Describing I-Junction Paul M. Pietroski University of Maryland Dept. of Linguistics, Dept. of Philosophy (extended version of the slides for this talk and a series of talks earlier this week)

A Large Project Meanings are simple they compose systematically in ways that emerge naturally for humans they are perceived rapidly and automatically There are decent first-pass theories of meaning/understanding Meaning relies on rudimentary linking of unsaturated conceptual “slots” Truth is complicated It depends on context in apparently diverse ways Making a claim truth-evaluable often requires work, especially if you want people to agree on which truth-evaluable claim got made There are paradoxes Truth requires fancy (Tarskian) variables

A Large Project Meanings are simple they compose systematically in ways that emerge naturally for humans they are perceived rapidly and automatically There are decent first-pass theories of meaning/understanding Meaning relies on rudimentary linking of unsaturated conceptual “slots” Truth is complicated It depends on context in apparently diverse ways Making a claim truth-evaluable often requires work, especially if you want people to agree on which truth-evaluable claim got made There are paradoxes Truth requires fancy (Tarskian) variables

An Elementary Case Study a phrase like ‘brown cow’ is somehow conjunctive to a first approximation… ‘brown cow’ indicates a concept like BROWN(_) & COW(_) ‘brown cow’ applies to an individual thing x if and only if x is brown AND x is a cow to a second approximation… ‘brown cow’ indicates COW(_) & BROWN-FOR-A-COW(_) ‘brown cow’ applies to an individual thing x if and only if x is a cow AND x is brown for a cow

An Elementary Case Study a phrase like ‘brown cow’ is somehow conjunctive to a first approximation… ‘brown cow’ indicates a concept like BROWN(_) & COW(_) ‘brown cow’ applies to an individual thing x if and only if x is brown AND x is a cow to a second approximation… ‘big ant’ indicates ANT(_) & BIG-FOR-AN-ANT(_) ‘big ant’ applies to an individual thing x if and only if x is an ant AND x is big for an ant

An Elementary Case Study a phrase like ‘brown cow’ is somehow conjunctive to a first approximation… ‘brown cow’ indicates a concept like: BROWN(_) & COW(_) ‘Ernie speak yesterday’ (as in ‘I heard Ernie speak yesterday’) indicates a concept like: AGENT(_, ERNIE) & SPEAK(_) & YESTERDAY(_) already many questions… what kind(s) of conjunction? how is such conjunction implemented in human psychology

Kinds of Conjoiners If P and P* are propositions (sentences with no free variables), then: &(P, P*) is true iff P is true and P* is true If S and S* are sentential expressions (with zero or more free variables) then for any sequence of domain entities σ: & (S, S*) is satisfied by σ iff S is satisfied by σ, and S* is satisfied by σ If M and M* are monadic predicates, then for each entity x: 1 &(M, M*) applies to x iff M applies to x and M* applies to x If D and D* are dyadic predicates, then for each ordered pair : 2 &(D, D*) applies to iff D applies to and so does D*

An Elementary Case Study a phrase like ‘brown cow’ is somehow conjunctive to a first approximation… ‘brown cow’ indicates a concept like: BROWN(_) & COW(_) ‘Ernie speak yesterday’ (as in ‘I heard Ernie speak yesterday’) indicates a concept like: AGENT(_, ERNIE) & SPEAK(_) & YESTERDAY(_) already many questions… what kind(s) of conjunction are we appealing to here? if we don’t know, that’s bad if we’re appealing to Tarski’s conjunction, are we saying that humans use this kind of conjunction to understand ‘brown cow’?

‘I’ Before ‘E’ Frege: each Function determines a "Course of Values" Church: function-in-intension vs. function-in-extension --a procedure that pairs inputs with outputs in a certain way --a set of ordered pairs (no instances of and where y ≠ z) Chomsky: I-language vs. E-language --a procedure, implementable by child biology, that pairs phonological structures (PHONs) with semantic structures (SEMs) --a set of pairs

I-Language/E-Language function in Intensionimplementable procedure that pairs inputs with outputs function in Extensionset of input-output pairs |x – 1| + √(x 2 – 2x + 1) {…(-2, 3), (-1, -2), (0, 1), (1, 0), (2, 1), …} λx. |x – 1| = λx. + √(x 2 – 2x + 1) λx. |x – 1| ≠ λx. + √(x 2 – 2x + 1) Extension[λx. |x – 1|] = Extension[λx. + √(x 2 – 2x + 1)]

Going Back To Church given a procedure P1 that maps each α to a β, and a procedure P2 that maps each β to an Ω, there is a procedure P3 that maps each α to an Ω in this sense, procedures compose (and some can be compiled) but a mind might implement P1 via certain representations/operations, and implement P2 via different representations/operations, yet lack the capacity to use outputs of P1 as inputs to P2 if s1 and s2 are recursively specifiable sets, and s1 pairs each α with a β, and s2 pairs each β with a Ω, then some recursively specifiable set s3 pairs each α with an Ω sets don’t compose: s3 is no more complex than s1 or s2 but procedural descriptions of sets might compose

Going Back To Church given a procedure P1 that maps each α to a β, and a procedure P2 that maps each Ψ to an Ω, there is a procedure P3 that maps each α to an Ω specifying a procedure in the lambda calculus (without cheating) tells us that the outputs in can be computed in the Church-Turing sense, given the inputs (and any posited capacities/oracles) this raises questions like those pressed by Marr (in the study of vision) and Chomsky (in the study of language) what kind of algorithm is needed to compute the outputs from the inputs? but MERELY specifying a procedure, by using familiar formal notation, tells us nothing about how the procedure in represented/implemented by human psychology

an I-language in Chomsky’s sense: the expression-generator generates semantic instructions; and executing these instructions yields concepts that can be used in thought  complex concepts that are available for use

 e.g., BROWN(_) & COW(_) But which concept of conjunction is invoked here?

Kinds of Conjoiners If P and P* are propositions (sentences with no free variables), then: &(P, P*) is true iff P is true and P* is true If S and S* are sentential expressions (with zero or more free variables) then for any sequence of domain entities σ: &(S, S*) is satisfied by σ iff S is satisfied by σ, and S* is satisfied by σ If M and M* are monadic predicates, then for each entity x: 1 &(M, M*) applies to x iff M applies to x and M* applies to x If D and D* are dyadic predicates, then for each ordered pair : 2 &(D, D*) applies to iff D applies to and so does D*

The Bold Tarskian Ampersand &(Fx, Gx) is satisfied by (a sequence) σ iff Fx is satisfied by σ, and Gx is satisfied by σ &(Rxx’, Gx’) is satisfied by σ iff Rxx’ is satisfied by σ, and Gx’ is satisfied by σ &(Fx, Gx’) is satisfied by σ iff Fx is satisfied by σ, and Gx’ is satisfied by σ &(Rxx’, Gx’’) is satisfied by σ iff Rxx’ is satisfied by σ, and Gx’ is satisfied by σ &(Wxx’x’’, Rx’’’x’’’’) is satisfied by σ iff Wxx’x’’ is satisfied by σ, and Rx’’’x’’’’ is satisfied by σ The adicity of &(S, S*) can exceed that of either conjunct but think about ‘from under’, which does NOT have these readings: Fxx’ & Ux’’x’’’, Fxx’ & Ux’x, etc.

Frege-to-Tarski Fregean Judgment: Unsaturated(saturated) Planet(Venus) Number(Two) Precedes( ); Precedes(Two, Three) First-Order Judgment-Frames: Unsaturated(_) Planet(_) Number(_) Precedes(_, Three); Precedes(Two, _); Precedes(_, _)

Frege-to-Tarski Tarskian Variables (first-order): x, x', x'', … Tarskian Sentences: Planet(x), Planet(x'),... Precedes(x, x'), Precedes(x', x), Precedes(x, x), … any variable can "fill" any slot of a first-order Judgment-Frame Sentences (open or closed) satisfied by sequences: σ satisfies Number(x'') iff σ (x'') is a number σ satisfies Precedes(x'', x''') iff σ(x'') precedes σ(x''') σ satisfies Precedes(x'', x''') & Number(x'') iff σ satisfies Precedes(x'', x''') and σ satisfies Number(x'’)

The Bold Tarskian Ampersand &(Fx, Gx) is satisfied by σ iff Fx is satisfied by σ, and Gx is satisfied by σ &(Rxx’, Gx’) is satisfied by σ iff Rxx’ is satisfied by σ, and Gx’ is satisfied by σ &(Fx, Gx’) is satisfied by σ iff Fx is satisfied by σ, and Gx’ is satisfied by σ &(Rxx’, Gx’’) is satisfied by σ iff Rxx’ is satisfied by σ, and Gx’ is satisfied by σ &(Wxx’x’’, Rx’’’x’’’’) is satisfied by σ iff Wxx’x’’ is satisfied by σ, and Rx’’’x’’’’ is satisfied by σ The adicity of &(S, S*) can exceed that of either conjunct Do humans naturally employ any such conjoiner?

Kinds of Conjoiners If P and P* are propositions (sentences with no free variables), then: &(P, P*) is true iff P is true and P* is true If S and S* are sentential expressions (with zero or more free variables) then for any sequence of domain entities σ: &(S, S*) is satisfied by σ iff S is satisfied by σ, and S* is satisfied by σ If M and M* are monadic predicates, then for each entity x: 1 &(M, M*) applies to x iff M applies to x and M* applies to x If D and D* are dyadic predicates, then for each ordered pair : 2 &(D, D*) applies to iff D applies to and so does D*

Kinds of Conjoiners (now using y instead of x’) Note the difference between 2 &(D, D*) and &(Pxy, Qxy) no need for variables in the former, and hence no analogs of: &(Pxy, Qyx); &(Pyx, Qxy); &(Pxx, Qxx); &(Pxx, Qxy);...; &(Pyy, Qyy) We could stipulate that 2 +(D, D*) applies to iff D applies to and D* applies to. But this still leaves no freedom with regard to variable positions There is a big difference between (1) a mind that can fill any unsaturated slot with any variable, and (2) a mind that has "unsaturated" concepts like D(_, _) but cannot fill the slots with variables and create open sentences

One More Conjoiner If D is a dyadic predicate, and M is a monadic predicate, then for each entity x: ^(D, M) applies to x iff for some entity y, D applies to and M applies to y _____ | | ^(D, M)  [D(_, _)^M(_)] |___________| Note the difference between ^(D, M) and &(Pxy, Qy) in the former, the “slots” are not independent; no analogs of &(Pxy, Qz) We could define other “mixed” conjunctions. But ^(D, M) is a simple one: its monadic conjunct is closed, leaving another monadic predicate

A Versatile (but simple) Conjoiner If D is a dyadic predicate, and M is a monadic predicate, then for each entity x: ^(D, M) applies to x iff for some entity y, D applies to and M applies to y _____ | | ^(D, M)  [D(_, _)^M(_)] |___________| a separate talk to show that given plausible lexical meanings, and a limited form of abstraction that is required on any view, this can handle ‘quickly eat (sm) grass’ ‘saw cows eat grass’ ‘think I saw most of the cows that ate every bit of grass in the field

General Point phrases indicate complex concepts already many questions… what kinds of complexity? how is this complexity implemented in human psychology?

Summary: Elementary Case Study a phrase like ‘brown cow’ is somehow conjunctive to a first approximation… ‘brown cow’ indicates a concept like BROWN(_) & COW(_) ‘brown cow’ applies to an individual thing x if and only if x is brown AND x is a cow already many questions… what kind of conjunction Monadic: 1 &(M, M*) applies to x iff M applies to x and M* applies to x Tarskian: &(S, S*) is satisfied by σ iff S is satisfied by σ, and S* is satisfied by σ Another Suggestion: ^(Dyadic, Monadic) how is the conjunction implemented in human psychology