2026-09-14
Agenda
Let’s take a moment to talk about how it went. Many of you have said it was stressful. At least one of you felt it was their “low of the week”.
Are there things you learned from the process?
Thoughts about how it could be done better?
Agenda
Argument
Agenda
Frege and Russell both thought we could reduce mathematics to logic. With the right definitions, claims like,
reduce to claims like,
(This project ran into unexpected difficulty. The current consensus is that no such reduction is possible.)
Contrast:
Neither is nonsense. So (following Frege) they both express a sense or thought.
The sense of (1) determines different truth values in different circumstances. That is, the truth of (1) depends not just on the sense, but also on the world.
The sense of (2) determines the same truth value, true, in every possible circumstance. That is, the truth of (2) depends only on the sense, not the world.
Logical truths like (2) are tautologies. They are not nonsense, but they have no substantive content.
They are true in virtue of meaning (sense) alone.
(3) is not a logical truth:
But, if we “unpack” the meaning of ‘bachelor’, we get,
And (4) is a logical truth.
Analytic truths reduce to logical truths, and so are also tautologies.
The truth of an analytic truth depends on (a) the definitions of any non-basic terms, like ‘bachelor’ and (b) logic.
Language makes contact with experience in the form of basic observation sentences, e.g.,
‘This’ refers to some immediate object of sense experience, and ‘is red’ categorizes it by how it appears.
More abstract sentences, like,
reduce, via definition, to basic observation sentences.
This leaves a residue of sentences that are no basic observation sentences and do not reduce to basic observation sentences.
These are either nonsense (metaphysics and theology) or analytic (mathematics).
One [dogma] is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact.
The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience.
Agenda
For each of the following, identify the extension and the intension:
Agenda
For the theory of meaning the most conspicuous question is as to the nature of its objects: what sort of things are meanings? They are evidently intended to be ideas, somehow — mental ideas for some semanticists, Platonic ideas for others. Objects of either sort are so elusive, not to say debatable, that there seems little hope of erecting a fruitful science about them. It is not even clear, granted meanings, when we have two and when we have one; it is not clear when linguistic forms should be regarded as synonymous, or alike in meaning, and when they should not. If a standard of synonymy should be arrived at, we may reasonably expect that the appeal to meanings as entities will not have played a very useful part in the enterprise. (22)
Once the theory of meaning is sharply separated from the theory of reference, it is a short step to recognizing as the business of the theory of meaning simply the synonymy of linguistic forms and the analyticity of statements; meanings themselves, as obscure intermediary entities, may well be abandoned. (22-23)
Agenda
There are those who find it soothing to say that the analytic statements of the second class reduce to those of the first class, the logical truths, by definition; ‘bachelor’, e.g., is defined as ‘unmarried man’. But how do we find that ‘bachelor’ is defined as ‘unmarried man’? (24)
The lexicographer is an empirical scientist, whose business is the recording of antecedent facts; and if he glosses ‘bachelor’ as ‘unmarried man’ it is because of his belief that there is a relation of synonymy between these forms, implicit in general or preferred usage prior to his own work. […] . Certainly the “definition” which is the lexicographer’s report of an observed synonymy cannot be taken as the ground of the synonymy (24)
Just what it means to affirm synonymy, just what the interconnections may be which are necessary and sufficient in order that two linguistic forms be properly describable as synonymous, is far from clear; but, whatever these interconnections may be, ordinarily they are grounded in usage. Definitions reporting selected instances of synonymy come then as reports upon usage. (25)
In explication the purpose is not merely to paraphrase the definiendum into an outright synonym, but actually to improve upon the definiendum by refining or supplementing its meaning.
Examples:
Such a definition still owes its explicative function to pre-existing synonymies (25)
Do you see why?
Quine goes on about this at some length. I’ll give you an example on the board, of a language that just has ‘or’ and ‘not’ as its primitives, and defines ‘and’ and ‘if…then…’ as abbreviations for sentences containing only ‘or’ and ‘not’.
But these correlations are not arbitrary. They are supposed to show how the primitive notations can accomplish all purposes, save brevity and convenience, of the redundant language. Hence the definiendum and its definiens may be expected, in each case, to be related in one or another of the three ways lately noted. The definiens may be a faithful paraphrase of the definiendum into the narrower notation, preserving a direct synonymy as of antecedent usage; or the definiens may, in the spirit of explication, improve upon the antecedent usage of the definiendum; or finally, the definiendum may be a newly created notation, newly endowed with meaning here and now. (27)
Only the last case plausibly gives rise to a new synonymy.
Agenda
A natural suggestion, deserving close examination, is that the synonymy of two linguistic forms consists simply in their interchangeability in all contexts without change of truth value; interchangeability, in Leibniz’s phrase, salva veritate. (27)
Truths which become false under substitution of ‘unmarried man’ for ‘bachelor’ are easily constructed with help of ‘bachelor of arts’ or ‘bachelor’s buttons’. Also with help of quotation, thus: ‘Bachelor’ has less than ten letters. Such counterinstances can, however, perhaps be set aside by treating the phrases ‘bachelor of arts’ and ‘bachelor’s buttons’ and the quotation ‘’bachelor’’ each as a single indivisible word and then stipulating that the interchangeability salva veritate which is to be the touchstone of synonymy is not supposed to apply to fragmentary occurrences inside of a word. (27-28)
Here is a schema for testing for synonymy of ‘A’ and ‘B’:
This works for ‘bachelor’ and ‘unmarried man’. It does not work for ‘creature with a heart’ and ‘creature with a kidney’.
Note that this depends on how we understand ‘necessarily’.