a computational perspective.

Too, logic has grown beyond Frege's logicist program. Defeasible logic deviates from classic logic in several ways, one of which is the weakening of the logical consequence with defeasible entailment and the resolution of logical contradiction by justification-based defeat. Paradoxes are species of contradiction generally avoided in the real world by actual thinkers (more often when they shouldn't be), and so my perspective of paradox is colored by the naturalized epistemology psychologism entails, defeasibility (SEP), and dialetheism (SEP). All three of these approaches do much to mitigate the mystery of the paradox. And again, these are all mentalist concepts, and I'm not sure how partially denying meaning and mental events, or even interpreting behavior sheds much light on paradox.

That being said what we agree on is truth-conditional semantics impoverishes meaning. To me, it's a disease of the mind to assume that meaning is solely determined by truth or falsity and idea in currency in the 19th and early 20th century but thoroughly dashed. Wittgenstein's notion of the language-game lays the foundation in the linguistic turn for revising semantics beyond the pedantry of the formal sciences. I've yet to come across a contemporary AI thinker who isn't looking to broaden semantics. The common thread of contemporary thinkers is simply to embrace later Wittgenstein's language-game which you don't seem to mention, which is strongly behavioristic in its interpretation. And in the philosophy of science, the failure of the logical positivists to reduce science to truth-conditional sentences and observation sentences puts truth-conditional semantics in the box it belongs: an abstraction that has utility in clarifying human thought, but does not determine it in any real sense.

Taken as a whole, I'd read up on the essence of language-games, embodied cognition, and constructionism. You used the phrase "bearer of truth" which is received language, but it's metaphorical. Truth-determination is a response to the stimulus of a logical proposition, and it is a constructive one, necessarily mentalistic and explainable by neuroscience and psychology. That's the essence of Tarski's definition of truth from Metasemantics. He invokes use-mention distinction to show that semantics finds its roots in a deeper level of discourse, the physical. Of course, to convey the idea, he has to use language, but the meaning one makes of a T sentence is an association to the intuitive truth that semantics is deeper than truth and logic, which are partial and useful aspects only. That you're looking to somehow cast out truth-conditional semantics by appeals to second-order logics seems to be conceptually counterproductive since the chief tool of the behaviorist is to resolve mental states and associated labels to dispositions of behavior.

At best second-order logic can be understood as temporarily providing support to the meaning of the first-order logic, providing a temporary foundation of sorts, but one that itself is rooted in experience and behavior of Sprachspiel. Mathematicians play this game all the time when they build theories, but then rely on axioms that are intuitive. It's the intuition that does the heavy lifting in making sense of the world.

But, you might persist, in which case you're interested in some sort of foundationalism. My sole work on second-order logic is:

Foundations without Foundationalism (GB) by Steward Shapiro

What it seems to me is that you're involved in nothing more than trying to reinvent a mental representation that involves some truth conditions by other truth conditions. How is that any different than model theory in principle? And still, all model-theoretic models still rely on natural language axioms that appeal to intuition.

Besides the articles, I'd take a good look at semantics from outside the Fregean, neo-Platonic thinking of the formal sciences and look to the natural sciences, particularly more contemporary philosophy of language:

• Philosophy of Language (GB) by Szabo and Thomason. This book is the most comprehensive I've found on the topic's philosophical foundations. I have a couple of others, but they're all oriented towards a truth-conditional semantics.
• Philosophical Investiations (GB) by Ludwig Wittgenstein (this version has the original German side-by-side). This is the man. His conversion from early to late thinking to me embodies the essence of the linguistic turn.
• Women, Fire, and Dangerous Things (GB) by George Lakoff. This advances Langacker and Rosch's work tremendously and moves away from Chomskian thinking with finality.
• Cognitive Linguistics: An Introduction (GB) by Evans and Green. This is explication on metaphysical speculation on the foundations of semantics, and is a big read.
• Foundations of Language (GB) by Ray Jackendoff. This is a tough read, but it's a model on the origins of truth conditions, syntax, and semantics based on second generation cognitive science. It's real nuts and bolts stuff.

Follow your intuitions, but be aware that if your head is in Tarskian logic as a model for semantics, you're almost 100 years behind the curve. If you have a naturalistic epistemology, then your understanding of paradox should be rooted in the findings of philosophers of language, not that of mathematicians and logicians. Those formal sciences are just useful formal abstractions derived from broader psychological principles. Viel glück!