Philosophical Reasoning Introduction to Elementary Logic I. Deduction / Induction Distinction Murali Ramachandran University of Sussex

Definition An argument is a collection of propositions, one of whichthe conclusionis putatively supported (backed-up) by the othersthe premises.

Definitions A deductively valid argument is one where it is impossible for the premises to be true and the conclusion false; i.e. is one which could not have true premises and false conclusion. When an argument is valid, we say the premises entail the conclusion.

Argument A 1) Singh and Patel went to the party. 2) The party was a success if Patel or Jones went. 3)Hence, the party was a success. Valid

Argument B 1) Vince is a nerd if Brian is. 2) Brian isnt a nerd unless he supports United. 3) Brian doesnt support United. 4)Hence, Vince is a nerd. Invalid

Propositions p and q are logically equivalent if p entails q and q entails p. So e.g. the following two statements are logically equivalent: 1) Hillary is in New York or in London. 2) If Hillary isnt in NY, she is in London, and if she isnt in London, she is in NY.

Inductive Arguments An inductively strong argument is one whose premises would provide positive support for the conclusion if they were true the premises render the conclusion more likely. An invalid argument that is not even inductively strong is called inductively weak.

Argument C 1) Kev is an animal-rights activist and Beth is a butcher. 2) So, if either is a vegetarian, Kev is. Invalid but inductively strong

Argument D 1) Malcolm is an accountant. 2) Beth was nearly bored to death by some accountants at a party once. 3) So, Beth will find Malcolm boring too. Invalid and inductively weak

Important point An argument may be (deductively) valid or inductively strong even if some (or all) of its premises are false! To say an argment is valid (or inductively strong) is to make a claim about how the premises are related to the conclusion; one is not thereby claiming the premises or conclusion to be true.

1) All vegetarians are healthy. 2) Babette is a vegetarian. 3) So, Babette is healthy. Valid argument, but premise (1) is false.

1) Most violinists are vegetarians. 2) Yehudi is a violinist. 3) So, Yehudi is a vegetarian. Inductively strong, but premise (1) false.

Difference between valid arguments and inductively strong ones Whether an argument is valid or invalid is knowable a priori; but whether an argument is inductively strong or weak often depends on background knowledge. Consider e.g. Arguments A and C; the latters strength stems from our knowledge of butchers and animal-rights activists.

Defeasibility of inductive strength Inductively strong arguments can be made weaker by adding further premises (and vice versa). Their strength is defeasible. E.g. suppose we added the premise that Beth comes from a long line of vegetarian butchers to argument C. The conclusion does not seem as compelling as before. Thus, inductive strength, unlike validity, admits of degrees.

Given our definition of validity, adding further premises to a valid argument cannot make it invalid. WHY???

1) All vegetarians are healthy. 2) Babette is a vegetarian. 3) Babette has cancer. 4) Therefore, Babette is healthy. Question: why is this still a valid argument?

1) 90% of children born in South India, have brown eyes. 2) R was born in South India. 3) Hence, it is likely that R has brown eyes. Is this a valid argument or merely inductively strong?

1) 90% of children born in South India, have brown eyes. 2) R was born in South India. 3) Most children born in Ambattur, S. India, have green eyes. 4) R was born in Ambattur. 5) Hence, it is likely that R has brown eyes.

1) Singh and Patel went to the party. (S and P) 2) The party was a success if Patel or Jones went. (Q if P or J) 3) Hence, the party was a success. (Q) Any argument with the same shape will be valid: S and P, Q if P or J; hence, Q

So, some valid arguments are valid purely in virue of their shape. They are said to have a valid logical form. Formal logic is the study of logical form and it is this we shall be concerned with for the remainder of the course, since it provides a fundamental and comparatively easy starting point.