Lecture 15: First Order Logic

Components of Language: Names & Predictors

  • An important aspect of how we use our ordinary language is:

  • 1: Refer to people or things (objects)

  • 2: To assert properties or characteristics of the people or things we have named

  • To achieve this: We can use (proper) names to single out objects CONT

At the level of world: Objects & CONT

Eg. Sylvester is a cat

  • Sylvester is the name

  • Take the name out, and no sentance but psedosentance/unsaturated sentence

  • CONT

Predicates (Common nouns)

Proper names pick out single objects

  • When we use a name to fill in blank use T or F

Eg. Analysis of relational sentances

“Socrates is older than Plato”

names: Socrates & Plato

Predicate: “Is older than”

  • This analysis can be done with many relations

  • CONT

  • We can turn relations into properties

  • Socrates is older than ___ is a one place predicate

Terminology

  • To distringish btwn prediates that have a doff number of places we speak their arity: unary binary ternary quaternary etc. or in general, n-ary

  • CONT

Symbolization

  • Names: Lowercase letters

  • Predicates: Uppercase letters

Atomic Sentences

  • Putting togetehr one of the 2 names in any of the gaps will then generate atomic sentences

  • a) including the ones we had before in our TFL: ‘H & S’ are now symbolized by H(m) and “S(s)”

  • This way of doing things already brings enormous benefits

  • We can make related sentences about different things:

  • 1. We can form both H(m) and CONT

Propositional connectives

  • New kind of atomic sentences

Beginnings of Conceptual analysis

  • You can make a smaller predicate to represent a bigger one as to save time

Difficulties with symbolization

  • benefits of learning symbolic logic: To symbolize sentence, you must understand a sentence

  • By learning how to symbolize English sentences, CONT

Combined predicates

  • Sometimes we have single predicates that appear to be combined predicates but linguistic form in English doesn’t always make this clear

  • Combined predicates:

    • Mandy is an inspiring hiker (mandy inspires and is a hiker)

    • Mandy is a hiker and Mandy is not from Calgary (

    • Mandy is an international soccer player (Mandy is international and a soccer player) ***clearly this is not correct and is a flaw of TFL

Pronouns

CONT

Symbolization as translation

  • It is useful to CONT

Assigning truth values

  • Bc atomic sentences in FOL have internal structure, we msut take this into consideration CONT

Interpretation

  • Assignment of truth values to atomic sentences depends of 3 items

  • 1. The selection of what is sometimes called a domain or discourse CONT

    • eg domain of all people, domain of all wizards, ect.

CONT

Truth value assignment

  • Given such an interpretation the assignment of a truth value to an atomic srntence is then a relatively straightforward matter

    • Suppose the domain specified is that of students in the current logic class

    • Suppose ‘m’ names Mandy , one of those students in the class who love hiking

    • Then atomic sentence ‘H(m)’ will receive value T CONT

  • It depends on other decisions we have made w respect to our interpretation

  • 1. The domain

  • 2. Named object