PHL 3000 Lecture Notes - Lecture 5: Propositional Calculus, Intensive Animal Farming, Logical Form
Module 5
Propositional Logic: Form vs. Content
Compare These Two Arguments
• To see the difference between form and content of argument
• Argument 1
o If animals can feel pain, then it is wrong to raise them in factory farms
o Animals can feel pain
o Therefore, it is wrong to raise them in factory farms
• Argument 2
o If Bob is in Atlanta, then Bob is in Georgia.
o Bob is in Atlanta
o Therefore, Bob is in Georgia
• These two arguments are about different things
o the content differs
• They both reason in the same way
o Share an argument form
o They both start by establishing a connection between two statements: if the first is
true, then so is the second
▪ They both go on to assert that the first is true
▪ Finally, they both conclude that, given these two things, the second
sentence must be true
▪ If A then B. B. Therefore A.
Form vs. Content
• Evaluate an argument by assessing the premises
o Ask whether the premises are true or acceptable
o That depends on the content of the argument, what the statements are about
• Or by assessing the connection between premises and conclusion
o ask whether the premises, if true, provide a reason to believe the conclusion
o For deductive arguments, determining this depends most on the form of the
argument
Simple vs. Complex Statements
• Recall the definition of a statement: a statement is any sentence that is capable of being
true or false
o A complex statement is any statement that has other statements as parts
o A simple statement is any statement that does not contain other statements as a
part
• Examples
o John went to the store – simple
o John did not go to the store – complex
▪ There is a simple statement and it is false
o John and Mary went to the store – complex
▪ John went to the store. Mary went to the store.
o John and Mary went to the store together – simple
▪ Cannot simplify it
o John went to the store or stayed home – complex
▪ Combined with or
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o John stayed home only if he did not go to the store, if Mary was at home –
complex
▪ Many different parts
The Logic of Statements: Propositional Logic
• A system examines the logical relationships that exist between statements
• This system is sometimes referred to as “propositional logic”
• The logic governing words such as:
o If…then
o Not
o Or
o And
o If and only if
• These words all serve one function: they are attached to one or more statements in order
to yield a more complex statement
Summary
• The content of an argument is what the argument is about. Assessing the premises
focuses on the content of the argument
• The form of an argument depends on how the simple statements in the argument are
logically related to each other. When asking whether an argument is valid, our main focus
will be the form of the argument
5.2 The Language of Propositional Logic
The Language of Propositional Logic
• A language that allows is to focus only on the form of both statements and arguments.
The language consists of the following elements:
o Capital Letters: variables that stand in for simple statements
o Logical Connectives: symbols that refer to concepts such as and, or, not,
if…then, if and only if
o Parenthesis: used for clarity
• The purpose of this language is to map certain features of everyday English in order to
help understand the logic of words like and, not, or, if…then, if and only if
o Like a map certain features will be distorted
o A perfectly detailed map that left nothing out would be just as large as the reality
it was meant to map. It would be useless.
• Just like any language, it has rules about which kinds of expressions are meaningful
within the language.
o Compare: “Dog hot has cool” is a sentence that contains English words that are all
meaningful, but makes no sense on its own because it violates rules of grammar
o So we need to know more about the elements of the language, we must also know
the rules for constructing meaningful expressions within Propositional Logic (PL)
Lower-case vs. Upper-case Letters
• Lower-case letters are not part of the language of PL. They are variables that range over,
and allow us to talk about, any well-formed formula (wff) of the language.
o ‘A’ refers to any statement in English that can be true or false
o ‘p’ refers to any expression in (PL) that is a well-formed formula (meaningful)
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Well-Formed Formulas
• Capital letters are wff of (PL)
o A is a well-formed formula
▪ Any sentence in English that is a statement
• If p is a wff (PL), then so is ~(p)
o ~(A)
o Applies to only one statement
• If p and q are wff (PL), then so is (p * q)
o (A * B) is a wff
o This rule says take anything that is any two well-formed formulas and you put a
dot between them and it becomes a wff
• If p and q are wff (PL), then so is (p c q) [backwards c]
o (A c B) – wff
o (C c D) – wff
o (A c B) c (C c D) – wff
• If p and q are wff (PL), then so is (p v q)
o (A v B) v (C v D)
o Or even:
▪ ( (A * B) v (C c D) )
▪ Because they can be separated and you can see how they fit the rules
• If p and q are wff (PL), then so is (p = q) [3 lines]
o (A = B) = (B = C) – wff
• Rules about parenthesis: outer-parenthesis can be dropped if there is no resulting
ambiguity
o ~(A) can be ~A
o (A * B) can be A * B
o (A * B) c C cannot lose parenthesis
• Examples of wff:
o A
▪ Refer to rule number 1
o ~A
▪ Refer two rule number 2
o (A c B)
o ((A c B) v C)
o [((A c B v C) = A]
o (A * ~A)
▪ A and not A but still wff
• Examples of formulas that are not wff:
o A c B c C
▪ No rule says you can combine three things together
o *CH
▪ Dot in the wrong place
o (((A * B) * C
▪ The extra parenthesis
o p * q
▪ lowercase letters
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