Predicate Logic Flashcards
Difference between Propositional and predicate logic
- So far, we have treated inference as made by appeal to connectives only. We treated all sentences as replaceable by schematic letters.
- But there are some deductive inferences where this is not the case, for example, the following, which is a syllogism:
- P1 Daphne is a dog P
- P2 All dogs are animals Q
_____________________
- Daphne is an animal R
- This argument is valid. But we can’t demonstrate its validity in propositional logic, because it seems to have the form: P, Q /R.
Solution: introduce more logical structure for sentences.
New sentence structure, what does it now consist of and how do now form it in logic
Inferences such as syllogisms are made by considering a unit of meaning smaller than an individual sentence.
These are things like predicates (‘is mortal’) relations (‘loves’) and quantifiers (‘All animals’ ‘someone’ ‘something’)
To represent sentences including them we dont use sentence letters. We divide them up into constants (ie names of things) and predicates (which expresses that a thing has a property or that there is some relationship between certain things.)
describe constants and predicates
how and in what order do we present them
Constants - names of individual things. We’ll also treat descriptions as names/constants (use lowercase letters to represent)
Predicates - Whnever you delete a name from a sentence you are now left with a predicate. Eg Delete Liz from Liz is cool and you are left with ‘is cool’ (use capital letters to represent)
It is convention to put the predicate before the constant (remember as it is predicate)
Eg. Cl
Remember to provide a key
Predicate logic allows us to express that two individuals have a common property
Negating predicate logic
If liz is a stud and chris is not a stud we can say Sl & ∼ Sc
Relational predicate - used for saying something is cooler, better, funnier, smarter etc.
We put the relational predicate first, followed by the names in order.
Liz is funnier than Chris: Flc
Some relations have more than two. Eg Ellie is between Liz and Chris
Belc
What is a syllogism?
A valid argument with two premises and one conclusion
All starting with either exactly one quantifier or a name.
Briefly what are the two types of Quantifier
Universal quantifier corresponds to ‘all’ or ‘every’
Existential quantifier corresponds to ‘some’ or ‘there is’ or ‘at least one’ or ‘exists’ (they mean the same thing in logic)
In more detail explain the two types of quantifier and how you write them
U_niversal quantifie_r; (All or every) we’ll write it ‘(x)’. It is more common to write it ‘∀x’ but this is hard to type. The ‘x’ occurring in it is a variable, and other variables can occur in it as well, e.g. ‘(y)’, ‘(z)’.
Existential quantifier; (Some/ there is/ at least one/ exists) we’ll write it ‘(Ex)’ (or ‘(Ey)’ or ‘(Ez)’). It is more common to write it ‘∃x’.
- So we have:
- (x)Fx : every x is such that x is F
- (Ex) Fx: some (at least one) x is such that x is F
Predicates v Constants v Variables
Predicates - The thing describing the subject of the sentence. Eg is cool. upper case letters
Constants -the name, lower case letters
Variables - non specific (not names) so most commonly pronouns.
We use the lower case letters x y z for variables.
Formalise - “If anyone comes to Harriet for help, she will help them.”
What is the variable here?
‘them’ is a variable here
- it stands for any person
- it tracks the previous ‘anyone’
- but ‘she’ stands for Harriet
- First step towards formalising:
- Any x is such that, if x is a person and x comes to Harriet for help, Harriet will help x.
How can you extend senctences using connectives and combine them?
Do the examples
- Something is F and G*
- All Fs are Gs (every x is such that if x is F then x is G)*
- Some x has R to every y*
- These sentences can be extended by using the connectives, and by combining them:
- (Ex) (Fx & Gx) Something is F and G
- (x)(Fx -> Gx) All Fs are Gs (every x is such that if x is F then x is G)
- (Ex)(y)Rxy Some x has R to every y
- Examples of English sentences with these forms:
- Some dog is furry.
- All dogs are furry.
What is a domain of discourse and why do we need it?
How do we present it?
In order to restrict the things we are discussing so the sentence is not ambiguous.
Eg (Ex)(y)Lxy Some x has L to every y
•Something loves everything? Or someone loves everyone? Ambiguous
A domain of discourse is the collection of the things we are discussing. In english we specify implicityly using context but in logic it is explicit.
We say the domain is a set.
We write D for domain and indicate sets of things using curly brackets.
EG. D: {Persons}
Restrict using domains
(Ex)(y)Lxy
- Some x has L to every y
- Something loves everything? Or someone loves everyone?
provide these two options^^
We restrict our domain explicitly or say our domain is unrestricted explicitly:
- Option 1:
- D: {everything}
- Lxy = x loves y
- Here ‘(Ex)(y)Lxy’ means that something loves everything
- Option 2:
- D: {persons}
- Lxy = x loves y
- In this case ‘(Ex)(y)Lxy’ means that someone loves everyone.
Truth and interpretation with domains
Difference between truth of existential quantifier and universal
- In propositional logic we just assigned ‘true’/‘false’ to sentences.
- In predicate logic, we explicate truth with reference to domains.
- We take a grammatical sentence consisting of constants and predicates to be true iff the things named have the property, or stand in the relation, which the sentence ascribes to them.
- A sentence beginning with an existential quantifier we take to be true just in case something in the domain has the property or stands in the relation ascribed to it.
- A sentence beginning with a universal quantifier we take to be true just in case everything in the domain has the property or stands in the relation ascribed to it.
