An inventory of the TYPES of things that weÕll assume make sense in our theory of interpretation.
The bottom level of the ontology. These are ÒthingsÓ in the most general possible sense. They are the things that we talk about, describe, and make claims about.
There are all SORTS of entities. (People, places, things, stuff, É)
á The top level of the ontology.
á They carry information.
á They can be true or false of a given situation.
á They can represent our beliefs and doubts, our convictions and suspicions, our knowledge, hopes, and fears.
These are Òsecond-orderÓ elements of the ontology.
The most important concept about properties is that entities HAVE (or lack) them.
One sort of proposition is the claim that some particular entity has some particular property.
Like properties, these are Òsecond-orderÓ elements.
Relations connect two or more entities. Each entity ÒHASÓ the relation with respect to the other entities in the relation. (relatum / relata)
We classify relations by the number of the relata (2-place, 3-place, etc.)
One sort of proposition is the claim that some set of entities stands in a particular relation to each other.
George W. Bush is the son of George H.W. Bush.
George H. W. Bush is the father of George W. Bush.
A simple idea but complex to define or relate to the above comments.
Propositions can be judged true (or false) of situations.
We will generally assume a set of two truth-values: True and False, usually represented as 1 and 0.
These combine and relate these objects.
For example, the combination of a property and an entity determines a proposition related by PREDICATION.
Design a representation system for lexical, phrasal, and sentence meaning which will:
á Support an account of semantics intuitions of sameness/difference and multiplicity of meaning.
á
Be grounded in intuitions about inference
(informativeness, redundancy, contradiction)
á Account for the distinction between deniable and undeniable implications
Now letÕs assume:
á
Sentence meanings are defined in terms of
propositions.
Declarative sentences are proposition-type expressions.
á The constituent phrases of sentences are assumed to have as their meanings objects from the ontology that can be combined into propositions.
á Lexemes are assigned meanings which are appropriately combined to provide meanings for the phrases that they
Accounting for same / different meaning:
á Sentences judged to have the same meaning should express the same proposition.
á Sentences that express different propositions should be judged to have different meanings.
How are propositions connected to inference?
Propositions can be judged true or false of particular situations.
So whatever a proposition is, it must determine the conditions under which it is judged true.
If propositions determine truth conditions, then we can define the necessary inferential notions.
Assuming that propositions necessarily determine truth conditions doesnÕt mean that we assume that propositions are just the truth conditions.
In pursuit of a formal theory that meets the inferential criteria, we will mostly just look at truth conditions.
But truth conditions are clearly not sufficient to define the appropriate notion of meaning for the sameness/difference criterion.
Can you think of examples that show why?
á Contingent (informative)
á Necessarily true (uninformative because redundant)
á Necessarily false (uninformative because contradictory)
á Logically independent
á Contraries (CanÕt both be true)
á Contradictories (Contraries, one of which must be true.)
á Logically equivalent propositions
á
Entailment
A entails B =
á Whenever A is true, B must be true.
á Every situation that satisfies A also satisfied B.
á It is impossible for A to be true and B to be false.
If sentence A expresses a proposition p that entails q, then the result of asserting A and then denying that q holds will be contradictory.
To test to see if sentence A has q as an undeniable implication, figure out a way to express the contradictory negation of q and then imagine someone saying ÒA and ÂqÓ.
If it sounds logically contradictory, then A entails q.
á Formula (WFF)
á
The letters p, q, r, etc. are used to represent atomic propositions.
They count as basic (atomic) expressions of the category formula.
1. The basic proposition letters are by themselves WFFs.
2. If f is a WFF, then Âf is a WFF.
3. If f and y are WFFs, then so are the following:
a) (f ô y)
b) (f ò y)
c) (f ¨y)
d) (f Ç y)
4. Nothing else is a WFF.
á An interpretation (model, evaluation) of the language is an assignment of a semantic value to every expression in the language.
á We will limit interpretations of this language to those in which formulas are assigned the truth values (1 or 0) as their semantic value.
á Semantics values of complex formulas are determined by the semantic values of their component (sub-)formulas in the following way:
V(Âf) = 1 iff V(f) = 0; [V(Âf) = 0 iff V(f) = 1]
V(f ô y) = 1 iff V(f) = 1 and V(y) = 1; otherwise itÕs 0.
V(f ò y) = 1 iff V(f) = 1 or V(y) = 1; otherwise itÕs 0.
V(f ¨y) = 1
iff V(f)
= 0 or V(y)
= 1; otherwise itÕs 0.
= 0 iff V(f) = 1 and V(y) = 0; otherwise itÕs 1.
V(f Ç y) = 1 iff V(f) = V(y);otherwise itÕs 0.
Discussion section tonight at 6:30 pm in C314.
(Presumably until about 8 pm.)