Statement Forms and Material Equivalence Kareem Khalifa Department of Philosophy Middlebury College

Overview Why this matters Material equivalence Tautologies, contradictions, and contingencies Sample exercises

Why this matters Identifying materially equivalent statements facilitates the understanding of otherwise- torturous prose. –Ex. Being neither honest nor worthy does not entail being either distrusted or disrespected. –Trusted and respected people can be both dishonest and unworthy. Good writing avoids tautologies and contradictions. –Say nothing false, but say something.

IMPORTANT!! Last class, we covered: 1.Testing for validity Today, we’ll be covering two new topics: 2.Material Equivalence 3.Tautologies, Contradictions, and Contingencies For EACH of these topics, you must look for DIFFERENT things on the truth-table. Make sure you understand these differences!

Today’s First Topic: Material Equivalence ( ,  ) General idea: “p  q” says that wherever p is true, q is true, and wherever p is false, q is false. In English, we represent “  ” as “if and only if” or “is necessary and sufficient for.” –Ex. A creature is a human if and only if it is a member of the species Homo sapiens. –Being human is necessary and sufficient for being a member of Homo Sapiens.

Truth table for  pq p  q TTT TFF FTF FFT

Relationship between  and  Recall: –“p only if q” = p  q –“p if q” = q  p So, since “p if and only if q” = p  q, then p  q = [(p  q) & (q  p)]. Similarly, –“p is sufficient for q” = p  q –“p is necessary for q” = q  p So, since “p is necessary and sufficient for q” = p  q, then p  q = [(p  q) & (q  p)].

Example (~q  p)  (p v q) pq~q ~q  p p v q (~q  p)  (p v q) TTF TFT FTF FFT TTTFTTTF T T T F T T T T

ATTENTION!! We have completed the first topic for today: material equivalence. We are moving on to today’s 2 nd topic: tautologies, contradictions, and contingencies. These two ideas require different uses of the truth-table.

Tautologies Tautologies are statements such that there is no way that they can be false. Where there’s a row, there’s a way. So, tautologies are statements such that there is no row on a truth table in which they are false.

Example: p  p “It ain’t over ‘til it’s over” If it’s over, then it’s over. p p  p TT FT

What’s wrong with tautologies? They are completely uninformative. –A statement’s “informativeness” is proportional to its probability of being false. Ex. Someone is in Twilight 302. Ex. Heidi Grasswick is in Twilight 302. –A tautology has no way of being false. –So the probability of a tautology being false is zero. –So tautologies are completely uninformative.

Contradictions Contradictions are statements such that there is no way that they can be true. Where there’s a row, there’s a way. So, contradictions are statements such that there is no row on a truth table in which they are true.

Example P~PP & ~P TFF FTF

Contingent statements Contingencies are statements such that there is some way that they can be true and some way that they can be false. Where there’s a row, there’s a way. So, contingencies are statements such that there is some row on a truth table in which they are true, and some row on a truth table in which they are false.

Example You will receive either an A or a B. ABA v B TTT TFT FTT FFF

Exercise B2 p  [(p  q)  q] pq TT TF FT FF p  q(p  q)  q p  [(p  q)  q] TFTTTFTT TTTFTTTF TTTTTTTT TAUTOLOGY

Exercise B6 (p  p)  (q & ~q) pq~q p  p q & ~q (p  p)  (q & ~q) TTF TFT FTF FFT TTTTTTTT FFFFFFFF FFFFFFFF CONTRADICTION

Exercise C3 [(p  q)  r]  [(q  p)  r] pqr p  qq  p(p  q)  r (q  p)  r [(p  q)  r]  [(q  p)  r] TTT TTF TFT TFF FTT FTF FFT FFF FFFF TTTT TTTTTTTT FFFF TTTT TTTTTTTT F F F T TTTTTT T F F F TTTTTT TTTT TTTFTTTF CONTINGENT TFTTTFTT