This week I have registered my MA thesis in Philosophy, titled "Inferentialism and Inductive Definitions". On this page I collect references, additional material, and ideas. I am using a blog instead of a LaTex file for this collection, as a reminder for myself not to resort to 'academic' or dense prose.

The first step when introducing a formal language is to define its alphabet & grammar (syntax). Which symbols are part of the language? How are the words and sentences (aka formulas, expressions) of the language formed? Next, one has to explain how the language is to be understood. What do the words and sentences mean? How should we interpret them? The study of meaning is semantics.

The approach to semantics used in my thesis, (Logical) Inferentialism (also called proof-theoretic semantics), is a non-standard account of meaning for formal languages. It makes sense to first recall the received approach, for pointing out essential differences. The standard approach to semantics in logic is to identify the meaning of a sentence with its truth conditions, thereby reducing sentence-meaning to the notion of truth. To know what a sentence means is to be able to recognize the conditions given which the sentence would be true. Following Tarski, truth is analyzed as truth in a model. This received approach is so ubiquitous nowadays, that 'to give a semantics' usually just means to give a basic model theory for the language in question.

• We briefly outline the usual semantics for formal logic and point out basic features of this account.
• Along the way we comment on conditions that an reasonable semantics has to satisfy and about the questions that semantics is supposed to answer.
• We then compare and contrast this to the operational understanding of the formal languages used for computer programming. What are the requirements here? To what extend are the expectations we have about semantics dependent on our agenda, aims or values?
• What is meaning? <-

Truth-conditional semantics for sentences:

• Tarski semantics: Give Model, map words to objects ("meanings", "semantic values"). Sentences are mapped to values true or false by a model (satisfaction relation). Extensional meaning = denotation = truth value

• Kripke semantics: Give set of models, each maps words to objects. Sentences are mapped to subsets of set of models (those in which they are satisfied (=true)). Intensional meaning = sense_Fr? = truth conditions

• Sentence-meaning explained in terms of truth. Understand what a sentence means = be aware of the conditions for its truth.

Operational semantics?

operational (CS) vs denotational (Logic) semantics.

The inferentialist story, broadly following Dummet. Also logical inferentialism, BHK and relizability, type theory etc as formalizations of bhk.

Model-theoretic semantics (as presented by Tarski) provides a formal account of the meaning of the logical vocabulary (\(\land, \lor, \forall, \ldots\)). In summary:

Given the usual language of first order logic \(\mathcal L\), we first define the notion of a model \(\mathfrak M_{\mathcal L}\). A model fixes the domain of individuals, and assigns an extension to the non-logical vocabulary of the language (=specifies what the variable, predicate and function symbols refer to). In other words, the model fixes the extension of the non-logical part of the language.

One can then specify the meaning of the logical vocabulary by stating their recursive satisfaction conditions (or truth-in-a-model conditions or simply truth conditions), for example

If \(\mathfrak M_{\mathcal L}\) is a model, then

the meaning of \(\varphi \land \psi\), i.e. the truth condition is thus a function that yields a truth value, given a model:

In this sense, Tarski's model theoretic semantics, is a semantic theory only of the logical vocabulary. It is not concerned with the meaning of the non-logical symbols. The connectives of first-order logic are truth-functional or extensional, that is, their truth conditions do not depend on the meaning (for sentences: truth conditions) of the non-logical constituents, but only on their extension (for sentences: truth value). Hence, it is sufficient that a model specifies the extension of the non-logical vocabulary to state the truth-conditions (meaning) of the 1st order connectives.

When extending the semantic theory to also account for the meaning of modalities such as necessity, one has to move to possible worlds semantics (Kripke models). Here, the semantic value assigned to a proposition by a Kripke model is no longer merely a truth value (extension) but a truth condition, i.e. a subset of worlds where the proposition is true (meaning, intension). This is so because modal vocabulary is intensional that is, it's meaning does not only depend on the extension of its argument (truth value) but on its meaning (truth-condition).

Further extension of this formal approch to semantics include: indexicals, .., .. and some require further refinement of the semantic value assigned to vocabulary, sentences of the language. It turns out, that only the meaning of the logical connectives of first order logic can be accounted for with a merely extensional model theory that only fixes the reference, but not the meaning of the non-logical. In fact, this is how Tarski defined what logical vocabulary is: that part of language, the meaning of which does not depend on the meaning, but merely the extension of its constituents.

Where does this leave theories formulated in first order logic?

Important such theories include (the first order axiomatisation of) Peano Arihmetic, and ZFC set theory. These theories involve certain non-logical symbols, such as \(+,S,\in\), and come with a set of axioms that inform us about the behavior about those symbols. The axioms, however, fall short of specifying the meaning (truth conditions) of the new connectives. In fact, they even fall short of specifying the extension of the new connectives, and necessarily so. The theories, if, consistent have different models and each model assigns its own extension to e.g. the \(\in\)-predicate.

Model-theory of ZFC does not provide a semantic theory of set theory, when semantic theory is understood as providing a formal account of meaning. It is not concerned with the truth-conditions of the \(\in\) predicate. A model of ZFC does not fix the meaning of \(\in\), merely its extension. And the axiomatic theory does not fix a model, and thus only partially informs us about the extension of \(\in\).

In summary, while Tarski's model theoretic definition of truth provides a semantic theory (truth conditions!) for the logical connectives, and, through certain extensions, for modal connectives and indexicals, etc., it does not account for the meaning of non-logical symbols.

Model theory is not a semantic theory for any axiomatic theory built in the language of first order logic. The meaning of non-logical constituents of such a theory is syntactically described via the axioms. Nowadays, where "to give a semantics" just means to giva a basic model theory, one has to be careful to avoid confusion:

Model theory of ZFC is not a semantic theory of set theory, because the 'semantic' values a model assigns are merely extensions not meanings. It does not attempt to fix truth-conditions of the sentences of set theory and it does not attempt to account for the meaning of \(\in\). It is a syntactic theory of \(\in\). Regarding set theory, the usual extension for assigning truth-conditions to atoms/sentences, namely the move to possible worlds semantics is possible without reinterpretation of that semantics, because the truths of set theory are considered necessary, which in that semantics means true at every world and this already fixes the semantic value such that all truths of set theory get assigned the same meaning, same for all falsehoods.

In particular, ZFC and its model theory is not a formal account of the semantics of mathematics.

When using the notion of a "standard model of set theory", that supposedly fixes the meaning of membership, this is not very satisfactory: Mathematical meaning, with mathematical theorems being necessary, shouldn't vary with models. Next it only fixes extension, and lastly an undefined notion of standard model is murky and subjective. How do i know my standard model is the same as that of others?

ToMaybe:

• Theory of meaning vs theory of reference
• Tarski semantics is theory of meaning for propositional/1st order connectives
• ... but merely theory of reference for axiomatic theories formulated in it (models only fix extension of the non-logical symbols)? Because not concerened with meaning (truth-conditions) of the non-logical.
• A formal account of meaning based on extensional theory is too limited for all but logic; some things (modalities, ...) can be given truth-conditions in a possible worlds semantics.

Tarski's semantics accounts for the meaning of the logical symbols. It can be extended to account for the meaning of other fragments of language, such as modal vocabulary (using Kripke models) or indexicals.

Similarly, we may wish to provide a semantics for the fragment of language that is the language of Mathematics.

The inferentialist account is bites with several orthodoxies in philosophy: - Truth as more basic than Proof - Truth as bivalent - The analysis of logical consequence in terms of truth (necessary truth-preservation) - Meaning=truth-conditions - yields just intuitionistic logic; inferentialist analysis of truth leave truth as an epistemic notion. - Fails to explain the referential/denotational aspect of meaning.

Logical Inferentialism = The meaning of logical symbols is fixed by their rules.

We take \(\to\) and \(\forall\) to be primitives.

What restrictions need to be imposed on the rules of a connective for it to be 'meaningful' (= a well-formed definition)?

- GE-Rules (local)
- Conservativeness (global)
- Stability ?
- Normal forms ?
- Monotonicity/strict positivity ?
- ..
- PSH's 'reductive' vs. foundational analysis

-> Leaves uneasy feeling that doesn't actually work.

But, rather: there are many ways to make it work, but it requires making (philosophically unjustified) choices. No canonical way.